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+On the Convergence of No-Regret Learning Dynamics in
+Time-Varying Games
+Ioannis Anagnostides1, Ioannis Panageas2, Gabriele Farina3, and Tuomas Sandholm4
+1,3,4Carnegie Mellon University
+2University of California Irvine
+4Strategy Robot, Inc.
+4Optimized Markets, Inc.
+4Strategic Machine, Inc.
+{ianagnos,gfarina,sandholm}@cs.cmu.edu, and ipanagea@ics.uci.edu
+January 27, 2023
+Abstract
+Most of the literature on learning in games has focused on the restrictive setting where
+the underlying repeated game does not change over time.
+Much less is known about the
+convergence of no-regret learning algorithms in dynamic multiagent settings. In this paper,
+we characterize the convergence of optimistic gradient descent (OGD) in time-varying games by
+drawing a strong connection with dynamic regret. Our framework yields sharp convergence
+bounds for the equilibrium gap of OGD in zero-sum games parameterized on the minimal first-order
+variation of the Nash equilibria and the second-order variation of the payoff matrices, subsuming
+known results for static games. Furthermore, we establish improved second-order variation
+bounds under strong convexity-concavity, as long as each game is repeated multiple times. Our
+results also apply to time-varying general-sum multi-player games via a bilinear formulation of
+correlated equilibria, which has novel implications for meta-learning and for obtaining refined
+variation-dependent regret bounds, addressing questions left open in prior papers. Finally, we
+leverage our framework to also provide new insights on dynamic regret guarantees in static games.
+arXiv:2301.11241v1  [cs.LG]  26 Jan 2023
+
+1
+Introduction
+Most of the classical results in the literate on learning in games—exemplified by, among others,
+the work of Hart and Mas-Colell [HM00], Foster and Vohra [FV97], and Freund and Schapire
+[FS99]—rest on the assumption that the underlying repeated game remains invariant throughout the
+learning process. Yet, in many learning environments that is unrealistic [Duv+22; Zha+22; Car+19;
+MS21]. One such class is settings where the underlying game is actually changing, such as routing
+problems on the internet [MO11], online advertising auctions [LST16], and dynamic mechanism
+design [Pap+22; DMZ21]. Another such class consists of settings in which many similar games need
+to be solved [Har+22]. For example, one may want to solve variations of a game for the purpose
+of sensitivity analysis with respect to the modeling assumptions used to construct the game model.
+Another example is solving multiple versions of a game any one of which might be faced in the future.
+Despite the considerable interest in such dynamic multiagent environments, much less is known
+about the convergence of no-regret learning algorithms in time-varying games. No-regret dynamics
+are natural learning algorithms that have desirable convergence properties in static settings. Also,
+the state-of-the-art algorithms for finding minimax equilibria in two-player zero-sum games are based
+on advanced forms of no-regret dynamics [FKS21; BS19a]. Indeed, all the superhuman milestones
+in poker have used them in the equilibrium-finding module of their architectures [Bow+15; BS17;
+BS19b].
+In this paper, we seek to fill this knowledge gap by understanding properties of no-regret dy-
+namics in time-varying games. In particular, we primarily investigate the convergence of optimistic
+gradient descent (OGD) [Chi+12; RS13] in time-varying games. Unlike traditional no-regret learning
+algorithms, such as (online) gradient descent, OGD has been recently shown to exhibit last-iterate
+convergence in static (two-player) zero-sum games [Das+18; GPD20; COZ22; GTG22]. For the more
+challenging scenario where the underlying game can vary in every round, a fundamental question
+arises: Under what conditions on the sequence of games does OGD (with high probability) approximate
+the sequence of Nash equilibria?
+1.1
+Our Results
+In this paper, we build a new framework that enables us to characterize the convergence of OGD
+in time-varying games. Specifically, our first contribution is to identify natural variation measures
+on the sequence of games whose sublinear growth guarantees that almost all iterates of OGD are
+(approximate) Nash equilibria in time-varying (two-player) zero-sum games (Corollary 3.6). More
+precisely, in Theorem 3.5 we derive a sharp non-asymptotic characterization of the equilibrium gap of
+OGD as a function of the variation measures we identify: the minimal first-order variation of the Nash
+equilibria and the second-order variation of the payoff matrices. It is a compelling property, in light of
+the multiplicity of Nash equilibria, that the variation of the Nash equilibria is measured in terms of the
+most favorable—i.e., one that minimizes the variation—such sequence. Additionally, we show that
+our convergence bounds can be further improved by considering a variation measure that depends on
+the deviation of approximate Nash equilibria of the games, a measure that could be arbitrarily smaller
+than the one based on (even the least varying) sequence of exact Nash equilibria (Proposition 3.3).
+From a technical standpoint, our analysis revolves around a new connection we draw between the
+convergence of OGD in time-varying games and dynamic regret. In particular, the first key observation
+is that dynamic regret is always nonnegative under any sequence of Nash equilibria (Property 3.2).
+By combining that property with a dynamic RVU bound—in the sense of Syrgkanis et al. [Syr+15]—
+1
+
+that we derive (Lemmas 3.1 and A.1), we obtain in Theorem 3.4 a variation-dependent bound for
+the second-order path length of OGD in time-varying games. In turn, this leads to our main result,
+Theorem 3.5, discussed above. As such, we extend the regret-based framework of Anagnostides et al.
+[Ana+22b] from static to time-varying games. In the special case of static games, our result reduces
+to a tight T −1/2 rate. It is worth stressing that Property 3.2 is in fact more general, being intricately
+tied to the admission of a minimax theorem (Property A.3), and applies even under a sequence of
+approximate Nash equilibria—with slackness that gently degrades with the approximation thereof.
+Moreover, for strongly convex-concave time-varying games, we obtain a refined second-order
+variation bound on the sequence of Nash equilibria, as long as each game is repeated multiple
+times (Theorem 3.8); this is inspired by an improved second-order bound for dynamic regret under
+analogous conditions due to Zhang et al. [Zha+17]. As a byproduct of our techniques, we point out
+that any no-regret learners are approaching a Nash equilibrium under strong convexity-concavity
+(Proposition 3.9). Those results apply even in non-strongly convex-concave settings by suitably
+trading-off the magnitude of a regularizer that makes the game strongly convex-concave. This offers
+significant gains in the meta-learning setting as well, wherein each game is repeated multiple times.
+Next, we extend our results to time-varying general-sum multi-player games via a bilinear formu-
+lation of correlated equilibria. As such, we recover similar convergence bounds parameterized on the
+variation of the correlated equilibria (Theorem 3.12). To illustrate the power of our framework, we
+immediately recover natural and algorithm-independent similarity measures for the meta-learning
+setting (Proposition A.13) even in general games (Corollary A.22), thereby addressing an open
+question of Harris et al. [Har+22]. Our techniques also imply new per-player regret bounds in
+zero-sum and general-sum games (Corollaries 3.7 and A.23), the latter addressing a question left
+open by Zhang et al. [Zha+22]. We further parameterize the convergence of (vanilla) gradient descent
+in time-varying potential games in terms of the deviation of the potential functions (Theorem 3.10).
+Finally, building on our techniques in time-varying games, we investigate the best dynamic-regret
+guarantees possible in static games. Although this is a basic question, it has apparently eluded
+prior research. We first show that instances of optimistic mirror descent guarantee O(
+√
+T) dynamic
+per-player regret (Proposition 3.13), matching the known rate of (online) gradient descent but for
+the significantly weaker notion of external regret. We further point out that O(log T) dynamic
+regret is attainable, but in a stronger two-point feedback model. In stark contrast, even obtaining
+sublinear dynamic regret for each player is precluded in general-sum games (Proposition 3.15). This
+motivates studying a relaxation of dynamic regret that constrains the number of switches in the
+comparator, for which we derive accelerates rates in general games (Theorem 3.16) by leveraging the
+techniques of Syrgkanis et al. [Syr+15] in conjunction with our dynamic RVU bound (Lemma 3.1).
+1.2
+Further Related Work
+Even in static (two-player) zero-sum games, the pointwise convergence of no-regret learning algo-
+rithms is a tenuous affair. Indeed, traditional learning dynamics within the no-regret framework, such
+as (online) mirror descent, may even diverge away from the equilibrium; e.g., see [SAF02; MPP18;
+Vla+20; GVM21]. Notwithstanding, the empirical frequency of no-regret learners is well-known
+to approach the set of Nash equilibria in zero-sum games [FS99], and the set of coarse correlated
+equilibria in general-sum games [HM00]—a standard relaxation of the Nash equilibrium [MV78;
+Aum74]. Unfortunately, those classical results are of little use beyond static games, thereby offering
+a crucial impetus for investigating iterate-convergence in games with a time-varying component—a
+ubiquitous theme in many practical scenarios of interest [DMZ21; Pap+22; Ven21; Gar17; Van10;
+2
+
+RG22; PKB22; YH15; RJW21].
+Indeed, there has been a considerable effort endeavoring to extend the scope of traditional
+game-theoretic results to the time-varying setting, approached from a variety of different stand-
+points [LST16; Zha+22; Car+19; MO11; MS21; Duv+22]. In particular, our techniques in Section 3.1
+share similarities with the ones used by Zhang et al. [Zha+22], but our primary focus is very different:
+Zhang et al. [Zha+22] were mainly interested in obtaining variation-dependent regret bounds, while
+our results revolve around iterate-convergence to Nash equilibria. We stress again that minimizing
+regret and approaching Nash equilibria are two inherently distinct problems, although connections
+have emerged [Ana+22b], and are further cultivated in this paper.
