diff --git "a/2dFQT4oBgHgl3EQfFzW1/content/tmp_files/load_file.txt" "b/2dFQT4oBgHgl3EQfFzW1/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/2dFQT4oBgHgl3EQfFzW1/content/tmp_files/load_file.txt" @@ -0,0 +1,2352 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf,len=2351 +page_content='Consensus based optimization with memory effects: random selection and applications Giacomo Borghi∗ Sara Grassi† Lorenzo Pareschi† February 1, 2023 Abstract In this work we extend the class of Consensus-Based Optimization (CBO) metaheuris- tic methods by considering memory effects and a random selection strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' The proposed algorithm iteratively updates a population of particles according to a consensus dynamics inspired by social interactions among individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' The consensus point is computed taking into account the past positions of all particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' While sharing features with the popular Parti- cle Swarm Optimization (PSO) method, the exploratory behavior is fundamentally different and allows better control over the convergence of the particle system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' We discuss some im- plementation aspects which lead to an increased efficiency while preserving the success rate in the optimization process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' In particular, we show how employing a random selection strat- egy to discard particles during the computation improves the overall performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Several benchmark problems and applications to image segmentation and Neural Networks training are used to validate and test the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' A theoretical analysis allows to recover convergence guarantees under mild assumptions of the objective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' This is done by first approximating the particles evolution with a continuous-in-time dynamics, and then by taking the mean-field limit of such dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Convergence to a global minimizer is finally proved at the mean-field level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Keywords: consensus-based optimization, stochastic particle methods, memory effects, ran- dom selection, machine learning, mean-field limit Contents 1 Introduction 2 2 Consensus-based optimization with memory effects 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='1 Particles update rule .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' 5 ∗RWTH Aachen University, Institute for Geometry and Applied Mathematics, Aachen, Germany (borghi@eddy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='rwth-aachen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='de) †University of Ferrara, Department of Mathematics and Computer Science & Center for Modelling Computing and Statistics, Ferrara, Italy (sara.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='grassi@unife.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='it, lorenzo.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' 23 5 Conclusions 25 A Proofs 25 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='1 Notation and auxiliary lemmas .' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='4 Proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='2 and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' 30 1 Introduction Meta-heuristic algorithms are recognized as trustworthy, easy to understand and to adapt op- timization methods which have been widely applied to a several fields such as Machine Learn- ing [28], path planning [29] and image processing [45], to name a few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Starting form a set of possible solutions, a meta-heuristic algorithm typically updates such set iteratively by combining deterministic and stochastic choices, often inspired by natural phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Exploration of the search space and exploitation of the current knowledge are the two fundamental mechanisms driving the algorithm iteration [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Examples of established meta-heuristic algorithms are given by Genetic Algorithm (GA) [17,42], Simulated Annealing (SA) [25], Particle Swarm Optimiza- tion (PSO) [24] and Differential Evolution (DE) [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' We refer to [21] for a complete literature review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Consensus-Based Optimization (CBO) is a class of gradient-free meta-heuristic algorithms inspired by consensus dynamics among individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' After its introduction [34] it has gained popularity among the mathematical community due to its robust mathematical framework [3,9, 16,19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' In CBO algorithms, a population of particles concentrates around a consensus point given by a weighted average of the particles position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' In the computation of such consensus point, more importance is given to those particles attaining relatively low values of the objective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' The exploration mechanism is introduced by randomly perturbing the particles positions at each iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Particles which are close to the consensus point are subject to small perturbations, while those that are far from it display a more exploratory behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' In this work, following the recent analysis in [14], we study a Consensus-Based Optimization algorithm with Memory Effects (CBO-ME) where the consensus point is computed among the whole history of the particles positions and not just on the positions of the current iteration, as 2 in the original CBO method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' This is done by keeping track of the best position found so far by each particle, and computing the consensus point among these “personal” bests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' While sharing common elements with PSO, such as convergence to a promising point and the presence of personal bests, CBO-ME differs in the way the exploration mechanism is implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Indeed, in CBO-ME, as in CBO algorithms, the stochastic behavior is given by adding Gaussian noise to the particles dynamics and can be tuned independently on the exploitation mechanisms, leading to a better control over the particles convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Therefore, while in classical PSO methods it is the balance between local best and global best that governs the optimization strategy, in CBO methods it is the balance between exploration and exploitation mechanisms that determines the choice of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' We recall that a generalization of PSO methods that allows leveraging the same flexibility in searching the global minimum as in CBO algorithms has been recently presented in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Many real-life problems, especially those regarding Machine Learning, require to optimize a large number of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Therefore, it essential to design fast algorithm to save computa- tional time and memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' This is a major weakness of swarm-based methods, which require a set of particles to minimize the problem, unlike gradient-based methods that can work on a single particle trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' For methods based on a collection of particles, existing algorithms can be improved by discarding particles whenever the system has a prominent exploitative behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' This is sometimes referred as “natural selection strategy” in the DE literature [27,40] and aims to discard the non-promising solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Inspired by particle simulations techniques where it is important to preserve the particles distribution, we examine a “random selection strategy” where particles are discarded randomly based on the local consensus achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' We will discuss such implementation aspects by testing CBO-ME against high-dimensional learning problems and theoretically analyze the impact of the random selection strategy on the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' In partic- ular, we prove that if the full particle system is expected to converge towards a solution to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='1), so will the reduce one, provided a sufficient number of particles remains active.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Note that, such analysis can be generalized to other particle dynamics and may be of independent interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Owing to the convergence analysis of CBO algorithms [3, 9, 10, 19] and recent analysis of PSO [14, 20] we are able to prove convergence of the algorithm under mild assumption on the objective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' This is done by first approximating the algorithm with a continuous-in- time dynamics and secondly by giving a probabilistic description to the particles system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' By assuming propagation of chaos [41], particles are considered to behave independently according to the same law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' This allows to reduce the possible large system of equations to a single partial differential equation: the so-called mean-field model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Such model is then analyzed to recover convergence guarantees under precise assumption on the objective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Developed in the field of statistical physics, this approach has shown be fruitful in studying particle-based meta- heuristic algorithms [9,10,20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' We note that convergence in mean-field law was recently proved in [37] in an independent work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Section 2 is devoted to the introduction of the CBO-ME algorithm with random selection and comparison with CBO methods without memory effects and PSO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' In Section 3 validate the proposed methods against several benchmark problems and two Machine Learning tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Theoretical convergence guarantees and analysis of the random selection strategy are summarized in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Some final remarks are given in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Technical details of the theoretical analysis are given in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' 3 2 Consensus-based optimization with memory effects In this section, we present the Consensus-Based Optimization algorithm with Memory Effects (CBO-ME) to solve problems of the form x∗ ∈ argmin x∈Rd F(x) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='1) where Rd, d ∈ N is the, possibly large, search domain for the continuous function F ∈ C(Rd, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' We will do so by highlighting similarities and differences between classical CBO methods and PSO algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='1 Particles update rule At each iteration step k and for every particle i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' , N, we store its position xk i and its best position found so far yk i = argmin{F(xk 1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' , F(xk N)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' The best positions are used to compute a consensus point ¯yα,k = N � i=1 ωk i yk i with ωk i = e−αF(yk i ) �N j=1 e−αF(yk j ) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='2) which approximate the global best solution ¯yα,k among all particles and all times for α > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Indeed, thanks to the choice of the weights ωk i , we have that ¯yα,k −→ ¯y∞,k := argmin{F(yk 1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' , F(yk N)} (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='3) as α → ∞, provided that there is only one global best position among {yk 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' , yk N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Such ap- proximation was first introduced for CBO methods [34] as it leads to more amenable theoretical analysis, but it also allows more flexibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Indeed, relatively small values of α are typically used at the beginning of the computation to promote exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Large values of α, on the other hand, lead to better exploitation of the computed solutions and to higher accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' We note that the weights used in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='2) correspond in statistical mechanics to the Boltzmann-Gibbs distribution associated with the energy F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' In this context, α plays the role of the inverse of the system temperature T and the limit α → ∞ corresponds to T → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Once the consensus point ¯yα,k is computed, the particle positions are then updated according to the law xk+1 i = xk i + λ � ¯yα,k − xk i � + σ � ¯yα,k − xk i � ⊗ θk i (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='4) with θk i ∈ Rd randomly sampled from the normal distribution (θk i ∼ N(0, Id)) and where ⊗ is the component-wise product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' The update rule is characterized by a deterministic component of strength λ ∈ (0, 1) promot- ing concentration around the consensus point ¯yα,k and a stochastic component of strength σ > 0 promoting exploration of the search space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' As the latter depends on the difference (¯yα,k − xk i ), the random behavior is stronger for particles which are far form the consensus point, whereas it is weaker for those that are close to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Also, such exploration resemble an anisotropic diffu- sive behavior exploring every coordinate direction at a different rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' This approach was first proposed in [4] in the context of CBO methods and has been proved to suffer less from the curse of dimensionality with the respect to the originally proposed isotropic diffusion given by σ∥¯yα,k − xk i ∥2θk i with θk i being again a normally distributed d-dimensional vector [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='2 Random selection strategy When the particle system concentrates around the consensus point, showing a mostly exploita- tive behavior, we employ a particle selection strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Discarding particles introduces additional stochasticity to the system, while reducing the computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Following the approach sug- gested in [7], we check the evolution of the system variance to decide how many particles to (eventually) discard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' For a given set of particles z = {zi}i∈J, the system variance is given by var(z) := 1 |J| � j∈J ∥zj − m(z)∥2 2 with m(z) := 1 |J| � i∈J zi , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='5) where |J| indicates the cardinality of I, that is, the number of particles in this context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Let Ik ⊆ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' , N} be the set of active particles at step k and Nk = |Ik|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' To decide how many particles to select, we compare the variance of the particle system before the position update (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='4), xk = {xk}i∈Ik and after it, ˜xk+1 = {xk+1 i }i∈Ik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Then, the number Nk+1 of particles we select for the next iteration is given by ˜Nk+1 = � Nk � 1 + µ var(˜xk+1) − var(xk+1) var(xk+1) �� Nk+1 = min � max � ˜Nk+1, Nmin � , Nk � (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='6) ⌊z⌋ being the integer part of a number z and Nmin ∈ N the smallest amount of particles we allow to have.