+Another closely related direction is on meta-learning in games [Har+22], wherein each game
+can be repeated for multiple iterations. Such considerations are motivated in part by a number of
+use-cases in which many “similar” games—or multiple game variations—ought to be solved [BS16],
+such as Poker with different stack-sizes. While the meta-learning problem is a special case of our
+general setting, our results are strong enough to have new implications for meta-learning in games,
+even though the algorithms considered herein are not tailored to operate in that setting.
+Finally, although our focus is on the convergence of OGD in time-varying games, some of our
+results—namely, the ones formalized in Appendix A.1.7—can be viewed as part of an ongoing effort
+to characterize the class of variational inequalities (VIs) that are amenable to efficient algorithms;
+see [DDJ21; CZ22; Azi+20; BMW21; CP04; DL15; MV21; MRS20; Nou+19; Son+20; YKH20;
+Das22], and references therein. We also highlight that the techniques used to establish last-iterate
+convergence even in monotone (time-invariant) settings are particularly involved [GPD20; COZ22;
+GTG22]; the simplicity of our framework, therefore, in the more challenging time-varying regime is
+a compelling aspect of this paper.
+2
+Preliminaries
+Notation
+We let N := {1, 2, . . . , } be the set of natural numbers. For a number p ∈ N, we let
+[[p]] := {1, . . . , p}. For a vector w ∈ Rd, we use ∥w∥2 to represent its Euclidean norm; we also
+overload that notation so that ∥ · ∥2 denotes the spectral norm when the argument is a matrix.
+For a two-player zero-sum game, we denote by X ⊆ Rdx and Y ⊆ Rdy the strategy sets of the two
+players—namely, Player x and Player y, respectively—where dx, dy ∈ N represent the corresponding
+dimensions. It is assumed that X and Y are nonempty convex and compact sets. For example, in
+the special case where X := ∆dx and Y := ∆dy—each set corresponds to a probability simplex—the
+game is said to be in normal form. Further, we denote by DX the ℓ2-diameter of X, and by ∥X∥2
+the maximum ℓ2-norm attained by a point in X. We will always assume that the strategy sets
+remain invariant, while the payoff matrix can change in each round. For notational convenience, we
+will denote by z := (x, y) the concatenation of x and y, and by Z := X × Y the Cartesian product
+of X and Y. In general n−player games, we instead use subscripts indexed by i ∈ [[n]] to specify
+quantities related to a player. Superscripts are typically reserved to identify the time index. Finally,
+to simplify the exposition, we use the O(·) notation to suppress time-independent parameters of
+the problem; precise statements are given in Appendix A.
+Dynamic regret
+We operate in the usual online learning setting under full-feedback. Namely, at
+every time t ∈ N the learner decides on a strategy x(t) ∈ X, and then observes a utility x �→ ⟨x, u(t)
+x ⟩,
+for u(t)
+x ∈ Rdx. Following Daskalakis, Deckelbaum, and Kim [DDK11], we will insist on allowing only
+3
+
+O(1) previous utilities to be stored; this will preclude trivial exploration protocols when learning in
+games.
+A strong performance benchmark in this online setting is dynamic regret, defined for a time
+horizon T ∈ N as follows:
+DReg(T)
+x (s(T)
+x ) :=
+T
+�
+t=1
+⟨x(t,⋆) − x(t), u(t)
+x ⟩,
+(1)
+where s(T)
+x
+:= (x(1,⋆), . . . , x(T,⋆)) ∈ X T is the sequence of comparators; setting x(1,⋆) = x(2,⋆) = · · · =
+x(T,⋆) in (1) we recover the standard notion of (external) regret (denoted simply by Reg(T)
+x ), which is
+commonly used to establish convergence of the time-average strategies in static two-player zero-sum
+games [FS99]. On the other hand, the more general notion of dynamic regret, introduced in (1),
+has been extensively used in more dynamic environments; e.g., [Zha+20; Zha+17; Jad+15; Ces+12;
+HS09]. We also let DReg(T)
+x
+:= maxs(T )
+x
+∈X T DReg(T)
+x (s(T)
+x ). While ensuring o(T) dynamic regret is
+clearly hopeless in a truly adversarial environment, Section 3.4 reveals that non-trivial guarantees
+are possible when learning in zero-sum games.
+Optimistic gradient descent
+Optimistic gradient descent (OGD) [Chi+12; RS13] is a no-regret
+algorithm defined with the following update rule:
+x(t) := ΠX
+�
+ˆx(t) + ηm(t)
+x
+�
+,
+ˆx(t+1) := ΠX
+�
+ˆx(t) + ηu(t)
+x
+�
+.
+(OGD)
+Here, η > 0 is the learning rate; ˆx(1) := arg minˆx∈X ∥ˆx∥2
+2 represents the initialization of OGD;
+m(t)
+x
+∈ Rdx is the prediction vector at time t, and it is set as m(t)
+x
+:= u(t−1)
+x
+when t ≥ 2, and
+m(1)
+x
+:= 0dx; and finally, ΠX (·) represents the Euclidean projection to the set X, which is well-
+defined, and can be further computed efficiently for structured sets, such as the probability simplex.
+For our purposes, we will posit access to a projection oracle for the set X, in which case the update
+rule (OGD) is efficiently implementable.
+In a multi-player n-player game, each Player i ∈ [[n]] is associated with a utility function
+ui :×
+n
+i=1 Xi → R. We recall the following fundamental definition [Nas50].
+Definition 2.1 (Approximate Nash equilibrium). A joint strategy profile (x⋆
+1, . . . , x⋆
+n) ∈×
+n
+i=1 Xi is
+an ϵ-approximate Nash equilibrium (NE), for an ϵ ≥ 0, if for any Player i ∈ [[n]] and any possible
+deviation x′
+i ∈ Xi,
+ui(x⋆
+1, . . . , x⋆
+i , . . . , x⋆
+n) ≥ ui(x⋆
+1, . . . , x′
+i, . . . , x⋆
+n) − ϵ.
+3
+Convergence in Time-Varying Games
+In this section, we formalize our results regarding convergence in time-varying games. We organize
+this section as follows: First, in Section 3.1, we build the foundations of our framework by studying
+the convergence of OGD in time-varying bilinear saddle-point problems, culminating in the non-
+asymptotic characterization of Theorem 3.5; Section 3.2 formalizes our improvements under strong
+convexity-concavity; we then extend our results (in Section 3.3) to time-varying multi-player general-
+sum and potential games; and finally, Section 3.4 concerns dynamic regret guarantees in static games.
+4
+
+3.1
+Bilinear Saddle-Point Problems
+We first study an online learning setting wherein two players interact in a sequence of time-
+varying bilinear saddle-point problems. More precisely, we assume that in every repetition t ∈ [[T]]
+the players select a pair of strategies (x(t), y(t)) ∈ X × Y. Then, Player x receives the utility
+u(t)
+x
+:= −A(t)y(t) ∈ Rdx, where A(t) ∈ Rdx×dy represents the payoff matrix at the t-th repetition;
+similarly, Player y receives the utility u(t)
+y
+:= (A(t))⊤x(t) ∈ Rdy. The proofs of this subsection are
+included in Appendix A.1.
+Dynamic RVU bound
+The first key ingredient that we need is the property of regret bounded by
+variation in utilities (RVU), in the sense of Syrgkanis et al. [Syr+15], but with respect to dynamic
+regret; such a bound is established below.
+Lemma 3.1 (RVU bound for dynamic regret). Consider any sequence of utilities (u(1)
+x , . . . , u(T)
+x )
+up to time T ∈ N. The dynamic regret (1) of OGD with respect to any sequence of comparators
+(x(1,⋆), . . . , x(T,⋆)) ∈ X T can be bounded by
+D2
+X
+2η + DX
+η
+T−1
+�
+t=1
+∥x(t+1,⋆) − x(t,⋆)∥2+η
+T
+�
+t=1
+∥u(t)
+x − m(t)
+x ∥2
+2
+− 1
+2η
+T
+�
+t=1
+�
+∥x(t) − ˆx(t)∥2
+2 + ∥x(t) − ˆx(t+1)∥2
+2
+�
+.
+(2)
+In the special case of external regret—x(1,⋆) = x(2,⋆) = · · · = x(T,⋆)—(2) recovers the bound for
+OGD of Syrgkanis et al. [Syr+15]. The key takeaway from Lemma 3.1 is that the overhead of dynamic
+regret in (2) grows with the first-order variation of the sequence of comparators. In Lemma A.1
+we also articulate an extension of Lemma 3.1 for the more general optimistic mirror descent (OMD)
+algorithm under a certain class of Bregman divergences.
+Having established Lemma 3.1, we next point out a crucial property: by selecting a sequence of
+Nash equilibria (recall Definition 2.1) as the comparators, the sum of the players’ dynamic regrets
+is always nonnegative:
+Property 3.2. Suppose that Z ∋ z(t,⋆) = (x(t,⋆), y(t,⋆)) is an ϵ(t)-approximate Nash equilibrium of
+the t-th game. Then, for s(T)
+x
+= (x(t,⋆))1≤t≤T and s(T)
+y
+= (y(t,⋆))1≤t≤T ,
+DReg(T)
+x (s(T)
+x ) + DReg(T)
+y
+(s(T)
+y
+) ≥ −2
+T
+�
+t=1
+ϵ(t).