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Then, a subset Ik+1 ⊂ Ik, |Ik+1| = Nk+1, of particles is randomly selected to continue the computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' The parameter µ ∈ [0, 1] regulates the mechanism: for µ = 0 there is no particle discarding, while for µ = 1 the maximum number of particles is discarded if the variance is decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' As we will see in Section 3, this random selection mechanism dramatically reduces the computational time without affecting the algorithm performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' We will also theoretically analyze this aspect in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='3, where we show that convergence properties are preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' As stopping criterion, we keep a counter n on how many times ∥¯yα,k+1 − ¯yα,k∥2 is smaller than a certain tolerance δstall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' If this happens for more than a given nstall number of times in a row, we assume the particles system found a solution and stop the computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' A maximum number of iteration kmax representing the computational budget is also given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' The proposed CBO-ME is summarized in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' In the meta-heuristic literature, particles are usually discarded depending on their objective value, in a way that particles with high values are more likely to be discarded [27,40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' The proposed strategy does not add a further heuristic strategy but simply cut down the algorithm complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Also, the convergence properties are in this way expected to be preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' We note that, on the other hand, there is no straightforward way to generate particles and, at the same time, preserve the particle system distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='3 Comparison with CBO and PSO What distinguishes CBO-ME from plain CBO, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='g [4,34], is clearly the introduction of the best positions {yk i }N i=1 and the fact that the consensus point is calculated among them and not 5 Algorithm 1: Consensus-Based Optimization with Memory Effects (CBO-ME) Input: F, N0, kmax, λ, σ, α, nstall and δstall;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' 1 Inizialize N0 particle positions xi 0, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' , N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' 2 y0 i ← x0 i for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' , Nk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' 3 Compute yα,0 according to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' 4 k ← 0, n ← 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' 5 while k < kmax and n < nstall do 6 for i = 1 to Nk do 7 θk i ∼ N(0, Id);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' 8 Compute xk+1 i according to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' 9 if F(xk+1 i ) < F(yk i ) then 10 yk+1 i ← xk+1 i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' 11 else 12 yk+1 i ← yk i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' 13 end 14 end 15 Compute ¯yα,k+1 according to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' 16 if ∥¯yα,k+1 − ¯yα,k∥2 < δstall then 17 n ← n + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' 18 else 19 n ← 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' 20 end 21 Compute Nk+1 according to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='6);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' 22 if Nk+1 < Nk then 23 Randomly discard Nk+1 − Nk particles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' 24 k ← k + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' 25 end 26 return ¯yα,k, F(¯yα,k) just among the particle positions {xk i }N i=1 at that given time k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Indeed, the classical CBO update rule without memory effects (and with anisotropic diffusion and projection step) is given by xk+1 i = xk i + λ � ¯xα,k − xk i � + σ � ¯xα,k − xk i � ⊗ θk i (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='7) where ¯xα,k is defined consistently with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='2) (by substituting yk i with xk i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' As we will see in the numerical tests, the use of memory effects improves the algorithm performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Since alignment towards personal bests yk i and towards the global best ¯y∞,k are also the fundamental building blocks of PSO algorithms, we highlight now the main differences and similarities between PSO and CBO-ME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' For completeness, we recall the canonical PSO method, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' [36], using the notation of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='4) for easier comparison � xk+1 i = xk i + vk+1 i vk+1 i = wvk i + C1 � yk i − xk i � ⊗ ˆθk i,1 + C2 � ¯y∞,k − xk i � ⊗ ˆθk i,2 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='8) 6 where vk i are the particles velocities, w, C1, C2 > 0 are the algorithm parameters and θk i,1, θk i,2 are uniformly sampled from [0, 1]d (ˆθk i,1, ˆθk i,2) ∼ Unif([0, 1]d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Several variants and improvements have been proposed starting from the above dynamics, but a complete review is beyond the scope of this paper and we refer to the recent survey [47] for more references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' We are interested in highlighting the main differences between (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='4) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='8) regarding the stochastic components: in CBO-ME deterministic and stochastic steps are de-coupled and tuned by two different parameters (λ and σ), while in PSO they are coupled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Indeed, in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='8), deterministic and stochastic components are both controlled by the same parameter: C1 in the case of personal best dynamics and C2 for the global best one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' By splitting the term C2 � ¯y∞,k − xk i � ˆθk i,2 into a deterministic step and a zero-mean term we obtain C2 � ¯y∞,k − xk i � ⊗ ˆθk i,2 = C2 2 � ¯y∞,k − xk i � + C2 2 � ¯y∞,k − xk i � ⊗ θk i,2 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='9) with θk i,2 = 2ˆθk i,2 − 1, θk i,2 ∼ Unif([−1, 1]d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Suggested in [14], such rewriting highlights how increasing the alignment strength towards the global best (by increasing C2) necessary increases the stochasticity of the system as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' In (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='4) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='7), on the other hand, one is allowed to tune the exploration and exploitation behaviors separately, by either changing parameters λ or σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Clearly, CBO-ME also differs from PSO due to its first-order dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Having the aim of resembling birds flocking, the first PSO algorithm [24] was proposed as a second-order dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' The inertia weight w, introduced later in [39], became an essential parameter to prevent early convergence of the swarm and to increase the global exploration behavior, especially at the beginning of the computation, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' [31, 39] and reviews [18, 36, 47] for more references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' We note that several other strategies have proposed to improve PSO exploration behavior, see, for example, [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' As already mentioned, in CBO methods convergence and exploration are de-coupled and can be tuned separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Therefore, to keep the algorithm more amenable to theoretical analysis, we consider a simpler first-order dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' We note that a CBO dynamics with inertia mechanism was proposed in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Similarly, we found the contribution given by the personal best alignment non-essential and difficult to tune.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Thus, the lack of alignment towards personal best in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Replacing alignment towards personal best with gaussian noise was also suggested in [48] where authors proposed the Accelerated PSO (APSO) algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Further studied in [11,49], APSO also allows to de-couple the stochastic component from the deterministic one and the noise is heuristically tuned to decrease during the computation (as in Simulated Annealing [25]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' In CBO methods, the noise strength automatically adapts as it depends on the distance from the consensus point, which is also different for every particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' For completeness, we note that many other variants of PSO have been proposed to include the explorative behavior, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Chaotic PSO [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' 3 Numerical results Having discussed the fundamental features of the CBO dynamics with memory effects, we now validate Algorithm 1 and compare its performance with plain CBO and PSO algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' We will test the methods against several benchmark optimization problems and analyze the impact of the 7 Name Objective function F(x) Search space x∗ F(x∗) Ackley −20 exp � −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='2 � 1 d �d i=1 (xi)2 � − exp � 1 d �d i=1 cos (2π(xi)) � + 20 + e [−32, 32]d (0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' , 0) 0 Griewank 1 + �d i=1 (xi)2 4000 − �d i=1 cos � xi i � [−600, 600]d (0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' , 0) 0 Rastrigin 10d + �d i=1 � (xi)2 − 10 cos (2π(xi)) � [−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='12, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='12]d (0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' , 0) 0 Rosenbrock 1 − cos � 2π ��d i=1 (xi)2 � + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='1 ��d i=1 (xi)2 [−5, 10]d (1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' , 1) 0 Salomon 1 − cos � 2π ��d i=1 (xi)2 � + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='1 ��d i=1 (xi)2 [−100, 100]d (0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' , 0) 0 Schwefel 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='20 �d i=1 |xi| [−100, 100]d (0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' , 0) 0 XSY random �d i=1 ηi|xi|i, ηi ∼ Unif([0, 1]) [−5, 5]d (0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' , 0) 0 XSY 4 ��d i=1 sin2(xi) − e − �d i=1(xi)2� e − �d i=1 sin2 √ |xi| [−10, 10]d (0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' , 0) −1 Table 1: Considered benchmark test functions for global optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' For each function, the corresponding search space and global solution is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' random selection technique on the convergence speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' We also employ 1 to solve problems arising form applications, such as image segmentation and training of a machine learning architectures for function approximation and image recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='1 Tests on benchmark problems We test the proposed algorithm against different optimization problems, by considering 8 bench- mark objective functions, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' [22], which we report in Table 1 for completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' The search space dimension is set to d = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' As in plain CBO methods, we expect the most important parameters are those governing the balance between the exploitative behavior (λ in this case) and the explorative one (σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' In particular, we are interested in the algorithm performance as we change the ratio between λ and σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Therefore, in the first experiment we fix λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='01, while considering different values of σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' The parameter α is adapted during the computation: starting form α0 = 10, it increases according to the law α = α0 · k · log2(k) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='1) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' 1 shows the accuracy and the objective value reached for σ ∈ [0, 2] after kmax = 104 algorithm iterations with N = 200 particles, with no random selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' The optimal value for σ is clearly problem-dependent, but we note that the optimal values for the problems considered all fall within a relative small range (underlined in gray in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' 1 we infer that a good value for all benchmark problems considered is given by σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Using this value, we now compare CBO-ME, with plain CBO and the standard PSO (with and without alignment towards personal best) for different population sizes N = 50, 100, 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' We keep the random selection mechanism off by setting µ = 0 and use the same 8 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='5 2 < 10!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='10 10!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='5 100 105 1010 ky,;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='k !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' x$k1 Rastrigin Ackley Griewank Rosenbrock Salomon Schwefel XSY 4 XSY random (a) ∥¯yα,k − x∗∥∞ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='5 2 < 10!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='10 10!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='5 100 105 1010 F(7y,;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='k) Rastrigin Ackley Griewank Rosenbrock Salomon Schwefel XSY 4 XSY random (b) F(¯yα,k) Figure 1: Optimization on benchmark functions using CBO-ME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Behavior of the expec- tation error and fitness value for different values of σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Here λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='01 and α is adaptive, with α0 = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' The particle population is N = 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Grey bands (of values [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='70, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='05] for the error and [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='65, 1] for the fitness) show the range in which the minima of the different benchmark functions fall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' The dotted line marks the visually estimate pseudo-optimal value σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Results averaged on 250 runs, are obtained with kmax = 104 iterations and without stopping criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' previously chosen parameters when memory effects are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' For plain CBO, without memory effects, we set σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='71 ≈ √ 2/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Concerning PSO, we use the solver provided by the MATLAB Global Optimisation Toolbox (particleswarm), changing the maximum number of iterations and the stall condition to the one used for CBO methods, to make the results comparable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' The remaining parameters are kept as described in the relative documentation [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' We set kmax = 104, δstall = 10−4 and consider a run successful when either ∥¯yα,k − x∗∥∞ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='1 or |F(¯yα,k) − F(x∗)| < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='01 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='2) Table 2 reports success rate, final error given by ∥¯yα,k −x∗∥∞, mean objective function value and total number of iterations, averaged over 250 runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' In addition to the classic PSO method, where the acceleration coefficients are chosen to be equal C1 = C2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='49, Table 2 also shows the results when only the alignment towards global best is considered in PSO (C1 = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' While CBO already manages to find the global minimizer in most of the problems considered, we note that it fails when Rastrigin, Rosenbrock or XSY random functions are optimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' CBO- ME, on the other hand, is able to solve the optimization problem correctly even in these cases if the population size N is large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' CBO seems to achieve greater accuracy in some cases, such as with Schwefel 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='20 and Salomon objectives, at the cost of more iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Standard PSO in many cases fails to solve the problem, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Rastrigin, Salomon or XSY 4 functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' PSO success rate is also lower among all problems, with the exception of the Schwefel 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='20 benchmark problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Considering only global adjustment seems to show advantages with respect to the classical PSO method,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' except in the case of Ackley where setting C1 = 0 decreases the success rate or,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' in the case of XSY 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Salomon or Rastrigin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' where convergence is not achieved even for C1 = 0 Consensus methods,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' however,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' seem to perform better in terms of both success 9 (a) Error: ∥¯yα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='k − x∗∥∞ (b) Fitness Value: F(¯yα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='k) Figure 2: Optimization of Ackley function for different values of the random selection parameter µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' where the initial particle population is N 0 = 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' We report error (on the left) and fitness values (on the right) as the number of function evaluations increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Parameters are set as λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='01, σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='8, α adaptive starting from α0 = 10 and following the law α = α0 · k · log2(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Results are averaged over 250 runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' rate and speed up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' In addition, for most problems, the population size N seems not to play a significant role in the algorithms performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' This further motivates the introduction of the random selection strategy described in the Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='1 in order to save computational costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' In the third experiment, we test the proposed random selection mechanism (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='6) for different values of the parameter µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' We recall that with µ = 0 we have no particles removal, while as µ increases, more particles are likely to be discarded when the system variance decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' The initial population is set to N0 = 200, while the minimum number of particles to Nmin = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Results are reported in Tables 3 and 4 in terms of: success rate, error, objective value, weighted number of iterations, given by witer = kend � k=1 Nk N0 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='3) and percentage of Computational Time Saved (CTS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Results show that relative large values of µ allow to reach fast convergence without affecting the algorithm performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' The values of µ considered in Table 4 as different from those in Table 3 as in our experiments, the Rastrigin problem allows for larger values of µ, while the Rosenbrock one seems to be more sensitive to the selection mechanism with respect to the other objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' In both cases, a suitable value of µ reduces the computational time with almost no impact in terms of accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='s 2 and 3 show error and fitness value as a function of the number of fitness evaluation during the algorithm computation, for the Ackley and Rastrigin problem respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Several values of µ are considered to display how the random selection mechanism affects the convergence speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Initial particle population is set to N0 = 104 and particles evolve for kmax = 104 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' We note how convergence speed increases as µ increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' 10 CBO (σ = √ 2/2) CBO-ME (σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='8) PSO PSO (C1 = 0) N = 50 N = 100 N = 200 N = 50 N = 100 N = 200 N = 50 N = 100 N = 200 N = 50 N = 100 N = 200 Ackley Rate 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='3% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='0% 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='6% 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='6% 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='3% 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='0% 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='0% 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='3% Error 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='11e-06 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='03e-06 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='55e-06 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='39e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='73e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='74e-06 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='97e-09 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='56e-11 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='16e-12 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='30e-08 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='90e-10 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='96e-13 Favg 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='18e-04 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='81e-05 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='30e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='54e-04 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='96e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='99e-05 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='24e-09 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='65e-11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='94e-12 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='06e-08 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='70e-10 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='01e-13 Iterations 954.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='3 778.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='2 678.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='1 997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='7 724.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='3 626.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='9 493.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='8 420.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='6 391.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='4 502.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='0 436.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='0 390.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='1 Griewank Rate 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='0% 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='0% 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='6% 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='3% 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='0% 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='7% 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='0% Error 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='20e-02 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='21e-02 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='24e-02 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='13e-02 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='16e-02 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='25e-02 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='34e-02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='56e-02 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='45e-03 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='17e-01 1.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='21e-06 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='85e-08 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='17e-08 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='35e-04 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='28e-04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='34e-04 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='22e-04 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='11e-04 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='45e-04 Iterations 10000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='0 10000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='0 10000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='0 10000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='0 10000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='0 10000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='0 10000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='0 10000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='0 10000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='0 10000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='0 10000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='0 10000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='0 XSY 4 Rate 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='0% 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='3% 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='7% 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='3% 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='7% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='0% Error 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='07e-01 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='48e-01 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='16e-01 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='44e-01 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='35e-01 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='95e-01 Favg 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='79e-07 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='78e-07 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='46e-07 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='58e-06 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='56e-07 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='43e-07 Iterations 10000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='0 10000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='0 10000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='0 9677.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='5 9128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='4 8943.