+In particular, if ϵ(t) = 0 for all t ∈ [[T]], we have
+DReg(T)
+x (s(T)
+x ) + DReg(T)
+y
+(s(T)
+y
+) ≥ 0.
+(3)
+In fact, as we show in Property A.3, Property 3.2 applies even in certain (time-varying) nonconvex-
+nonconcave min-max optimization problems, and it is a consequence of the minimax theorem;
+Property 3.2 also holds for time-varying variational inequalities (VIs) that satisfy the so-called MVI
+property (see Remark A.4). For comparison, it is evident that under a static sequence of two-player
+zero-sum games, it holds that Reg(T)
+x
++ Reg(T)
+y
+≥ 0.
+5
+
+Next, let us introduce some natural measures of the games’ variation. First, the first-order
+variation of the Nash equilibria is defined for T ≥ 2 as
+V(T)
+NE :=
+inf
+z(t,⋆)∈Z(t,⋆),∀t∈[[T]]
+T−1
+�
+t=1
+∥z(t+1,⋆) − z(t,⋆)∥2,
+(4)
+where Z(t,⋆) is the (nonempty) set of Nash equilibria of the t-th game. We recall that there can be a
+multiplicity of Nash equilibria [van91]; as such, a compelling feature of the variation measure (4) is
+that it depends on the most favorable sequence of Nash equilibria—one that minimizes the first-order
+variation.
+It is also important to point out the well-known fact that Nash equilibria can change abruptly
+even under a “small” perturbation in the payoff matrix (see Example A.5), which is a caveat of the
+variation (4). To address this, and in accordance with Property 3.2, we consider a more favorable
+variation measure, defined as
+V(T)
+ϵ−NE := inf
+�T−1
+�
+t=1
+∥z(t+1,⋆) − z(t,⋆)∥2 + C
+T
+�
+t=1
+ϵ(t)
+�
+,
+for a sufficiently large parameter C > 0; the infimum above is subject to ϵ(t) ∈ R≥0 and z(t,⋆) ∈ Z(t,⋆)
+ϵ(t)
+for all t ∈ [[T]], where we denote by Z(t,⋆)
+ϵ(t)
+the set of ϵ(t)-approximate NE. It is evident that
+V(T)
+ϵ−NE ≤ V(T)
+NE since one can take ϵ(1) = · · · = ϵ(T) = 0; in fact, V(T)
+ϵ−NE can be arbitrarily smaller:
+Proposition 3.3. For any T ≥ 4, there is a sequence of T games such that V(T)
+NE ≥ T
+2 while
+V(T)
+ϵ−NE ≤ δ, for any δ > 0.
+Moreover, we also introduce a quantity that captures the variation of the payoff matrices:
+V(T)
+A
+:=
+T−1
+�
+t=1
+∥A(t+1) − A(t)∥2
+2,
+(5)
+where we recall that here ∥ · ∥2 denotes the spectral norm. Unlike (4), the variation measure (5)
+depends on the second-order variation (of the payoff matrices), which could translate to a lower-order
+impact compared to (4) (see, e.g., Corollary A.8). We stress that while our convergence bounds
+will be parameterized based on (4) and (5), the underlying algorithm—namely OGD—will remain
+oblivious to those variation measures.
+We are ready now to establish a refined bound on the second-order path-length of OGD in
+time-varying zero-sum games.
+Theorem 3.4 (Detailed version in Theorem A.6). Suppose that both players employ OGD with learning
+rate η ≤
+1
+4L in a time-varying bilinear saddle-point problem, where L := maxt∈[[T]] ∥A(t)∥2. Then, for
+any T ∈ N, the second-order path length �T
+t=1
+�
+∥z(t) − ˆz(t)∥2
+2 + ∥z(t) − ˆz(t+1)∥2
+2
+�
+can be bounded by
+O
+�
+1 + V(T)
+ϵ−NE + V(T)
+A
+�
+.
+(6)
+6
+
+It is worth noting that when the deviation of the payoff matrices is controlled by the deviation of
+the players’ strategies, in the sense that �T−1
+t=1 ∥A(t+1)−A(t)∥2
+2 ≤ W 2 �T−1
+t=1 ∥z(t+1)−z(t)∥2
+2 for some
+parameter W ∈ R>0, the variation measure V(T)
+A
+in (6)—and in the subsequent convergence bounds—
+can be entirely eliminated; see Corollary A.8. The same, in fact, applies under an improved prediction
+mechanism (Remark A.12), but that prediction is not implementable in our online learning setting.
+Armed with Theorem 3.4, we are ready to establish Theorem 3.5. The key observation is that the
+Nash equilibrium gap at the t-th game can be bounded in terms of the quantity ∥z(t)− ˆz(t)∥2+∥z(t)−
+ˆz(t+1)∥2, which in turn allows us to use (6) to bound the cumulative (squared) Nash equilibrium
+gaps across the sequence of games; the aforeclaimed property was established in [Ana+22b] for
+static games (Claim A.9), but readily extends to our setting as well, and in fact applies to any
+member of OMD under a smooth regularizer. Below, we use the notation EqGap(t)(z(t)) ∈ R≥0 to
+represent the Nash equilibrium gap of the joint strategy profile z(t) ∈ Z at the t-th game.
+Theorem 3.5 (Main result; detailed version in Theorem A.10). Suppose that both players em-
+ploy OGD with learning rate η =
+1
+4L in a time-varying bilinear saddle-point problem, where L :=
+maxt∈[[T]] ∥A(t)∥2. Then,
+T
+�
+t=1
+�
+EqGap(t)(z(t))
+�2
+= O
+�
+1 + V(T)
+ϵ−NE + V(T)
+A
+�
+,
+(7)
+where (z(t))1≤t≤T is the sequence of joint strategy profiles produced by OGD.
+We next state some immediate consequences of this result. (Item 2 below follows from (7) by
+Jensen’s inequality.)
+Corollary 3.6. In the setting of Theorem 3.5,
+1. If at least a δ-fraction of the iterates of OGD have at least ϵ > 0 Nash equilibrium gap, then
+ϵ2δ ≤ O
+�
+1
+T
+�
+V(T)
+ϵ−NE + V(T)
+A
++ 1
+��
+;
+2. The average Nash equilibrium gap of OGD is bounded as O
+��
+1
+T
+�
+V(T)
+ϵ−NE + V(T)
+A
++ 1
+��
+.
+In particular, in terms of asymptotic implications, if limT→+∞
+V(T )
+ϵ−NE
+T
+, limT→+∞
+V(T )
+A
+T
+= 0, then
+(i) for any ϵ > 0 the fraction of iterates of OGD with at least an ϵ Nash equilibrium gap converges to
+0; and (ii) the average Nash equilibrium gap of the iterates of OGD converges to 0.
+In the special case where V(T)
+ϵ−NE, V(T)
+A
+= O(1), Theorem 3.5 recovers the T −1/2 rate of OGD in
+static bilinear saddle-point problems. It is also worth pointing out that Theorem 3.5 readily extends
+to more general time-varying variational inequality problems as well (Remark A.4).
+We also state below another interesting consequence of Theorem 3.4, which bounds each player’s
+individual regret parameterized based on the variation measures.
+Corollary 3.7 (Detailed version in Corollary A.11). In the setup of Theorem 3.4, it holds that
+Reg(T)
+x , Reg(T)
+y
+= O
+�
+1
+η + η(V(T)
+NE + V(T)
+A )
+�
+.
+7
+
+The O(·) notation here is considered in the regime η ≪ 1. Hence, selecting optimally the learning
+rate gives an O(
+�
+V(T)
+NE + V(T)
+A ) bound on the individual regret of each player; while that optimal
+value depends on the variation measures, which are not known to the learners, there are techniques
+that would allow bypassing this [Zha+22]. Corollary 3.7 can also be readily parameterized in
+terms of the improved variation measure V(T)
+ϵ−NE. Finally, in Appendix A.1.7 we highlight certain
+implications of our framework on solving (static) general VIs.
+Meta-Learning
+Our results also have immediate applications in the meta-learning setting [Har+22].
+More precisely, meta-learning in games is a special case of time-varying games which consists of a
+sequence of H ∈ N separate games, each of which is repeated for m ∈ N consecutive rounds, so that
+T := m × H. The central goal in meta-learning is to obtain convergence bounds parameterized by
+the similarity of the games; identifying suitable similarity metrics is a central question in that line
+of work.
+In this context, we highlight that Theorem 3.5 readily provides a meta-learning guarantee
+parameterized by the following notion of similarity between the Nash equilibria:
+inf
+z(h,⋆)∈Z(h,⋆),∀h∈[[H]]
+H−1
+�
+h=1
+∥z(h+1,⋆) − z(h,⋆)∥2,
+(8)
+where Z(h,⋆) is the set of Nash equilibria of the h-th game in the meta-learning sequence,1 as well
+as the similarity of the payoff matrices—corresponding to the term V(T)
+A
+in (7). In fact, under a
+suitable prediction—the one used by Harris et al. [Har+22]—the dependence on V(T)
+A
+can be entirely
+removed; see Proposition A.13 for our formal result. A compelling aspect of our meta-learning
+guarantee is that the considered algorithm is oblivious to the boundaries of the meta-learning. We
+further provide some novel results on meta-learning in general-sum games in Section 3.3.