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='2 10000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='0 10000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='0 10000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='0 10000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='0 10000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='0 10000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='0 Table 2: Comparison between classical CBO, CBO-ME and standard PSO with and with- out alignment towards personal best on benchmark problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' The solver particleswarm available in the MATLAB Global Optimisation Toolbox was used for the results concerning the PSO method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Optimal choice of parameters, different for each method, are used for the CBO algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Same stopping criterion and definition of success, see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='2), were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Performance metric considered: success rate (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='2)), error (∥¯yα,k−x∗∥∞), fitness value F(¯yα,k) and number of iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Results are averaged over 250 runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' 11 µ = 0 µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='05 µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='1 µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='2 Ackley Rate 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='0% Error 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='84e-06 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='16e-06 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='54e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='34e-05 Favg 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='30e-05 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} 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+page_content='1% 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='6 % 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='8% Griewank Rate 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='0% Error 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='35e-02 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='22e-02 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='32e-02 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='28e-02 Favg 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='82e-02 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='20e-02 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='72e-02 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='70e-02 witer 635.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='4 395.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='9 204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='6 184.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='6 CTS 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='8% 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='2% 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='3% Schwefel 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='20 Rate 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='0% Error 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='76e-07 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='08e-07 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='21e-07 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='73e-08 Favg 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='37e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='93e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='58e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='74e-05 witer 467.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='7 359.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='8 318.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='7 172.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='4 CTS 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='4% 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='9% 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='1% Salomon Rate 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='0% Error 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='11e-02 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='35e-02 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='74e-02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='75e-02 Favg 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='34e-01 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='43e-01 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='07e-01 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='26e-01 witer 2456.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='5 1595.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='8 1289.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='2 913.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='0 CTS 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='7% 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='1% 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='7% XSY random Rate 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='0% Error 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='50e-02 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='62e-02 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='89e-02 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='08e-02 Favg 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='97e-07 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='75e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='48e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='06e-04 witer 10000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='0 2642.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='3 1755.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='7 1123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='7 CTS 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='6% 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='4% 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='7% XSY 4 Rate 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='0% Error 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='30e-01 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='78e-01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='35e-01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='37e-01 Favg 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='17e-05 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='28e-06 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='41e-06 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='55e-06 witer 8943.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='2 3910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='4 1890.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='2 1060.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='1 CTS 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='9% 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='1% 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='4% Table 3: CBO-ME algorithm with random selection of particles tested against different benchmark functions with different values of µ, which regulates the random selection mech- anism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' The system is initialized with N0 = 200 particles and σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Performance metric considered: success rate (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='2)), error (∥¯yα,k − x∗∥∞), fitness value F(¯yα,k), weighted iteration (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='3), and Computational Time Saved (CTS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Results are averaged over 250 runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' µ = 0 µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='1 µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='2 µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='5 Rastrigin Rate 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='0% Error 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='14e-05 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='12e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='77e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='24e-05 Favg 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='23e-06 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='19e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='98e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='27e-06 witer 1161.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='1 719.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='6 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='5 111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='2 CTS 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='1% 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='9% 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='4% µ = 0 µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='01 µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='02 µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='05 Rosenbrock Rate 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='0% 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='4% 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='0% Error 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='55e-02 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='23e-02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='66e-02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='341e-02 Favg 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='20e-03 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='23e-03 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='10e-03 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='24e-03 witer 3172.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='3 852.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='9 347.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='8 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='5 CTS 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='1% 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='1% 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='4% Table 4: CBO-ME algorithm with particle reduction tested against Rastrigin and Rosen- brock functions with an higher diffusion parameter σ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='1 and for different values of µ , which regulates the random selection mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' The system is initialized with N0 = 200 particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Performance metric considered: success rate (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='2)), error (∥¯yα,k − x∗∥∞), fitness value F(¯yα,k), weighted iteration (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='3), and Computational Time Saved (CTS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' 12 (a) Error: ∥¯yα,k − x∗∥∞ (b) Fitness Value: F(¯yα,k) Figure 3: Optimization of Rastigin function for different values of the random selection parameter µ where the initial particle population is N0 = 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' We report error (on the left) and fitness values (on the right) as the number of function evaluations increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Parameters are set as λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='01, σ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='1, α adaptive starting from α0 = 10 and following the law α = α0 · k · log2(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Results are averaged over 250 runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='2 Applications In this section, we propose some applications of the proposed optimization algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' First we consider a image segmentation problem using multi-thresholding, then we use the CBO- ME to train a Neural Network (NN) architecture to approximate functions and perform image classification on MNIST database of handwritten digits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='1 Image segmentation To perform image segmentation, we use a threshold detection technique, namely, the multidimen- sional Otsu algorithm [32,44] in order to compare the results to similar optimization algorithm, such as the Modified PSO in [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' In the Otsu algorithm, every pixel of the image is assigned to one of the possible L grayscale values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' We denote with ηi the number of pixel with gray level i, 1 ≤ i ≤ L and Npix = �L i=1 ηi the total number of pixels [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Then, the image is divided into object C0 with gray-level [1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' , l] and background C1 with gray-level [l + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' , L] by inserting a threshold l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' The probabilities of class occurrence and the class mean level for the object, respectively, are given by ω0(l) = l � i=1 pi, pi = ηi Npix µ0(l) = l � i=1 ipi ω0(k) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' For the background, the class occurrence probabilities and the class mean level are given by ω1(l) = L � i=l+1 pi, pi = ηi Npix 13 µ1(l) = L � i=l+1 ipi ω1(k) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' As in [32], the best threshold l∗ is obtained when the variance formula f(l) = ω0(l) ω1(l) (µ0(l) − µ1(l))2 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='4) between object group and background reaches its maximum value, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' l∗ = argmaxlf(l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' The problem is then reduced to a threshold problem, which we can solve with optimization methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Since segmentation is a trivial one-dimensional problem, we consider an extension of Otsu’s technique to the multidimensional case [44] to test capabilities of method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Assuming we want to optimize the choice of d thresholds, we require d + 1 classes of different gray-scales (C0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' , Cd) with relative probabilities of occurrence classes defined as ω0(l1) = l1 � i=1 pi , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' , ωd(ld) = L � i=ld+1 pi, pi = ηi Npix and classes mean levels µ0(l1) = �l1 i=1 ipi ω0 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' , µd(ld) = �L i=ld+1 ipi ωd , The optimal thresholds (ˆl1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' , ˆld) are those that satisfy ˆl1 < · · · < ˆld and maximise f(l1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' ld) = d � i=1 ωi(li)µ2 i (li) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='5) For the experiment, we chose d = 5 thresholds and compare the segmentation performed by Otsu’s method, solved with both standard PSO and CBO-ME, with segmentation obtained by dividing the greyscale into d + 1 uniformly spaced intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' For PSO, we use to the default parameters in the particleswarm function in the MATLAB Global Optimisation Toolbox, while for CBO-ME we used optimal parameters found in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='1 and exploit the random selection technique to speed up the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' We report the results on two sample images, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='s 4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' We fix kmax = 103 and average results over 250 runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' As in [2], we evaluate multi-thresholding segmentation through the Peak Signal to Noise Ratio (PSNR) computed as: PSNR = 20 · log10 � 255 RMSE � where RMSE is the Root Mean-Squared Error, defined as RMSE = � � � � 1 Npix Nrow � i=1 Ncol � j=1 [I(i, j) − S(i, j)]2 where Npix = Nrow · Ncol, I is the original image and S is the associated segmented image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' The higher the value of PSNR is, the greater the similarity between the clustered image and the original image is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='s 4,5, we note that the most accurate segmentation on details is obtained by the CBO-ME method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' This is quantitatively confirmed by the PSNR values reported in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' 14 (a) Original (b) Standard segmentation (c) Otsu seg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' (PSO) (d) Otsu seg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' (CBO-ME) Figure 4: Image segmentation of darkhair woman image (256 × 256 pixels) with standard segmentation and Otsu segmentation solved respectively by PSO (c) and by CBO-ME (d);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' results are averaged over 250 runs, with an initial population of N0 = 103 particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' (a) Original (b) Standard segmentation (c) Otsu seg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' (PSO) (d) Otsu seg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' (CBO-ME) Figure 5: Image segmentation of lake image (256×256 pixels) with standard segmentation and Otsu segmentation solved respectively by PSO (c) and by CBO-ME (d);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' results are averaged over 250 runs, with an initial population of N0 = 103 particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' cameraman lake lena peppers woman darkhair Standard segmentation 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='83 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='72 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='35 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='24 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='33 Otsu segmentation (PSO) 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='62 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='33 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='19 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='03 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='14 Otsu segmentation (CBO-ME) 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='22 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='44 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='72 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='28 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='57 Table 5: PSNR values to evaluating the advantages of the method in optimising threshold values in 5 sample images known in literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' For these results, we compared the Otsu segmentation solved by the proposed CBO-ME method with the classical PSO method with equispaced thresholding segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Experiments are performed with d = 5 thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' 15 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='5 1 2 1 0 1 2 (a) 2000 epochs 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='5 1 2 1 0 1 2 (b) 3000 epochs 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='5 1 2 1 0 1 2 (c) 5000 epochs 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='5 1 2 1 0 1 2 (d) 8000 epochs Figure 6: Approximating smooth function u1 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='8) using a network with n = 50 and m = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' The learning rate is λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='2 and we initially use N0 = 500 particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' The others parameters are set as λ = 1, σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='8 and α adaptive starting from α0 = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='2 Approximating functions with NN In this section, we use the proposed CBO-ME algorithm to train a NN architecture into approx- imating a function u : I → R, I ⊂ R with low regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' As in [5], we use a fully-connected NN with m layers f(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' θ) = (Lm ◦ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' L2 ◦ L1)(x) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='6) where each layer is given by Li = σ(W ix + bi) with σ(x) = 1/(1+exp(−x)) being the sigmoid function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' We use internal layers of dimension n, so W 1 ∈ Rn×1, b1 ∈ R, W m ∈ R1×n, bm ∈ Rd and W i ∈ Rn×n for all i = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' , m − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' In (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='6), all DNN parameters are collected in θ = {W i, bi}m i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' As loss function which need to be minimized, we consider the L2-norm between the target function u and its NN approximation f(· ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' θ) F(θ) := ∥f(· ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' θ) − u∥L2(I) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='7) Again, similarly to [5], we test the method against the following two functions: u1(x) = sin(2πx) + sin(8πx2) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='8) 16 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='5 1 2 1 0 1 2 (a) 2000 epochs 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='5 1 2 1 0 1 2 (b) 3000 epochs 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='5 1 2 1 0 1 2 (c) 5000 epochs 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='5 1 2 1 0 1 2 (d) 8000 epochs Figure 7: Approximating non-smooth u2 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='9) function using a network with n = 50, m = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' The learning rate is λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='2 and we use initially N0 = 500 particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' The others parameters are set as λ = 1, σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='8 and α adaptive starting from α0 = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' u2(x) = � � � � � 1 if x < − 7 8, − 1 8 < x < 1 8, x > 7 8 −1 if 3 8 < x < 5 8, − 5 8 < x < − 3 8, 0 otherwise .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='9) We note that u1 is smooth, while u2 is discontinuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Parameters of the CBO-ME algorithm have been set to λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='01, σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='8, as in the previous sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Parameter α is adapted during the computation as in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='1 and random selection mechanism is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' We employ m = 3 layers with internal dimension n = 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Results are displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='s 6 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' We note that smooth function u1 is well-approximated already after 5000 epochs, while convergence is slower for the discontinuous step function u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='3 Application on MNIST dataset We now employ the proposed algorithm to train a NN architecture to solve a image classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' We will consider the MNIST dataset [26] composed of handwritten digits in grayscale with 28 × 28 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' For better comparability with CBO methods without memory effects, we closely follow the experiment settings used in the literature [4,10,37], which we summarize below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' We consider a 1-layer NN where input images x ∈ R28×28 are first vectorized x �→ vec(x) ∈ R728 and then processed through a fully-connected layer with parameters θ = {W, b}, with 17 10 20 30 40 50 Epochs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='8 1 Accuracy on test data CBO-ME CBO 10 20 30 40 50 Epochs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='18 Loss CBO-ME CBO Figure 8: Performance during training of shallow NN (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='10) on image classification (MNIST dataset) with CBO-ME optimizer and plain CBO without memory effects [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Training is performed by Algorithm 1 with N = 100 particles and no particle selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Cross-entropy loss function (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='11) and adaptive parameters strategy (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='12) were used in the training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' W ∈ R10×728, b ∈ R10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' That is, the network is given by fSNN(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' θ) = softmax (ReLU (Wvec(x) + b) ) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='10) where ReLU(z) = max{z, 0} (component-wise) and softmax(z) = (ez1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' , ezn)/(� i ezi) are the commonly activation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' During the training, batch regularization is performed after ReLU is applied in order to speed up convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Given a training set {(xm, ℓm)}M m=1, xm ∈ R28×28, ℓm ∈ {0, 1}10 made of M image-label tuples we train the model by minimizing the categorical cross-entropy loss F(θ) = 1 M M � m=1 � − 10 � i=1 ℓm i log(fi(xm, θ)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='11) We employ a population on N = 100 particles throughout the entire computation, initially sampled from the standard normal distribution N(0, Id).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Following the mini-batch approach suggested in [4], the consensus points ¯yα,k is computed only among a random subset of nN = 10 particles, but all particles are updated at each step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' The training data is divided in batches of nF = 60 images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' The drift parameter is set to λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='01, while σ and α are adapted during the computation after each epoch as σepoch = σ0/ log2 (epoch + 2) αepoch+1 = 2 · αepoch+1 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='12) starting form σ0 = √ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='04, and α0 = 50, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' 8 shows the results in terms of loss function (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='11) over the test data set and the accuracy reached in the classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' While challenging state-of-the-art training methods 18 is beyond the scope of the experiment, we note how high-dimensional data optimization tasks can be solved with as little as N = 100 particles by the proposed method, obtaining results comparable with the literature on CBO methods [4, 10, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Also, we remark that parameters have not been tuned extensively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' 4 Theoretical analysis A strength of CBO algorithms lays on the possibility of theoretically analyze the particle system by relying on a mean-field approximation of the dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' We will illustrate in this section how to derive such approximation and present the main theoretical result regarding the convergence of the particle system towards a solution to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='1), in case of no selection mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Next, we will study the impact of the random selection strategy on the convergence properties of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Technical details are left to Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='1 Mean-field approximation First, we note that a simple update rule for the personal bests yk i is given by yk+1 i = yk i + 1 2 � xk+1 i − yk i � S(xk+1 i , yk i ) , with S(x, y) = 1 + sign (F(y) − F(x)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='1) As in [14], we approximate it for β ≫ 1 as yk+1 i = yk i + ν 2 � xk+1 i − yk i � Sβ(xk+1 i , yk i ) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='2) with Sβ(x, y) being a continuous approximation of S(x, y) as β → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' By choosing ν = 1 we get (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='1) with the only difference of having Sβ instead of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' As for ¯yα with respect to ¯y∞, this is needed to make the update rule easier to handle mathematically, but it does have an impact on the performance for large values of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' With the aim of deriving a continuous-in-time reformulation of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='4) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='2), we introduce a single parameter ∆t > 0 which controls the step length of all involved update mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' By performing the rescaling λ ← λ∆t , σ ← σ √ ∆t , ν ← ν∆t to get the update rules � xk+1 i = xk i + λ∆t � ¯yα,k − xk i � + σ √ ∆t � ¯yα,k − xk i � ⊗ θk i yk+1 i = yk i + (ν∆t/2) � xk+1 i − yk i � Sβ(xk+1 i , yk i ) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='3) which differ form the original formulation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='4), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='1) only due to the use of Sβ instead of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' As already noted in [14], the iterative process (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='3) corresponds to an Euler-Maruyama scheme applied to a system of Stochastic Differential Equations (SDEs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Indeed, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='3) corre- sponds