+3.2
+Strongly Convex-Concave Games
+In this subsection, we show that under additional structure we can significantly improve the variation
+measures established in Theorem 3.4. More precisely, we first assume that each objective function
+f(x, y) is µ-strongly convex with respect to x and µ-strongly concave with respect to y. Our second
+assumption is that each game is played for multiple rounds m ∈ N, instead of only a single round;
+this is akin to the meta-learning setting. The key insight is that, as long as m is large enough,
+m = Ω(1/µ), those two assumptions suffice to obtain a second-order variation bound in terms of
+the sequence of Nash equilibria,
+S(H)
+NE :=
+H−1
+�
+h=1
+∥z(h+1,⋆) − z(h,⋆)∥2
+2,
+(9)
+where z(h,⋆) is a Nash equilibrium of the h-th game. This significantly refines the result of Theo-
+rem 3.4, and is inspired by the improved dynamic regret bounds obtained by Zhang et al. [Zha+17].
+Below we sketch the key ideas of the improvement; the proofs are included in Appendix A.2.
+In this setting, it is assumed that Player x obtains the utility u(t)
+x := −∇xf(t)(x(t), y(t)) at every
+time t ∈ [[T]], while its regret will be denoted by Reg(T)
+L,y; similar notation applies for Player y. The
+1In accordance to Theorem 3.5, (8) can be refined using a sequence of approximate Nash equilibria.
+8
+
+first observation is that, focusing on a single (static) game, under strong convexity-concavity the
+sum of the players’ regrets are strongly nonnegative (Lemma A.15):
+Reg(m)
+L,x(x⋆) + Reg(m)
+L,y (y⋆) ≥ µ
+2
+m
+�
+t=1
+∥z(t) − z⋆∥2
+2,
+(10)
+for any Nash equilibrium z⋆ ∈ Z of the game. In turn, this can be cast in terms of dynamic regret
+over the sequence of the h games (Lemma A.16). Next, combining those dynamic-regret lower
+bounds with a suitable RVU-type property leads to a refined second-order path length bound as
+long as that m = Ω(1/µ), which in turn leads to our main result below. Before we present its
+statement, let us introduce the following measure of variation of the gradients:
+V(H)
+∇f :=
+H−1
+�
+h=1
+max
+z∈Z ∥F (h+1)(z) − F (h)(z)∥2
+2,
+(11)
+where let F : z := (x, y) �→ (∇xf(x, y), −∇yf(x, y)). This variation measure is analogous to V(T)
+A
+we introduced in (5) for time-varying bilinear saddle-point problems.
+Theorem 3.8 (Detailed version in Theorem A.18). Let f(h) : X × Y be a µ-strongly convex-concave
+and L-smooth function, for h ∈ [[H]]. Suppose that both players employ OGD with learning rate
+η = min
+�
+1
+8L, 1
+2µ
+�
+for T repetitions, where T = m × H and m ≥
+2
+ηµ. Then,
+T
+�
+t=1
+�
+EqGap(t)(z(t))
+�2
+= O(1 + S(H)
+NE + V(H)
+∇f ),
+where S(H)
+NE and V(H)
+∇f are defined in (9) and (11).
+Our techniques also imply improved regret bounds in this setting, as we formalize in Corol-
+lary A.19.
+There is another immediate but important implication of (10): any no-regret algorithm in a
+(static) strongly convex-concave setting ought to be approaching the Nash equilibrium; in contrast,
+this property is spectacularly false in (general) monotone settings [MPP18].
+Proposition 3.9. Let f : X × Y → R be a µ-strongly convex-concave function. If players incur
+regrets such that Reg(T)
+L,x + Reg(T)
+L,y ≤ CT 1−ω, for some parameters C > 0 and ω ∈ (0, 1], then for any
+ϵ > 0 and T >
+�
+2C
+µϵ2
+�1/ω
+there is a pair of strategies z(t) ∈ Z such that ∥z(t) − z⋆∥2 ≤ ϵ, where z⋆
+is a Nash equilibrium.
+The insights of this subsection are also of interest in general monotone settings by incorpo-
+rating a strongly convex regularizer; tuning its magnitude allows us to trade-off between a better
+approximation and the benefits of strong convexity-concavity revealed in this subsection.
+3.3
+General-Sum Multi-player Games
+Next, we turn our attention to general-sum multi-player games. For simplicity, in this subsection
+we posit that the game is represented in normal form, so that each Player i ∈ [[n]] has a finite set of
+available actions Ai, and Xi := ∆(Ai). The proofs of this subsection are included in Appendix A.3.
+9
+
+Potential Games
+First, we study the convergence of (online) gradient descent (GD) in time-
+varying potential games (see Definition A.20 for the formal description).2 In our time-varying
+setup, it is assumed that each round t ∈ [[T]] corresponds to a different potential game described
+with a potential function Φ(t). We further let d : (Φ, Φ′) �→ maxz∈×n
+i=1 Xi (Φ(z) − Φ′(z)), so that
+V(T)
+Φ
+:= �T−1
+t=1 d(Φ(t), Φ(t+1)); we emphasize the fact that d(·, ·) is not symmetric. Analogously
+to Theorem 3.5, we use EqGap(t)(z(t)) ∈ R≥0 to represent the NE gap of the joint strategy profile
+z(t) := (x(t)
+1 , . . . , x(t)
+n ) at the t-th game.
+Theorem 3.10. Suppose that each player employs (online) GD with a sufficiently small learning
+rate. Then,
+T
+�
+t=1
+�
+EqGap(t)(z(t))
+�2
+= O(Φmax + V(T)
+Φ ),
+where Φmax is such that |Φ(t)(·)| ≤ Φmax for any t ∈ [[T]].
+We refer to Appendix B for some illustrative experiments.
+General games
+Unfortunately, unlike the settings considered thus far, computing Nash equilibria
+in general games is computationally hard [DGP08; CDT09]. Instead, learning algorithms are known
+to converge to relaxations of the Nash equilibrium, known as (coarse) correlated equilibria. For
+our purposes, we will employ a bilinear formulation of (coarse) correlated equilibria, which dates
+back to the seminal work of Hart and Schmeidler [HS89]. This will allow us to translate the results
+of Section 3.1 to general multi-player games.
+Specifically, correlated equilibria3 can be expressed via a game between the n players and a medi-
+ator. Intuitively, the mediator is endeavoring to identify a correlated strategy µ ∈ Ξ := ∆
+�×
+n
+i=1 Ai
+�
+for which no player has an incentive to deviate from the recommendation. In contrast, the players
+are trying to optimally deviate so as to maximize their own utility. More precisely, there exist
+matrices A1, . . . , An, with each matrix Ai depending solely on the utility of Player i, for which the
+bilinear problem can be expressed as
+min
+µ∈Ξ
+max
+(¯x1,...,¯xn)∈×n
+i=1 ¯
+Xi
+n
+�
+i=1
+µ⊤Ai ¯xi,
+(12)
+where ¯
+Xi := conv(Xi, 0); incorporating the 0 vector will be useful for our purposes. This zero-sum
+game has the property that there exists a strategy µ⋆ ∈ Ξ such that max¯xi∈ ¯
+Xi(µ⋆)⊤Ai ¯xi ≤ 0, for
+any Player i ∈ [[n]], which corresponds to a correlated equilibrium.
+Before we proceed, it is important to note that the learning paradigm considered here deviates
+from the traditional one in that there is an additional learning agent, resulting in a less decentralized
+protocol. Yet, the dynamics induced by solving (12) via online algorithms remain uncoupled [HM00],
+in the sense that each player obtains feedback—corresponding to the deviation benefit—that depends
+solely on its own utility.
+Now in the time-varying setting, the matrices A1, . . . , An that capture the players’ utilities can
+change in each repetition. Crucially, we show that the structure of the induced bilinear problem (12)
+2Unlike two-player zero-sum games, gradient descent is known to approach Nash equilibria in potential games.
+3The following bilinear formulation applies to coarse correlated equilibria as well (with different payoff matrices),
+but we will focus solely on the stronger variant (CE) for the sake of exposition.
+10
+
+is such that there is a sequence of correlated equilibria that guarantee nonnegative dynamic regret;
+this refines Property 3.2 in that only one player’s strategies suffice to guarantee nonnegativity, even
+if the strategies of the other player remain invariant. Below, we denote by DReg(T)
+µ
+the dynamic
+regret of the min player in (12), and by Reg(T)
+i
+the regret of each Player i up to time T ∈ N, so
+that the regret of the max player in (12) can be expressed as �n
+i=1 Reg(T)
+i
+.
+Property 3.11. Suppose that Ξ ∋ µ(t,⋆) is a correlated equilibrium of the game at any time t ∈ [[T]].
+Then,
+DReg(T)
+µ (µ(1,⋆), . . . , µ(T,⋆)) +
+n
+�
+i=1
+Reg(T)
+i
+≥ 0.
+As a result, this enables us to apply Theorem 3.5 parameterized on (i) the variation of the CE
+V(T)
+CE :=
+inf
+µ(t,⋆)∈Ξ(t,⋆),∀t∈[[T]]
+T−1
+�
+t=1
+∥µ(t+1,⋆) − µ(t,⋆)∥2,
+where Ξ(t,⋆) denotes the set of CE of the t-th game, and (ii) the variation in the players’ utilities
+V(T)
+A
+≤ �n
+i=1
+�T−1
+t=1 ∥A(t+1)
+i
+− A(t)
+i ∥2
+2; below, we denote by CeGap(t)(µ(t)) the CE gap of µ(t) ∈ Ξ
+at the t-th game.
+Theorem 3.12. Suppose that each player employs OGD in (12) with a suitable learning rate. Then,
+T
+�
+t=1
+�
+CeGap(t)(µ(t))
+�2
+= O(1 + V(T)
+CE + V(T)
+A ).
+There are further interesting implications of our framework that are worth highlighting. First,
+we obtain meta-learning guarantees for general games that depend on the (algorithm-independent)
+similarity of the correlated equilibria (Corollary A.22); that was left as an open question by Harris
+et al. [Har+22], where instead algorithm-dependent similarity metrics were derived. Further, by
+applying Corollary 3.7, we derive natural variation-dependent per-player regret bounds in general
+games (Corollary A.23), addressing a question left by Zhang et al. [Zha+22]; we suspect that
+obtaining such results—parameterized on the variation of the CE—are not possible without the
+presence of the additional player.
+3.4
+Dynamic Regret Bounds in Static Games
+Finally, in this subsection we switch gears by investigating dynamic regret guarantees when learning
+in static games. The proofs of this subsection are included in Appendix A.4.
+First, we point out that while traditional no-regret learning algorithms guarantee O(
+√
+T) external
+regret, instances of OMD—a generalization of OGD; see (13) in Appendix A—in fact guarantee O(
+√
+T)
+dynamic regret in two-player zero-sum games, which is a much stronger performance measure:
+Proposition 3.13. Suppose that both players in a (static) two-player zero-sum game employ OMD
+with a smooth regularizer. Then, DReg(T)
+x , DReg(T)
+y
+= O(
+√
+T).
+11
+
+In proof, the dynamic regret for each player under OMD with a smooth regularizer can be
+bounded by the first-order path length of that player’s strategies, which in turn can be bounded
+by O(
+√
+T) given that the second-order path length is O(1) (Theorem 3.4).
+In fact, Theo-
+rem 3.4 readily extends Proposition 3.13 to time-varying zero-sum games as well, implying that
+DReg(T)
+x , DReg(T)
+y
+= O
+�√
+T(1 + V(T)
+ϵ−NE + V(T)
+A )
+�
+.
+A question that arises from Proposition 3.13 is whether the O(
+√
+T) guarantee for dynamic
+regret of OMD can be improved in the online learning setting. Below, we point out a significant
+improvement to O(log T), but under a stronger two-point feedback model; namely, we posit that in
+every round each player can select an additional auxiliary strategy, and each player then gets to
+additionally observe the utility corresponding to the auxiliary strategies. Notably, this is akin to
+how the extra-gradient method works [Hsi+19] (also cf. [RS13, Section 4.2] for multi-point feedback
+models in the bandit setting).
+Observation 3.14. Under two-point feedback, there exist learning algorithms that guarantee
+DReg(T)
+x , DReg(T)
+y
+= O(log T) in two-player zero-sum games.
+In particular, it suffices for each player to employ OMD, but with the twist that the first strategy
+in each round is the time-average of OMD; the auxiliary strategy is the standard output of OMD.
+Then, the dynamic regret of each player will grow as O
+��T
+t=1
+1
+t
+�
+= O(log T) since the duality
+gap of the average strategies is decreasing with a rate of T −1 [RS13]. It is an interesting question
+whether Observation 3.14 can be improved to O(1).
+General-sum games
+In contrast, no (efficient) sublinear dynamic-regret guarantees are possible
+in general games:
+Proposition 3.15. Unless PPAD ⊆ P, any polynomial-time algorithm incurs �n
+i=1 DReg(T)
+i
+= Ω(T),
+even if n = 2, where Ω(·) here hides polynomial factors.
+Indeed, this follows since computing a Nash equilibrium to (1/poly) accuracy in two-player
+games is PPAD-hard [CDT09]. In fact, Proposition 3.15 applies beyond the online learning setting.
+This motivates considering a relaxation of dynamic regret, wherein the sequence of comparators is
+subject to the constraint �T−1
+t=1 1{x(t+1,⋆) ̸= x(t,⋆)} ≤ K − 1, for some parameter K ∈ N; this will
+be referred to as K-DReg(T)
+x . Naturally, external regret coincides with K-DReg(T)
+x
+under K = 1.
+In this context, we employ Lemma 3.1 to bound K-DReg(T) under OGD:
+Theorem 3.16 (Detailed version in Theorem A.24). Suppose that all n players employ OGD in an
+L-smooth game. Then, for any K ∈ N,
+1. �n
+i=1 K-DReg(T)
+i
+= O(K√nL);
+2. K-DReg(T)
+i
+= O(K3/4T 1/4n1/4√
+L), for i ∈ [[n]].
+One question that arises here is whether the per-player bound of O(K3/4T 1/4) (Item 2) can
+be improved to ˜O(K), where ˜O(·) hides logarithmic factors. The main challenge is that, even for
+K = 1, all known methods that obtain ˜O(1) [DFG21; PSS21; Ana+22a; Far+22] rely on non-
+smooth regularizers that violate the preconditions of Lemma A.1—our dynamic RVU bound that
+generalizes Lemma 3.1 beyond (squared) Euclidean regularization. It would also be interesting to give
+12
+
+a natural game-theoretic interpretation to the limit point of no-regret learners with K-DReg = o(T),
+even for a fixed K ∈ N; for K = 1, it corresponds to the fundamental coarse correlated equilibrium.
+At a superficial level, it seems to be related to the variant considered by Harrow, Natarajan, and
+Wu [HNW16].
+4
+Conclusions and Future Work
+In this paper, we developed a new framework for characterizing iterate-convergence of no-regret
+learning algorithms—primarily optimistic gradient descent (OGD)—in time-varying games. There
+are many promising avenues for future research. Besides closing the obvious gaps we highlighted
+in Section 3.4, it is important to characterize the behavior of no-regret learning algorithms in
+other fundamental multiagent settings, such as Stackelberg (security) games [Bal+15]. Moreover,
+our results operate in the full-feedback model where each player receives feedback on all possible
+actions. Extending the scope of our framework to capture partial-feedback models as well is another
+interesting direction for future work.
+Acknowledgements
+We are grateful to Vince Conitzer and Caspar Oesterheld for helpful feedback. This material is based
+on work supported by the National Science Foundation under grants IIS-1901403 and CCF-1733556
+and by the ARO under award W911NF2210266.
+References
+[Ana+22a]
+Ioannis Anagnostides, Gabriele Farina, Christian Kroer, Chung-Wei Lee, Haipeng Luo,
+and Tuomas Sandholm. “Uncoupled Learning Dynamics with O(log T) Swap Regret in
+Multiplayer Games”. In: NeurIPS 2022. 2022.
+[Ana+22b]
+Ioannis Anagnostides, Ioannis Panageas, Gabriele Farina, and Tuomas Sandholm. “On
+Last-Iterate Convergence Beyond Zero-Sum Games”. In: International Conference on
+Machine Learning, ICML 2022. Vol. 162. Proceedings of Machine Learning Research.
+PMLR, 2022, pp. 536–581.
+[Aum74]
+Robert Aumann. “Subjectivity and Correlation in Randomized Strategies”. In: Journal
+of Mathematical Economics 1 (1974), pp. 67–96.
+[Azi+20]
+Wa¨ıss Azizian, Ioannis Mitliagkas, Simon Lacoste-Julien, and Gauthier Gidel. “A
+Tight and Unified Analysis of Gradient-Based Methods for a Whole Spectrum of
+Differentiable Games”. In: The 23rd International Conference on Artificial Intelligence
+and Statistics, AISTATS 2020. Vol. 108. Proceedings of Machine Learning Research.
+PMLR, 2020, pp. 2863–2873.
+[Bal+15]
+Maria-Florina Balcan, Avrim Blum, Nika Haghtalab, and Ariel D. Procaccia. “Commit-
+ment Without Regrets: Online Learning in Stackelberg Security Games”. In: Proceedings
+of the Sixteenth ACM Conference on Economics and Computation, EC ’15. ACM, 2015,
+pp. 61–78.
+13
+
+[BMW21]
+Heinz H. Bauschke, Walaa M. Moursi, and Xianfu Wang. “Generalized monotone
+operators and their averaged resolvents”. In: Math. Program. 189.1 (2021), pp. 55–74.
+[Bow+15]
+Michael Bowling, Neil Burch, Michael Johanson, and Oskari Tammelin. “Heads-up
+Limit Hold’em Poker is Solved”. In: Science 347.6218 (Jan. 2015).
+[BS16]
+Noam Brown and Tuomas Sandholm. “Strategy-Based Warm Starting for Regret
+Minimization in Games”. In: AAAI Conference on Artificial Intelligence (AAAI). 2016.
+[BS17]
+Noam Brown and Tuomas Sandholm. “Superhuman AI for heads-up no-limit poker:
+Libratus beats top professionals”. In: Science (Dec. 2017), eaao1733.
+[BS19a]
+Noam Brown and Tuomas Sandholm. “Solving imperfect-information games via dis-
+counted regret minimization”. In: AAAI Conference on Artificial Intelligence (AAAI).
+2019.
+[BS19b]
+Noam Brown and Tuomas Sandholm. “Superhuman AI for multiplayer poker”. In:
+Science 365.6456 (2019), pp. 885–890.
+[Car+19]
+Adrian Rivera Cardoso, Jacob D. Abernethy, He Wang, and Huan Xu. “Competing
+Against Nash Equilibria in Adversarially Changing Zero-Sum Games”. In: Proceedings
+of the 36th International Conference on Machine Learning, ICML 2019. Vol. 97.
+Proceedings of Machine Learning Research. PMLR, 2019, pp. 921–930.