to a discretization of the system � dXi t = λ � ¯yα(ρN t ) − Xi t � dt + σ � ¯yα(ρN t ) − Xi t � ⊗ dBi t dY i t = ν(Xi t − Y i t )Sβ(Xi t, Y i t ) dt (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='4) 19 where, for convenience, we underlined above the dependence of the consensus point on the empirical distribution ρN t = � i δY i t (δy being the Dirac measure at y ∈ Rd) by using ¯yα(ρ) := � ye−αF(y)dρ(y) � e−αF(y)dρ(y) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='5) defined for any Borel probability measure ρ over Rd (ρ ∈ P(Rd)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' In this way, we generalized the definition introduced in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='2) to any ρ ∈ P(Rd), provided the above integrals exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' In (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='4), the random component of the dynamics is now described by N independent Wiener processes (Bi t)t>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' As before, we supplement the system with initial conditions Xi 0 ∼ ρ0, Y i 0 = Xi 0 for some ρ0 ∈ P(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' The continuous-in-time description (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='4) already simplifies the analytical analysis of the optimization algorithm, but still pays the price of a possible large number O(N) of equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' This issue is typically addressed by assuming that for large populations N, the particles become indistinguishable from one another and start behaving, in some sense, as a unique system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' More precisely, let F N(t) ∈ P(R(2d)N) denote the joint probability distribution of N tuples (Xi t, Y i t ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' We assume propagation of chaos [41] for large N ≫ 1, that is, we assume that the joint probability distribution decomposes as F N(t) = f(t)⊗N for some f(t) ∈ P(R2d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' System (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='4) becomes independent on the index i and hence every particle moves according to the mono-particle process d ¯Xt = λ(¯yα(¯ρt) − ¯Xt) dt + σ (¯yα(¯ρt) − ¯Xt) ⊗ d ¯Bt d ¯Yt = ν( ¯Xt − ¯Yt)Sβ( ¯Xt, ¯Yt) dt (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='6) where ¯ρt = Law( ¯Yt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Assume ( ¯Xt, ¯Yt) are initially distributed according to f0 = ρ⊗2 0 , by applying Itˆo formula we have that f(t) = Law( ¯Xi t, ¯Y i t ) satisfies ∂tf + ∇x · (λ(¯yα(¯ρ) − x)f) + ∇y · � ν(x − y)Sβ(x, y)f � = d � ℓ=1 ∂2 xℓ � σ(¯yα(¯ρ) − x)2 ℓf � (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='7) and initial data limt→0 f(t) = f0 in a weak sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Dynamics (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='6), or, equivalently, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='7), corresponds to the mean-field approximation of the particle system (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='4) as N → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' We remark that the above derivation has only been possible thanks to the approximations S ≈ Sβ and ¯y∞ ≈ ¯yα for large α and β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Well-posedness of the system is also granted by such approximations (proof details are given in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='1 (well-posedness of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='6)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' There exists a unique process ( ¯X, ¯Y ) ∈ C([0, T], Rd), T > 0 satisfying (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='4) with initial conditions ( ¯X0, ¯Y0) with ¯X0 ∼ ρ0 ∈ P4(Rd) and ¯Y0 = ¯X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Being mathematically tractable, we show next that the mean-field dynamics converges to a global solution to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='1) if F, Sβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' 20 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='2 Convergence in mean-field law We start by enunciating the necessary assumptions to the convergence result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='1 (Assumptions on F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' The objective function F ∈ C(Rd, R), satisfies: A1 there exists some constant LF > 0 such that |F(x) − F(x′)| ≤ LF � ∥x∥2 + ∥x′∥2 � ∥x − x′∥2, ∀ x, x′ ∈ Rd ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' A2 there exists uniquely x∗ ∈ Rd solution to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' A3 there exist η, R0 > 0 and γ ∈ (2, ∞) such that F(x) − inf F ≥ η ∥x − x∗∥γ ∞ ∀x ∈ Rd , ∥x − x∗∥∞ ≤ R0 F(x) − inf F ≥ η Rγ 0 ∀x ∈ Rd , ∥x − x∗∥∞ > R0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' A4 F is convex in a (possibly small) neighborhood {x ∈ Rd : ∥x − x∗∥∞ ≤ R1} of x∗ for some R1 < R0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' A5 There exists cg, R2 > 0 such that F(x) − inf F ≥ cg∥x − x∗∥2 2 ∀x ∈ Rd , ∥x − x∗∥2 > R2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='2 (Assumptions on Sβ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' The function Sβ ∈ C(R2d, [0, 2]), with β > 0 A6 has the following structure Sβ(x, y) = 2ψ (β(F(y) − F(x))) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='8) with ψ ∈ C1(R, [0, 1]) being an increasing function with Lipschitz constant Lψ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' A7 The value Sβ(x, y) is positive only when x is strictly better than y in terms of objective value F: Sβ(x, y) � ≥ 0 if F(x) < F(y) = 0 else .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Assuming uniqueness of global minimum is a typical assumption for analysis of CBO methods [9,10] and it is due to the definition of the consensus point ¯yα (or ¯xα in the case without memory mechanism).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Indeed, in presence of two global minima, ¯yα may be placed between them, no matter how large α is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Assumption A2 ensure to avoid such situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Furthermore, A3 also allows to give quantitative estimates on the difference between the global minimum and eventual local minima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' In the literature, such property is known as conditioning [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Requirements A4 and A7 ensure that if a personal best yk i enters such small neighborhood where F is convex, it will not leave it for the rest of the computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Condition A5 (quadratic growth at infinity) is needed for the well-posedness of the mean-field mono-particle process (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='6), see also [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' For an intuition of A3 and A4 we refer to Figure 9, where the Rastrigin function is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' 21 x$ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' R0 x$ x$ + R0 0 20 40 objective lower bound (A3) convex area (A4) Figure 9: Assumptions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='1 illustrated for Rastrigin function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' For example, such objective function satisfies A3 with η = 1, γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='8, R0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='42 and A4 with R1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='1 (Convergence in mean-field law).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Assume F satisfies A1–A5, Sβ satisfies A6, A7 for some β > 0 fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Let ( ¯Xt, ¯Yt)t≥0 be a solution to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='6) for t ∈ [0, T], with initial data ¯X0 ∼ ρ0 ∈ P4(Rd), Y0 = X0 such that x∗ ∈ supp(ρ0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Fix an accuracy ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' If 2λ > σ2, there exists a time T ∗ such that the expected ℓ2-error satisfies E � ∥ ¯XT ∗ − x∗∥2 2 � ≤ ε (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='9) provided T, α > 0 are large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' We refer to Appendix A for a proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' The mean-field mono-particle process (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='6) aims to approximate the algorithm iterative dynamics (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='3) for small time steps ∆t ≪ 1 and large particle populations N ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Therefore, convergence of the algorithm dynamics towards the global solution x∗ can be proven by coupling Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='1 with error estimates of such approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' For instance, assuming that all considered dynamics take place on a bounded set D ensures that the error introduced by the continuous-in-time particle system will be of order ∆t thanks to classical results on Euler-Maruyama schemes [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Likewise, considering a bounded dynamics allows to prove that the error introduced by the mean-field approximation is of order N−1 (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' [8, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='1], [9, Proposition 16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Let {(xk i , yk i )}N i=1 be given by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='3), {(Xi t, Y i t )}N i=1 be a solution (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='4) and {( ¯Xi t, ¯Y i t )}N i=1 be N-copies of a solution to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Altogether, one obtains the following error decomposition for K∆t = T ∗ E � 1 N N � i=1 ∥xK i − x∗∥2 2 � ≤ C � E � 1 N N � i=1 ∥xK i − Xi T ∗∥2 2 � + E � 1 N N � i=1 ∥Xi T ∗ − ¯Xi T ∗∥2 2 � + E � 1 N N � i=1 ∥ ¯Xi T ∗ − x∗∥2 2 � � ≤ CEM∆t + CMFAN−1 + ε 22 where C, CEM, CMFA are positive constant independent on N, ∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='3 Random selection analysis In this section, we analytically investigate the impact of randomly discarding particles during the computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' We are particularly interested in tracking the distance between a particle system {xk i , xk j }N0 i=1 evolving according to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='3) where no particles are discarded, and a second system {ˆxk i , ˆyk i }Ik, |Ik| = Nk where Nk − Nk+1 particles are discarded after update rule (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Clearly, we have that Nk+1 ≤ Nk and Ik+1 ⊆ Ik ⊆ I0 = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' , N0} for all k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Similarly to the analysis carried out in [15,16], we restrict to the simpler dynamics where, at every step k, the random variables θk i and ˆθk i used to generate such systems are the same for all particles: θk i = ˆθk j = θk ∼ N(0, Id) for all i ∈ Ik, j ∈ I0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='10) To compare particle systems with a different number of particles, we rely on their represen- tation as empirical probability measures and the notion of 2-Wasserstein distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' For {ˆxk i }i∈Ik and {xk i }N0 i=1 we consider, respectively, the following probability measures ρk Nk := 1 Nk � i∈Ik δˆxk i and ρk N0 := 1 N0 � i∈I0 δxk i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='11) Informally, the 2-Wasserstein distance W2(ρk Nk, ρk N0) quantifies the minimal effort needed to move the mass from distribution ρk Nk into ρk N0 (or vice versa) [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Let wij denote the amount of mass leaving particle xk i and going into ˆxk i : the cost of such movement is assumed to be given by wij∥xk i − ˆxk j ∥2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Therefore, if we indicate the set of all admissible couplings between the two discrete probability measures as Γ(ρk Nk, ρk N0) = � � �w ∈ RN0×Nk : Nk � j=1 wij = 1 N0 , N0 � i=1 wij = 1 Nk , wij ≥ 0, ∀ i, j � � � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='12) the 2- Wasserstein distance is defined as W2(ρk Nk, ρk N0) := min w∈Γ(ρk Nk,ρk N0) � �� i,j wij∥xk i − ˆxk j ∥2 2 � � 1 2 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='13) see, for instance, [38, Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Before providing estimates on (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='12), let us present a more general result on the impact that the random selection strategy has on an arbitrary particle distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='2 (Stability of random selection procedure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Let z = {zi}i∈I, |I| = N be an ensemble of particles and {zi}j∈Isel with Isel ⊆ I, |I| = Nsel a random sub-set of such ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Consider the associated empirical distributions µN and µNsel (defined consistently to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='11)), it holds E � W 2 2 (µN, µNsel) � ≤ 2 var(z) N − Nsel N − 1 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='14) where the expectation is taken with respect to the random selection of Isel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' 23 The proof is provided Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' We note how the system variance var(z) enters the error estimate due to the randomness of the selection, similar to the Law of Large Number error for random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' In particular, the smaller the particles variance is, the closer the reduced particle system will be to the original distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' This justifies the choice of Nk+1 proposed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='2 where we are allowed to discard particles only if the system shows a contractive behavior, see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' By iteratively applying Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='2 and by using suitable stability estimates of dynamics (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='3), we are able to bound the error introduced by the random selection procedure as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Proof details are a given in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Let {xk i , yk i }N0 i=1 be constructed according to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='3) were particles are not discarded, and {ˆxk i , ˆyk i }Ik, |Ik| = Nk where Nk−Nk+1 particles are discarded after update rule (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Assume (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='10) is satisfied and consider the probability measures (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' If {xk i , yk i }N0 i=1, {ˆxk i , ˆyk i }i∈Ik ⊂ BM(0) at all step k for some M > 