+[CDT09]
+Xi Chen, Xiaotie Deng, and Shang-Hua Teng. “Settling the Complexity of Computing
+Two-Player Nash Equilibria”. In: Journal of the ACM (2009).
+[Ces+12]
+Nicol`o Cesa-Bianchi, Pierre Gaillard, G´abor Lugosi, and Gilles Stoltz. “Mirror Descent
+Meets Fixed Share (and feels no regret)”. In: Advances in Neural Information Processing
+Systems 25: 26th Annual Conference on Neural Information Processing Systems 2012.
+2012, pp. 989–997.
+[Chi+12]
+Chao-Kai Chiang, Tianbao Yang, Chia-Jung Lee, Mehrdad Mahdavi, Chi-Jen Lu, Rong
+Jin, and Shenghuo Zhu. “Online optimization with gradual variations”. In: Conference
+on Learning Theory. 2012, pp. 6–1.
+[COZ22]
+Yang Cai, Argyris Oikonomou, and Weiqiang Zheng. “Tight Last-Iterate Convergence
+of the Extragradient Method for Constrained Monotone Variational Inequalities”. In:
+CoRR abs/2204.09228 (2022).
+[CP04]
+Patrick L. Combettes and Teemu Pennanen. “Proximal Methods for Cohypomonotone
+Operators”. In: SIAM J. Control. Optim. (2004), pp. 731–742.
+[CZ22]
+Yang Cai and Weiqiang Zheng. “Accelerated Single-Call Methods for Constrained
+Min-Max Optimization”. In: CoRR abs/2210.03096 (2022).
+[Das+18]
+Constantinos Daskalakis, Andrew Ilyas, Vasilis Syrgkanis, and Haoyang Zeng. “Training
+GANs with Optimism”. In: 6th International Conference on Learning Representations,
+ICLR 2018. OpenReview.net, 2018.
+[Das22]
+Constantinos Daskalakis. Non-Concave Games: A Challenge for Game Theory’s Next
+100 Years. 2022.
+14
+
+[DDJ21]
+Jelena Diakonikolas, Constantinos Daskalakis, and Michael I. Jordan. “Efficient Meth-
+ods for Structured Nonconvex-Nonconcave Min-Max Optimization”. In: The 24th
+International Conference on Artificial Intelligence and Statistics, AISTATS 2021.
+Vol. 130. Proceedings of Machine Learning Research. PMLR, 2021, pp. 2746–2754.
+[DDK11]
+Constantinos Daskalakis, Alan Deckelbaum, and Anthony Kim. “Near-optimal no-
+regret algorithms for zero-sum games”. In: Annual ACM-SIAM Symposium on Discrete
+Algorithms (SODA). 2011.
+[DFG21]
+Constantinos Daskalakis, Maxwell Fishelson, and Noah Golowich. “Near-Optimal No-
+Regret Learning in General Games”. In: Advances in Neural Information Processing
+Systems 34: Annual Conference on Neural Information Processing Systems 2021,
+NeurIPS 2021. 2021, pp. 27604–27616.
+[DGP08]
+Constantinos Daskalakis, Paul Goldberg, and Christos Papadimitriou. “The Complexity
+of Computing a Nash Equilibrium”. In: SIAM Journal on Computing (2008).
+[DL15]
+Cong D. Dang and Guanghui Lan. “On the convergence properties of non-Euclidean
+extragradient methods for variational inequalities with generalized monotone operators”.
+In: Comput. Optim. Appl. (2015), pp. 277–310.
+[DMZ21]
+Yuan Deng, Vahab Mirrokni, and Song Zuo. “Non-Clairvoyant Dynamic Mechanism
+Design with Budget Constraints and Beyond”. In: Proceedings of the 22nd ACM
+Conference on Economics and Computation. EC ’21. New York, NY, USA: Association
+for Computing Machinery, 2021, p. 369.
+[Duv+22]
+Benoit Duvocelle, Panayotis Mertikopoulos, Mathias Staudigl, and Dries Vermeulen.
+“Multiagent online learning in time-varying games”. In: Mathematics of Operations
+Research (2022).
+[Far+22]
+Gabriele Farina, Ioannis Anagnostides, Haipeng Luo, Chung-Wei Lee, Christian Kroer,
+and Tuomas Sandholm. “Near-Optimal No-Regret Learning for General Convex Games”.
+In: NeurIPS 2022. 2022.
+[FKS21]
+Gabriele Farina, Christian Kroer, and Tuomas Sandholm. “Faster Game Solving
+via Predictive Blackwell Approachability: Connecting Regret Matching and Mirror
+Descent”. In: Proceedings of the AAAI Conference on Artificial Intelligence. 2021.
+[FS99]
+Yoav Freund and Robert Schapire. “Adaptive game playing using multiplicative
+weights”. In: Games and Economic Behavior 29 (1999), pp. 79–103.
+[FV97]
+Dean Foster and Rakesh Vohra. “Calibrated Learning and Correlated Equilibrium”.
+In: Games and Economic Behavior 21 (1997), pp. 40–55.
+[Gar17]
+Daniel F. Garrett. “Dynamic mechanism design: Dynamic arrivals and changing values”.
+In: Games and Economic Behavior 104 (2017), pp. 595–612.
+[GPD20]
+Noah Golowich, Sarath Pattathil, and Constantinos Daskalakis. “Tight last-iterate
+convergence rates for no-regret learning in multi-player games”. In: Advances in
+Neural Information Processing Systems 33: Annual Conference on Neural Information
+Processing Systems 2020, NeurIPS 2020. 2020.
+[GTG22]
+Eduard Gorbunov, Adrien Taylor, and Gauthier Gidel. Last-Iterate Convergence of
+Optimistic Gradient Method for Monotone Variational Inequalities. 2022.
+15
+
+[GVM21]
+Angeliki Giannou, Emmanouil-Vasileios Vlatakis-Gkaragkounis, and Panayotis Mer-
+tikopoulos. “Survival of the strictest: Stable and unstable equilibria under regularized
+learning with partial information”. In: Conference on Learning Theory, COLT 2021.
+Vol. 134. Proceedings of Machine Learning Research. PMLR, 2021, pp. 2147–2148.
+[Har+22]
+Keegan Harris, Ioannis Anagnostides, Gabriele Farina, Mikhail Khodak, Zhiwei Steven
+Wu, and Tuomas Sandholm. “Meta-Learning in Games”. In: CoRR abs/2209.14110
+(2022).
+[HM00]
+Sergiu Hart and Andreu Mas-Colell. “A Simple Adaptive Procedure Leading to Corre-
+lated Equilibrium”. In: Econometrica 68 (2000), pp. 1127–1150.
+[HNW16]
+Aram W. Harrow, Anand Natarajan, and Xiaodi Wu. “Tight SoS-Degree Bounds for
+Approximate Nash Equilibria”. In: 31st Conference on Computational Complexity,
+CCC 2016. Ed. by Ran Raz. Vol. 50. LIPIcs. Schloss Dagstuhl - Leibniz-Zentrum f¨ur
+Informatik, 2016, 22:1–22:25.
+[HS09]
+Elad Hazan and C. Seshadhri. “Efficient Learning Algorithms for Changing Envi-
+ronments”. In: Proceedings of the 26th Annual International Conference on Machine
+Learning. ICML ’09. Association for Computing Machinery, 2009, pp. 393–400.
+[HS89]
+Sergiu Hart and David Schmeidler. “Existence of Correlated Equilibria”. In: Mathe-
+matics of Operations Research 14.1 (1989), pp. 18–25.
+[Hsi+19]
+Yu-Guan Hsieh, Franck Iutzeler, J´erome Malick, and Panayotis Mertikopoulos. “On
+the convergence of single-call stochastic extra-gradient methods”. In: Advances in
+Neural Information Processing Systems 32: Annual Conference on Neural Information
+Processing Systems 2019, NeurIPS 2019. 2019, pp. 6936–6946.
+[Jad+15]
+Ali Jadbabaie, Alexander Rakhlin, Shahin Shahrampour, and Karthik Sridharan.
+“Online Optimization : Competing with Dynamic Comparators”. In: Proceedings of
+the Eighteenth International Conference on Artificial Intelligence and Statistics. 2015,
+pp. 398–406.
+[LST16]
+Thodoris Lykouris, Vasilis Syrgkanis, and ´Eva Tardos. “Learning and Efficiency in
+Games with Dynamic Population”. In: Proceedings of the Twenty-Seventh Annual
+ACM-SIAM Symposium on Discrete Algorithms, SODA 2016. SIAM, 2016, pp. 120–129.
+[MO11]
+Ishai Menache and Asuman E. Ozdaglar. Network Games: Theory, Models, and Dynam-
+ics. Synthesis Lectures on Communication Networks. Morgan & Claypool Publishers,
+2011.
+[MPP18]
+Panayotis Mertikopoulos, Christos H. Papadimitriou, and Georgios Piliouras. “Cycles
+in Adversarial Regularized Learning”. In: Proceedings of the Twenty-Ninth Annual
+ACM-SIAM Symposium on Discrete Algorithms, SODA 2018. SIAM, 2018, pp. 2703–
+2717.
+[MRS20]
+Eric Mazumdar, Lillian J. Ratliff, and S. Shankar Sastry. “On Gradient-Based Learning
+in Continuous Games”. In: SIAM J. Math. Data Sci. (2020), pp. 103–131.