0, it holds E � W 2 2 � ρk Nk, ρk N0 �� ≤ C max h=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=',k var � ˜zh� N0 − Nk Nk − 1 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='15) where C = C(∆t, λ, σ, ν, β, α, k, LF, M) and ˜zh = {(ˆxh i , ˆyh i )}i∈Ih−1 describes the particle system just before the random selection procedure at step h ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' The expectation is taken with respect to the sampling of {θh}k h=1 and with respect to the selection procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' We can directly apply the above result to relate the expected ℓ2-errors of the two particle system, which we define as Err(k) = E � � 1 N0 � i∈I0 ∥xk i − x∗∥2 2 � � , Err(k) = E � � 1 Nk � i∈Ik ∥ˆxk i − x∗∥2 2 � � , that is, the discrete counterpart of the mean-field error E[∥ ¯Xi t − x∗∥2 2] studied in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' By definition of the Wasserstein-2 distance, we have Err(k) = E � W 2 2 (ρk N0, δx∗) � for any solution x∗ to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='1), and the same holds of Errsel(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' We then apply inequality W 2 2 (ρk Nk, δx∗) ≤ 2 � W 2 2 (ρk Nk, ρk N0) + W 2 2 (ρk N0, δx∗) � to obtain the following estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Under the assumptions of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='2, at all steps k, it holds Errsel(k) ≤ 2 � Err(k) + C max h=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=',k var(˜zh) N0 − Nk Nk − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='16) Before concluding the section, let us report some remarks concerning the theoretical results just presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' 24 Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='2 can be adapted to any other particle system with random selection, provided that the update rule is stable with respect to the 2-Wasserstein distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' In the proposed method, such stability was proved thanks to the approximation of the global best ¯y∞,k with ¯yα,k for α ≫ 1 (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='2)) and S(x, y) with Sβ(x, y) for β ≫ 1 in the personal best update (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Quantitative estimates on the variance decay can be used, if available, to improve the error bound in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='2, see also proof in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' The error introduced by a sub-sampling technique in a Monte Carlo integral approximation is expected to be of order 2 var(z) � 1 N − 1 − 1 Nsel − 1 � = 2 var(z) N − Nsel (N − 1)(Nsel − 1) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='17) see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Therefore, an additional factor of order 1/(Nsel − 1) seems to be missing in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' We remark, though, that Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='2 does not concern the Monte Carlo approximation of an integral quantity, but rather consider the 2-Wasserstein distance between discrete measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Numerical simulations suggest that estimates of order (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='17) do not hold on in this case, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' 5 Conclusions In this work, we studied a Consensus-Based Optimization algorithm with Memory Effects (CBO- ME) and random selection for single objective optimization problems of the form (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' While sharing common features with Particle Swarm Optimization (PSO) methods, CBO-ME differs on the way the particle system explore the search space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Its structure provides greater flexi- bility in balancing the exploration and exploitation processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' In particular, we implemented and analytically investigates a random selection strategy which allows to reduce the algorithm computational complexity, without affecting convergence properties and overall accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' This analysis is entirely general and, in perspective, applicable to other particle swarm-based opti- mization methods as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' The convergence analysis to the global minimum is carried out by relying on a mean-field approximation of the particle system and error estimates are given un- der mild assumptions on the objective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' We compared CBO-ME against CBO without memory effects and PSO against several benchmark problem and showed how the introduction of memory effects and random selection improves the algorithm performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Applications to image segmentation and machine learning problems are finally reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' A Proofs A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='1 Notation and auxiliary lemmas We will use the following notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' For any a ∈ R, |a| indicates the absolute value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' For a given vector b ∈ Rd, ∥b∥p indicates its p-norm, p ∈ [1, ∞];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' (b)ℓ its ℓ-th component;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' while diag(b) ∈ Rd×d 25 0 20 40 60 80 100 # particle selected !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Nsel " 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='5 2 N = 100;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' d = 3 squared Wasserstein dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' estimate (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='18) estimate (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='21) 0 20 40 60 80 100 # particle selected !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Nsel " 0 1 2 3 4 5 6 7 N = 100;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' d = 10 squared Wasserstein dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' estimate (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='18) estimate (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='21) Figure 10: Numerical validation of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='2 with different dimensions d = 3, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' N = 100 points are randomly, uniformly sampled over [0, 1]d to construct the empirical distribution µN and Nsel ∈ [2, N − 1] are discarded to obtain µNsel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' The experiment is repeated 500 times for all Nsel to obtain an approximation of E � W 2 2 (µN, µNsel) � (blue line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' In red, estimate provided by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='2 (RHS of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='14)), in yellow the one given equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Wasserstein distances are computed with the ot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='emd function provided by the Python Optimal Transport library [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' is the diagonal matrix with elements of b on the main diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Let a, b ∈ Rd, ⟨a, b⟩ denotes the scalar product in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' For a given closed convex set A ⊂ Rd, N(A, x), T (A, x) denote the normal and the tangential cone at x ∈ A respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' The ball or radius r centered at x ∈ Rd is indicated with Br(x) = {x ∈ Rd | ∥x∥2 ≤ r}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' All considered stochastic processes are assumed to take their realizations over the common probability space (Ω, ¯F, P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' P(Rd) is the set of Borel probability measures over Rd and Pq(Rd) = {µ ∈ P(Rd) | � ∥x∥q 2dµ < ∞} which we equip with the Wasserstein distance Wq, q ≥ 1, see [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' For a random variable X, X ∼ µ, µ ∈ P(Rd) indicates a sampling procedure such that P(X ∈ A) = µ(A) for any Borel set A ⊂ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' With Unif(A) ∈ P(Rd) we denote the uniform probability measure over a bounded Borel set A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Throughout the computations, C will denote an arbitrary positive constant, whose value may vary from line to line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Dependence on relevant parameters or variables, will be underlined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='1 ( [3, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Let F satisfy Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='1 (in particular the locally Lipschitz assumption A1) and ρ1, ρ2 ∈ P4(Rd) with � ∥x∥4 2dρ1 , � ∥x∥4 2dρ2 ≤ M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Then, the following stability estimate holds ∥¯yα(ρ1) − ¯yα(ρ2)∥2 ≤ C W2(ρ1, ρ2) for a constant C = C(α, LF, M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' 26 Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Under Assumptions A1 and A6, for any x1, x2, y1, y2 ∈ BM(0) and β > 0, it holds ∥(x1 − y1)Sβ(x1, y1) − (x2 − y2)Sβ(x2, y2)∥2 ≤ C (∥x1 − y1∥2 + ∥x2 − y2∥2) where C = C(β, LF, M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Thanks to the Lipschitz continuity of ψ, F and the choice of ψ (Assumptions A1 and A6), it holds |Sβ(x1, y1) − Sβ(x2, y2)| = |2ψ (β(F(y1) − F(x1)) − 2ψ(β(F(y2) − F(x2)) | ≤ 2β |F(y1) − F(x1) − F(y2) + F(x2)| ≤ 2βLF (∥x1 − x2∥2 + ∥y1 − y2∥2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Next, we have ∥(x1 − y1)Sβ(x1, y1) − (x2 − y2)Sβ(x2, y2)∥2 ≤ ∥(x1 − y1)Sβ(x1, y1) − (x2 − y2)Sβ(x1, y1)∥2 + (x2 − y2)Sβ(x1, y1) − (x2 − y2)Sβ(x2, y2)∥2 ≤ ∥(x1 − x2 + y2 − y1)Sβ(x1, y1)∥2 + ∥(x2 − y2) � Sβ(x1, y1) − Sβ(x2, y2) � ∥2 ≤ 2 (∥x1 − x2∥2 + ∥y1 − y2∥2) + 2M|Sβ(x1, y1) − Sβ(x2, y2)| ≤ C (∥x1 − x2∥2 + ∥y1 − y2∥2) with C = C(β, LF, M), where we used the first estimate to conclude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='2 Proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='1 Proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' The proof is based on the Leray–Schauder fixed point theorem [13, Chapter 11], and we follow closely the proof steps of [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' For any ξ ∈ C([0, T], Rd) there exists a unique process ( ˆXt, ˆYt) ∈ C([0, T], Rd) satisfying d ˆXt = λ(ξ(t) − ˆXt) dt + σ(ξ(t) − ˆXt) ⊗ d ˆBt d ˆYt = ν( ˆXt − ˆYt)Sβ( ˆXt, ˆYt) dt with Law( ˆX0) = Law( ˆY0) = ρ0 ∈ Rd, by the Lipschitz continuity of the coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' As a consequence, we have that f(t) := Law( ˆXt, ˆYt) satisfies d dt � φ df(t) = � � −λ⟨∇xφ, ξ(t) − x⟩ + � ℓ=1 ∂2φ ∂x2 ℓ (ξt) − y)2 ℓ − νSβ⟨∇yφ, y − x⟩ � df(t) for all φ ∈ C2 b (R2d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Therefore, let ¯ρ(t) = Law( ˆYt), we can set T ξ := ¯yα(¯ρ(·)) ∈ C([0, T], Rd) to define T : C([0, T], Rd) → C([0, T], Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' 27 Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' We prove now compactness of T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Thanks to ρ0 ∈ P4(Rd) and standard results for SDEs (see [1, Chapter 7]) we have boundedness of the forth moments E � ∥ ˆXt∥4 2 + ∥ ˆYt∥4 2 � ≤ c1 � 1 + E[∥ ˆX0∥4 2 + ∥ ˆY0∥4 2]ec2t� for some c1, c2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Therefore, we can apply Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='1 to obtain for any 0 < s < t < T, ∥¯yα(¯ρ(t)) − ¯yα(¯ρ(s))∥2 ≤ CW2 (¯ρ(t), ¯ρ(s)) ≤ ˜C|t − s|1/2 for some constants C, ˜C > 0, from which H¨older continuity of t �→ ¯yα(¯ρ(t) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Therefore, by T (C([0, T], Rd)) ⊂ C0, 1 2 ([0, T], Rd) �→ C([0, T], Rd) we get compactness of T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Consider ξ ∈ C([0, T], Rd) satisfying ξ = τT ξ, for τ ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Thanks to [3][Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='3] and boundedness of second moments, we obtain compactness of the set {ξ ∈ C([0, T], Rd) : ξ = τT ξ, τ ∈ [0, 1]} and by Leray–Schauder fixed point theorem there exists a fixed point for the mapping T and hence a solution to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Step 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Assume now there exist two solutions, ( ¯X1 t , ¯Y 1 t ) and ( ¯X2 t , ¯Y 2 t ) to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='6) with same Brownian process ¯Bt and initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Let ¯ρℓ = Law( ¯Y ℓ t ), ℓ = 1, 2, we have ∥ ¯X1 t − ¯X2 t ∥2 2 = � t 0 � ¯X1 s − ¯X2 s , ¯yα(¯ρ1(s)) − ¯yα(¯ρ2(s)) − ¯X1 s + ¯X2 s � dt + � t 0 � diag � ¯yα(¯ρ1(s)) − ¯X1 s � − diag � ¯yα(¯ρ2(s)) − ¯X2 s �� d ¯Bs .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='1) We note that all terms can be estimated by means of W 2 2 (¯ρ1(s), ¯ρ2(s)) and ∥ ¯X1 s − ¯X2 s ∥2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Similarly, ∥ ¯Y 1 t − ¯Y 2 t ∥2 2 can be bounded in terms ∥ ¯X1 s − ¯X2 s ∥2 2 thanks to the Lipschitz continuity of Sβ and Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Therefore, for some constant C > 0 ∥ ¯X1 t − ¯X2 t ∥2 2 + ∥ ¯Y 1 t − ¯Y 2 t ∥2 2 ≤ C � t 0 � ∥ ¯X1 s − ¯X2 s ∥2 2 + ∥ ¯Y 1 s − ¯Y 2 s ∥2 2 + W 2 2 (¯ρ1(s), ¯ρ2(s)) � ds from which, together with (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='1), follows for some ˜C > 0 E � ∥ ¯X1 t − ¯X2 t ∥2 2 + ∥ ¯Y 1 t − ¯Y 2 t ∥2 2 � ≤ E � ∥ ¯X1 0 − ¯X2 0∥2 2 + ∥ ¯Y 1 0 − ¯Y 2 0 ∥2 2 � e ˜C t by Gr¨onwall’s inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Since E � ∥ ¯X1 0 − ¯X2 0∥2 2 + ∥ ¯Y 1 0 − ¯Y 2 0 ∥2 2 � = 0, we proved uniqueness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='3 Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='1 Having proved there exists a solution ( ¯Xt, ¯Yt)t∈[0,T] to the mean-field process (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='6) we are here interested in studying the expected ℓ2-error given by E∥ ¯Xt − x∗∥2 2 where x∗ is the unique solution to the minimization problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='1), see Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' We do so by means of the following quantitative version of the Laplace principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' 28 Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='1 (quantitative Laplace principle [10, Proposition 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Let ρ ∈ P(Rd) be such that x∗ ∈ supp(ρ) and fix α > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' For any r > 0, define Fr = supx∈B∗r F(x) − F(x∗) with B∗ r := {x | ∥x − x∗∥∞ ≤ r} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Then, under Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='1, for any r ∈ (0, R0] and q > 0 such that q + Fr ≤ F∞ = ηRγ 0, it holds ∥yα(ρ) − x∗∥2 ≤ √ d(q + Fr)γ η + √ d exp(−αq) ρ(B∗r) � ∥x − x∗∥2 dρ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='2) We remark that RHS of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='2) can be made arbitrary small by taking large values of α and small values of q, r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' To apply Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='1 to all ¯ρ(t) = Law( ¯Yt), we need though to provide lower bounds on ¯ρ(t)(B∗ r) for any small radius r and times t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Let ¯ρ(t) = Law( ¯Yt), with ¯Yt evolving according to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='6) and limt→0 ¯ρ(t) = ρ0 with x∗ ∈ supp(ρ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Under Assumptions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='2 , it holds ¯ρ(t)(B∗ r) ≥ mr > 0, for all t ∈ [0, T] and for all r ≤ R0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Let δ = η min{R1, r}γ, we start by proving that the mass in the set Lδ = {x ∈ Rd | F(x) ≤ inf F + δ} is non-decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' We note that for this choice of δ, Lδ is convex due to Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Consider now (Ω, ¯F, P) to be the common probability space over which the considered processes take their realization and define Ωδ = {ω : ¯Y0(ω) ∈ Lδ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' By Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='2, Sβ( ¯Xt(ω), ¯Yt(ω)) = 0 whenever ¯Xt(ω) /∈ Lδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Therefore, it holds � ( ¯Xt(ω) − ¯Yt(ω))Sβ( ¯Xt(ω), ¯Yt(ω)) , n( ¯Yt(ω)) � � = 0 if ¯Xt(ω) /∈ Lδ ≤ 0 if ¯Xt(ω) ∈ Lδ for ¯Yt(ω) ∈ ∂Lδ for any n( ¯Yt(ω)) ∈ N(Lδ, x) from which follows that ¯Yt(ω) solves ¯Yt(ω) = ¯Y0(ω) + � t 0 ΠT (Lδ, ¯Ys(ω)) � ( ¯Xs(ω) − ¯Ys(ω))Sβ( ¯Xs(ω), ¯Ys(ω)) � ds for all ω ∈ Ωδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' As a consequence, if ¯Y0(ω) ∈ Lδ, ¯Yt(ω) ∈ Lδ for all t ≥ 0 and so ¯ρ(t)(B∗ r) = P( ¯Yt ∈ Lδ) ≥ P( ¯Y0 ∈ Lδ) =: mr for all t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' We conclude by noting that mr > 0 since x∗ ∈ supp(ρ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Next, we study the evolution of the error E∥ ¯Xt − x∗∥2 2 and, in particular, we try to bound it in terms of ∥¯yα(¯ρ(s)) − x∗∥2 and E∥ ¯Xt − x∗∥2 itself for s ∈ [0, t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' 29 Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' [10, Lemma 1] Let ( ¯Xt, ¯Yt) ∈ C([0, T], R2d) be the solution to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='6) with initial datum ¯X0 ∼ ρ0, ¯Y0 = ¯X0 for some time horizon T > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' For all t ∈ [0, T], it holds E∥ ¯Xt − x∗∥2 2 ≤ � t 0 � − (2λ − σ2)E∥ ¯Xs − x∗∥2 2 + √ 2(λ + σ2)E∥ ¯Xs − x∗∥2∥¯yα(¯ρ(s)) − x∗∥2 + σ2 2 ∥¯yα(¯ρ(s)) − x∗∥2 2 � ds (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='3) where ¯ρ(t) = Law( ¯Yt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' The above result, together with Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='3, leads to the convergence in mean-field law of the dynamics towards the solution to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' The proof can be carried out exactly as in [10, Theorem 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='4 Proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='2 and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='2 We start by collecting a preliminary result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Let {xk 1,i, yk 1,i}N1 i=1 and {xk 2,j, yk 2,j}N2 j=1 be two particle populations generated through update rules (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='3) with θk 1,i = θk 2,j = θk for all i, j and k ∈ Z+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' At any iteration step k and for any couple of indexes (i, j), it holds E � ∥xk+1 1,i − xk+1 2,j ∥2 2 + ∥yk+1 1,i − yk+1 2,j ∥2 2 � ≤ CE � ∥xk 1,i − xk 2,j∥2 2 + ∥yk 1,i − yk 2,j∥2 2 + ∥¯yα(¯ρk 1) − ¯yα(¯ρk 2)∥2 2 � where C = C(∆t, λ, σ, ν, β) is a positive constant and ¯ρk 1, ¯ρk 2 are the empiricial distributions associated with {yk 1,i}N1 i=1 and {yk 2,j}N2 j=1 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' For all k ∈ Z+ and i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' j E∥xk+1 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='i − xk+1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='j ∥2 2 ≤ E ���xk 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='i + λ∆t � ¯yα(¯ρk 1) − xk 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='i � + σ √ ∆t � ¯yα(¯ρk 1) − xk 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='i � ⊗ θk 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='i − � xk 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='j + λ∆t � ¯yα(¯ρk 2) − xk 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='j � + σ √ ∆t � ¯yα(¯ρk 2) − xk 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='j � ⊗ θk 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='j � ��� 2 2 ≤ 2E ��� � 1 − λ∆t − σ √ ∆t diag(θk) � (xk 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='i − xk 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='j) ��� 2 2 + 2E ��� � λ∆t + σ √ ∆t diag(θk) � � ¯yα(¯ρk 1) − ¯yα(¯ρk 2) ���� 2 2 ≤ 2(1 + σ2∆t)E∥xk 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='i − xk 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='j∥2 2 + 2(λ2∆t2 + σ2∆t)E∥¯yα(¯ρk 1) − ¯yα(¯ρk 2)∥2 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='4) where we also used that E[(θk)2 ℓ] = 1 for all ℓ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' , d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' We now bound ∥yk+1 1,i − yk+1 2,j ∥2 2 as E∥yk+1 1,i − yk+1 2,j ∥2 2 ≤ E ���yk 1,i + (ν∆t/2) � xk+1 i,1 − yk 1,i � Sβ(xk+1 1,i , yk 1,i) 30 − � yk 2,j + (ν∆t/2) � xk+1 2,j − yk 2,j � Sβ(xk+1 2,j , yk 2,j) � ��� 2 2 ≤ CE � ∥xk+1 i,1 − xk+1 j,2 ∥2 2 + ∥yk i,1 − yk j,2∥2 2 � (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='5) where we used Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='2 and C = C(∆t, β, ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' By combining (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='4) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='5) we get the desired estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Next, we show how the particle update rule (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='3) is stable with respect to the 2-Wasserstein distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='3 (Stability of update rule (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Let {xk 1,i, yk 1,i}N1 i=1, {xk 2,j, yk 2,j}N2 j=1 ⊂ BM(0), for some M > 0, be two particle populations generated through the update rules (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='3) with θk 1,i = θk 2,j = θk for all i, j and k ∈ Z+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Let µk 1, µk 2 ∈ P(R2d) the empirical probability measures defined as µk 1 := 1 N1 N1 � i=1 δ(xk 1,i,yk 1,i) , µk 2 := 1 N2 N2 � j=1 δ(xk 2,j,yk 2,j) , it holds E � W 2 2 (µk+1 1 , µk+1 2 ) � ≤ C1 E � W 2 2 (µk 1, µk 2) � , where C1 = C1(∆, λ, σ, ν, α, β, LF, M) is positive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Let Eθk[·] denote the expectation taken with respect to the sampling of θk only and w ∈ RN1×N2 be the optimal coupling between µk 1, µk 2, see (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='12) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Being w a sub- optimal coupling for µk+1 1 , µk+1 2 , it holds Eθk[W 2 2 (µk+1 1 , µk+1 2 )] ≤ Eθk � i,j wij � ∥xk+1 1,i − xk+1 2,j ∥2 2 + ∥yk+1 1,i − yk+1 2,j ∥2 2 � ≤ C � i,j wij � ∥xk 1,i − xk 2,j∥2 2 + ∥yk 1,i − yk 2,j∥2 2 � + ∥¯yα(¯ρk 1) − ¯yα(¯ρk 2)∥2 2 where we used the linearity of the expectation, estimates given by Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='4 and, to take the last term out of the sum, the fact that � ij wij = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' To estimate the distance between the two consensus points, we use Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='1 and note that the coupling w is sub-optimal for ¯ρk 1, ¯ρk 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' By Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='1, it follows ∥¯yα(¯ρk 1) − ¯yα(¯ρk 2)∥2 2 ≤ CW 2 2 (¯ρk 1, ¯ρk 2) ≤ C � i,j wij∥yk 1,i − yk 2,j∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Therefore, Eθk[W 2 2 (µk+1 1 , µk+1 2 )] ≤ C1 � i,j wij � ∥xk 1,i − xk 2,j∥2 2 + ∥yk 1,i − yk 2,j∥2 2 � = C1 W 2 2 (µk 1, µk 2) , thanks to the optimality of w, with C1 = C1(∆, λ, σ, ν, α, β, LF, M) being a positive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' One can conclude by taking the expectation of the above inequality with respect to the remaining sampling processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' 31 We now quantify the impact of the particle discarding step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' For notational simplicity, let us introduce zi = (xi, yi) ∈ R2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' As in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='13), the 2-Wasserstein distance is given by an optimal coupling between the full particle system {zi}i∈I and the reduced one {zj}j∈Isel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' We consider the following transportation of mass from µN to µNsel: if particle i has not been discarded, all its mass remains in xi, otherwise the mass is uniformly distributed among the selected particles to generate an admissible coupling w ∈ RN×Nsel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' This means that w is given by wij = � � � � � 1/N if j = i, i ∈ Isel 1/(N · Nsel) if i ∈ I \\ Isel, j ∈ Isel 0 else .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='6) We note that such coupling w satisfies the coupling conditions � j∈Isel wij = 1 N � i∈I wij = 1 Nsel , ∀ i ∈ I, j ∈ Isel (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='7) and that this choice will be in general sub-optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Therefore, it holds W 2 2 (µN, µNsel) ≤ � i∈I, j∈Isel wij∥zi − zj∥2 2 = 1 N � i∈Isel ∥zi − zi∥2 2 + 1 N · Nsel � i∈I\\Isel, j∈Isel ∥zi − zj∥2 2 = 1 N · Nsel � i,j∈I ∥zi − zj∥2 2 1i∈I\\Isel 1j∈Isel where 1i∈I = 1 if i ∈ I and zero otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Now, the probability of having i ∈ I \\ Isel is given by (N − Nsel)/N, while the probability of having j ∈ Isel (condition i ∈ I \\ Isel) is given by Nsel/(N − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Hence, we have E � 1i∈I\\Isel 1j∈Isel � = P [i ∈ I \\ Isel, j ∈ Isel] = (N − Nsel)Nsel N(N − 1) from which follows E � W 2 2 (µN, µNsel) � ≤ 1 N · Nsel � i,j∈I ∥zi − zj∥2 2 E � 1i∈I\\Isel 1j∈Isel � = 1 N · Nsel (N − Nsel)Nsel N(N − 1) � i,j∈I ∥zi − zj∥2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' The desired estimates can finally be obtained by noting that the variance can be computed as var(z) = 1/(2N2) � i,j∈I ∥zi − zj∥2 2, see definition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' 32 Finally, we are ready to provide a proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Let {(xk i , yk i )}i∈Ik, |Ik| = Nk be the sequence of particles generated by iteration (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='3) where additionally Nk+1 − Nk particles are discarded after each step k ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' We denote with µk Nk ∈ P(R2d) the empirical measure associated with such particle system given by µk Nk = 1 Nk � i∈Ik δ(xk i ,yk i ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' We also introduce the measures µk N0, k ≥ 0 corresponding to a particle system generated with the same initial conditions µ0 N0 but where no particle reduction occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Consistently, we define µh Nk, h > k to represent the particle system generated starting from µk Nk, after h − k iterations, with no random selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' The relation between such measures is summarized in the following diagram µ0 N0 → µ1 N0 → µ2 N0 → .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' → µk N0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' ��� µ1 N1 → µ2 N1 → .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' → µk N1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' ��� µ2 N2 → .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' → µk N2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' µk Nk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='8) where → indicates an iteration step (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='3) while ��� a particle reduction procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Therefore, we are interested in studying the distance between the main diagonal of such diagram µk Nk, cor- responding to the system with particle reduction, and the first line µk N0 where particle reduction is never performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' We note that the 2-Wasserstein distance between subsequent rows can be estimated thanks to Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='3 and Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Let ˜zh+1 denote the set of particles associated with the probability measure µh+1 Nh , that is, the particle systems before the selection procedure (up- per diagonal elements in scheme (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='8)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' By first applying Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='3 and, subsequently, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='2 to ˜zh+1, we obtain that for some constant C > 0 E � W 2 2 (µk Nk, µk N0) � ≤ C k−1 � h=0 E � W 2 2 � µk Nh, µk Nℓ+1 �� ≤ C k−1 � h=0 Ck−h+1 1 E � W 2 2 � µh+1 Nh , µh+1 Nh+1 �� ≤ 2 C k−1 � h=0 Ck−h+1 1 var � ˜zh+1� Nh − Nh+1 Nh − 1 33 ≤ C2 max h=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=',k var � ˜zh� 1 Nk − 1 k−1 � h=0 Nh − Nh+1 = C2 max h=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=',k var � ˜zh� N0 − Nk Nk − 1 with C2 = C2(∆t, λ, σ, ν, β, α, k, M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Finally, the desired estimate follows after noting that W 2 2 (ρk Nk, ρk N0) ≤ W 2 2 (µk Nk, µk N0) since ∥xk i − xk j ∥2 2 ≤ ∥(xk i , yk i ) − (xk j , yk j )∥2 2 for all couples of particles (i, j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' Acknowledgments This work has been written within the activities of GNCS group of INdAM (National Institute of High Mathematics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' acknowledges the partial support of MIUR-PRIN Project 2017, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' 2017KKJP4X “Innovative numerical methods for evolutionary partial differential equations and applications”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' The work of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through 320021702/GRK2326 “Energy, Entropy, and Dissipative Dynamics (EDDy)” and SFB 1481 “Sparsity and Singular Structures”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} +page_content=' acknowledges the support of the ESF PhD Grant “Mathematical and statistical methods for machine learning in biomedical and socio-sanitary applications”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQfFzW1/content/2301.13242v1.pdf'} 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