+[MS21]
+Panayotis Mertikopoulos and Mathias Staudigl. “Equilibrium Tracking and Convergence
+in Dynamic Games”. In: 2021 60th IEEE Conference on Decision and Control (CDC).
+IEEE, 2021, pp. 930–935.
+16
+
+[MV21]
+Oren Mangoubi and Nisheeth K. Vishnoi. “Greedy adversarial equilibrium: an efficient
+alternative to nonconvex-nonconcave min-max optimization”. In: STOC ’21: 53rd
+Annual ACM SIGACT Symposium on Theory of Computing, 2021. ACM, 2021, pp. 896–
+909.
+[MV78]
+H. Moulin and J.-P. Vial. “Strategically zero-sum games: The class of games whose
+completely mixed equilibria cannot be improved upon”. In: International Journal of
+Game Theory 7.3-4 (1978), pp. 201–221.
+[Nas50]
+John Nash. “Equilibrium points in N-person games”. In: Proceedings of the National
+Academy of Sciences 36 (1950), pp. 48–49.
+[Nou+19]
+Maher Nouiehed, Maziar Sanjabi, Tianjian Huang, Jason D. Lee, and Meisam Raza-
+viyayn. “Solving a Class of Non-Convex Min-Max Games Using Iterative First Order
+Methods”. In: Advances in Neural Information Processing Systems 32: Annual Confer-
+ence on Neural Information Processing Systems 2019. 2019, pp. 14905–14916.
+[Pap+22]
+Christos H. Papadimitriou, George Pierrakos, Alexandros Psomas, and Aviad Rubin-
+stein. “On the complexity of dynamic mechanism design”. In: Games and Economic
+Behavior 134 (2022), pp. 399–427.
+[PKB22]
+Jorge I. Poveda, Miroslav Krstic, and Tamer Basar. “Fixed-Time Seeking and Tracking
+of Time-Varying Nash Equilibria in Noncooperative Games”. In: American Control
+Conference, ACC 2022, Atlanta, GA, USA, June 8-10, 2022. IEEE, 2022, pp. 794–799.
+[PSS21]
+Georgios Piliouras, Ryann Sim, and Stratis Skoulakis. “Optimal No-Regret Learning in
+General Games: Bounded Regret with Unbounded Step-Sizes via Clairvoyant MWU”.
+In: arXiv preprint arXiv:2111.14737 (2021).
+[RG22]
+Aitazaz Ali Raja and Sergio Grammatico. “Payoff Distribution in Robust Coalitional
+Games on Time-Varying Networks”. In: IEEE Trans. Control. Netw. Syst. 9.1 (2022),
+pp. 511–520.
+[RJW21]
+Jad Rahme, Samy Jelassi, and S. Matthew Weinberg. “Auction Learning as a Two-
+Player Game”. In: 9th International Conference on Learning Representations, ICLR
+2021. 2021.
+[RS13]
+Alexander Rakhlin and Karthik Sridharan. “Optimization, learning, and games with
+predictable sequences”. In: Advances in Neural Information Processing Systems. 2013,
+pp. 3066–3074.
+[SAF02]
+Yuzuru Sato, Eizo Akiyama, and J. Doyne Farmer. “Chaos in learning a simple
+two-person game”. In: Proceedings of the National Academy of Sciences 99.7 (2002),
+pp. 4748–4751.
+[Sha12]
+Shai Shalev-Shwartz. “Online Learning and Online Convex Optimization”. In: Founda-
+tions and Trends in Machine Learning 4 (2012).
+[Son+20]
+Chaobing Song, Zhengyuan Zhou, Yichao Zhou, Yong Jiang, and Yi Ma. “Optimistic
+Dual Extrapolation for Coherent Non-monotone Variational Inequalities”. In: Ad-
+vances in Neural Information Processing Systems 33: Annual Conference on Neural
+Information Processing Systems 2020. 2020.
+17
+
+[Syr+15]
+Vasilis Syrgkanis, Alekh Agarwal, Haipeng Luo, and Robert E Schapire. “Fast conver-
+gence of regularized learning in games”. In: Advances in Neural Information Processing
+Systems. 2015, pp. 2989–2997.
+[Van10]
+Ngo Van Long. A survey of dynamic games in economics. Vol. 1. World Scientific,
+2010.
+[van91]
+Eric van Damme. Stability and perfection of Nash equilibria. Vol. 339. Springer, 1991.
+[Ven21]
+Xavier Venel. “Regularity of dynamic opinion games”. In: Games and Economic
+Behavior 126 (2021), pp. 305–334.
+[Vla+20]
+Emmanouil-Vasileios Vlatakis-Gkaragkounis, Lampros Flokas, Thanasis Lianeas, Panay-
+otis Mertikopoulos, and Georgios Piliouras. “No-Regret Learning and Mixed Nash
+Equilibria: They Do Not Mix”. In: Advances in Neural Information Processing Systems
+33: Annual Conference on Neural Information Processing Systems 2020, NeurIPS 2020.
+2020.
+[YH15]
+Maojiao Ye and Guoqiang Hu. “Distributed Seeking of Time-Varying Nash Equilibrium
+for Non-Cooperative Games”. In: IEEE Trans. Autom. Control. 60.11 (2015), pp. 3000–
+3005.
+[YKH20]
+Junchi Yang, Negar Kiyavash, and Niao He. “Global Convergence and Variance-Reduced
+Optimization for a Class of Nonconvex-Nonconcave Minimax Problems”. In: CoRR
+abs/2002.09621 (2020).
+[YM22]
+Yuepeng Yang and Cong Ma. “O(T −1) Convergence of Optimistic-Follow-the-Regularized-
+Leader in Two-Player Zero-Sum Markov Games”. In: arXiv preprint arXiv:2209.12430
+(2022).
+[Zha+17]
+Lijun Zhang, Tianbao Yang, Jinfeng Yi, Rong Jin, and Zhi-Hua Zhou. “Improved
+Dynamic Regret for Non-degenerate Functions”. In: Advances in Neural Information
+Processing Systems 30: Annual Conference on Neural Information Processing Systems
+2017. 2017, pp. 732–741.
+[Zha+20]
+Peng Zhao, Yu-Jie Zhang, Lijun Zhang, and Zhi-Hua Zhou. “Dynamic Regret of Convex
+and Smooth Functions”. In: Advances in Neural Information Processing Systems 33:
+Annual Conference on Neural Information Processing Systems 2020. 2020.
+[Zha+22]
+Mengxiao Zhang, Peng Zhao, Haipeng Luo, and Zhi-Hua Zhou. “No-Regret Learning
+in Time-Varying Zero-Sum Games”. In: International Conference on Machine Learn-
+ing, ICML 2022. Vol. 162. Proceedings of Machine Learning Research. PMLR, 2022,
+pp. 26772–26808.
+[Zin03]
+Martin Zinkevich. “Online Convex Programming and Generalized Infinitesimal Gradient
+Ascent”. In: International Conference on Machine Learning (ICML). Washington, DC,
+USA, 2003, pp. 928–936.
+18
+
+A
+Omitted Proofs
+In this section, we provide the proofs from Section 3.
+A.1
+Proofs from Section 3.1
+First, we start with the proof of Lemma 3.1 from Section 3.1. Before we proceed, it will be useful to
+express the update rule (OGD) in the following equivalent form:
+x(t) := arg max
+x∈X
+�
+Ψ(t)
+x (x) := ⟨x, m(t)
+x ⟩ − 1
+ηBφx(x ∥ ˆx(t))
+�
+,
+ˆx(t+1) := arg max
+ˆx∈X
+�
+ˆΨ(t)
+x (ˆx) := ⟨ˆx, u(t)
+x ⟩ − 1
+ηBφx(ˆx ∥ ˆx(t))
+�
+.
+(13)
+Here, Bφx(· ∥ ·) denotes the Bregman divergence induced by the (squared) Euclidean regularizer
+φx : x �→ 1
+2∥x∥2
+2; namely, Bφx(x ∥ x′) := φ(x)−φ(x′)−⟨∇φ(x′), x−x′⟩ = 1
+2∥x−x′∥2
+2, for x, x′ ∈ X.
+The update rule (13) for general Bregman divergences will be referred to as optimistic mirror descent
+(OMD).
+A.1.1
+Dynamic RVU Bounds
+We now show Lemma 3.1, the statement of which is recalled below for the convenience of the reader.
+Then, in Lemma A.1 we provide an extension of Lemma 3.1 to a broader class of regularizers.
+Lemma 3.1 (RVU bound for dynamic regret). Consider any sequence of utilities (u(1)
+x , . . . , u(T)
+x )
+up to time T ∈ N. The dynamic regret (1) of OGD with respect to any sequence of comparators
+(x(1,⋆), . . . , x(T,⋆)) ∈ X T can be bounded by
+D2
+X
+2η + DX
+η
+T−1
+�
+t=1
+∥x(t+1,⋆) − x(t,⋆)∥2 + η
+T
+�
+t=1
+∥u(t)
+x − m(t)
+x ∥2
+2 − 1
+2η
+T
+�
+t=1
+�
+∥x(t) − ˆx(t)∥2
+2 + ∥x(t) − ˆx(t+1)∥2
+2
+�
+.
+Proof. First, by (1/η)-strong convexity of the function Ψ(t)
+x
+(defined in (13)) for any time t ∈ [[T]],
+we have that
+⟨x(t), m(t)
+x ⟩ − 1
+2η∥x(t) − ˆx(t)∥2
+2 − ⟨ˆx(t+1), m(t)
+x ⟩ + 1
+2η∥ˆx(t+1) − ˆx(t)∥2
+2 ≥ 1
+2η∥x(t) − ˆx(t+1)∥2
+2,
+(14)
+where we used [Sha12, Lemma 2.8, p. 135]. Similarly, by (1/η)-strong convexity of the function ˆΨ(t)
+x
+(defined in (13)) for any time t ∈ [[T]], we have that for any comparator x(t,⋆) ∈ X,
+⟨ˆx(t+1), u(t)
+x ⟩ − 1
+2η∥ˆx(t+1) − ˆx(t)∥2
+2 − ⟨x(t,⋆), u(t)
+x ⟩ + 1
+2η∥x(t,⋆) − ˆx(t)∥2
+2 ≥ 1
+2η∥ˆx(t+1) − x(t,⋆)∥2
+2. (15)
+Thus, adding (14) and (15),
+⟨x(t,⋆) − ˆx(t+1), u(t)
+x ⟩ + ⟨ˆx(t+1) − x(t), m(t)
+x ⟩ ≤ 1
+2η
+�
+∥ˆx(t) − x(t,⋆)∥2
+2 − ∥ˆx(t+1) − x(t,⋆)∥2
+2
+�
+− 1
+2η
+�
+∥x(t) − ˆx(t)∥2
+2 + ∥x(t) − ˆx(t+1)∥2
+2
+�
+.
+(16)
+19
+
+We further see that
+⟨x(t,⋆) − x(t), u(t)
+x ⟩ = ⟨x(t) − ˆx(t+1), m(t)
+x − u(t)
+x ⟩ + ⟨x(t,⋆) − ˆx(t+1), u(t)
+x ⟩ + ⟨ˆx(t+1) − x(t), m(t)
+x ⟩. (17)
+Now the first term on the right-hand side can be upper bounded using the fact that, by (14) and (15),
+⟨x(t) − ˆx(t+1), m(t)
+x − u(t)
+x ⟩ ≥ 1
+η∥ˆx(t+1) − x(t)∥2
+2 =⇒ ∥ˆx(t+1) − x(t)∥2 ≤ η∥m(t)
+x − u(t)
+x ∥2,
+by Cauchy-Schwarz, in turn implying that ⟨x(t) − ˆx(t+1), m(t)
+x − u(t)
+x ⟩ ≤ η∥m(t)
+x − u(t)
+x ∥2
+2. Thus, the
+proof follows by combining this bound with (16) and (17), along with the fact that
+T
+�
+t=1
+�
+∥ˆx(t) − x(t,⋆)∥2
+2 − ∥ˆx(t+1) − x(t,⋆)∥2
+2
+�
+≤ ∥ˆx(1) − x(1,⋆)∥2
+2
++
+T−1
+�
+t=1
+�
+∥ˆx(t+1) − x(t+1,⋆)∥2
+2 − ∥ˆx(t+1) − x(t,⋆)∥2
+2
+�
+≤ D2
+X + 2DX
+T−1
+�
+t=1
+∥x(t+1,⋆) − x(t,⋆)∥2,
+where the last bound follows since
+∥ˆx(t+1) − x(t+1,⋆)∥2
+2 − ∥ˆx(t+1) − x(t,⋆)∥2
+2 ≤ 2DX
+���∥ˆx(t+1) − x(t+1,⋆)∥2 − ∥ˆx(t+1) − x(t,⋆)∥2
+���
+≤ 2DX ∥x(t+1,⋆) − x(t,⋆)∥2,
+where we recall that DX denotes the ℓ2-diameter of X.
+As an aside, we remark that assuming that m(t)
+x := 0 and ∥u(t)
+x ∥2 ≤ 1 for any t ∈ [[T]], Lemma 3.1
+implies that dynamic regret can be upper bounded by O
+��
+(1 + �T−1
+t=1 ∥x(t+1,⋆) − x(t,⋆)∥2)T
+�
+,
+for any (bounded)—potentially adversarially selected—sequence of utilities (u(1)
+x , . . . , u(T)
+x ), for
+η :=
+�
+D2
+X
+2T + DX
+�T −1
+t=1 ∥x(t+1,⋆)−x(t,⋆)∥2
+T
+, which is a well-known result in online optimization [Zin03];
+while that requires setting the learning rate based on the first-order variation of the (optimal)
+comparators, there are standard techniques that would allow bypassing that assumption.
+Next, we provide an extension of Lemma 3.1 to the more general OMD algorithm under a broad
+class of regularizers.
+Lemma A.1 (Extension of Lemma 3.1 beyond Euclidean regularization). Consider a 1-strongly
+convex continuously differentiable regularizer φ with respect to a norm ∥·∥ such that (i) ∥∇φ(x)∥∗ ≤ G
+for any x, and (ii) Bφx(x ∥ x′) ≤ L∥x − x′∥ for any x, x′. Then, for any sequence of utilities
+(u(1)
+x , . . . , u(T)
+x ) up to time T ∈ N the dynamic regret (1) of OMD with respect to any sequence of
+comparators (x(1,⋆), . . . , x(T,⋆)) ∈ X T can be bounded as
+Bφx(x(1,⋆) ∥ ˆx(1))
+η
++ L + 2G
+η
+T−1
+�
+t=1
+∥x(t+1,⋆) − x(t,⋆)∥+η
+T
+�
+t=1
+∥u(t)
+x − m(t)
+x ∥2
+∗
+− 1
+2η
+T
+�
+t=1
+�
+∥x(t) − ˆx(t)∥2 + ∥x(t) − ˆx(t+1)∥2�
+.
+20
+
+The proof is analogous to that of Lemma 3.1, and relies on the well-known three-point identity
+for the Bregman divergence:
+Bφx(x ∥ x′) = Bφx(x ∥ x′′) + Bφx(x′′ ∥ x′) − ⟨x − x′′, ∇φ(x′) − ∇φ(x′′)⟩.
+(18)
+In particular, along with the assumptions of Lemma A.1 imposed on the regularizer φx, (18) implies
+that the term �T−1
+t=1
+�
+Bφx(x(t+1,⋆) ∥ ˆx(t+1)) − Bφx(x(t,⋆) ∥ ˆx(t+1))
+�
+is equal to
+T−1
+�
+t=1
+�
+Bφx(x(t+1,⋆) ∥ x(t,⋆)) − ⟨x(t+1,⋆) − x(t,⋆), ∇φ(ˆx(t+1)) − ∇φ(x(t,⋆))⟩
+�
+≤ (L + 2G)
+T−1
+�
+t=1
+∥x(t+1,⋆) − x(t,⋆)∥,
+since Bφx(x(t+1,⋆) ∥ x(t,⋆)) ≤ L∥x(t+1,⋆) − x(t,⋆)∥ (by assumption) and
+⟨x(t+1,⋆) − x(t,⋆), ∇φ(ˆx(t+1)) − ∇φ(x(t,⋆))⟩≤ ∥x(t+1,⋆) − x(t,⋆)∥∥∇φ(ˆx(t+1)) − ∇φ(x(t,⋆))∥∗
+(19)
+≤
+�
+∥∇φ(ˆx(t+1))∥∗+ ∥∇φ(x(t,⋆))∥∗
+�
+∥x(t+1,⋆) − x(t,⋆)∥
+(20)
+≤ 2G∥x(t+1,⋆) − x(t,⋆)∥,
+(21)
+where (19) follows from the Cauchy-Schwarz inequality; (20) uses the triangle inequality for the
+dual norm ∥ · ∥∗; and (21) follows from the assumption of Lemma A.1 that ∥∇φ(·)∥∗ ≤ G. The rest
+of the proof of Lemma A.1 is analogous to Lemma 3.1, and it is therefore omitted. An important
+question is whether Lemma A.1 can be extended under any regularizer; as we explain in Section 3.4,
+this is the main obstacle to improving Theorem 3.16.
+A.1.2
+Nonnegativity of Dynamic Regret
+We next proceed with the proof of Property 3.2. To provide additional intuition, we first prove the
+following special case; the proof of Property 3.2 is then analogous.
+Property A.2 (Special case of Property 3.2). Suppose that Z ∋ z(t,⋆) = (x(t,⋆), y(t,⋆))) is a
+Nash equilibrium of the t-th game, for any time t ∈ [[T]]. Then, for s(T)
+x
+= (x(t,⋆))1≤t≤T and
+s(T)
+y
+= (y(t,⋆))1≤t≤T ,
+DReg(T)
+x (s(T)
+x ) + DReg(T)
+y
+(s(T)
+y
+) ≥ 0.
+Proof. Let v(t) := ⟨x(t,⋆), A(t)y(t,⋆)⟩ be the value of the t-th game, for some t ∈ [[T]]. Then, we
+have that v(t) = ⟨x(t,⋆), A(t)y(t,⋆)⟩ ≤ ⟨x, A(t)y(t,⋆)⟩ for any x ∈ X, since x(t,⋆) is a best response to
+y(t,⋆); similarly, v(t) = ⟨x(t,⋆), A(t)y(t,⋆)⟩ ≥ ⟨x(t,⋆), A(t)y⟩ for any y ∈ Y. Hence, ⟨x(t), A(t)y(t,⋆)⟩ −
+⟨x(t,⋆), A(t)y(t)⟩ ≥ 0, or equivalently, ⟨x(t,⋆), u(t)
+x ⟩ + ⟨y(t,⋆), u(t)
+y ⟩ ≥ 0. But given that the game is
+zero-sum, it holds that ⟨x(t), u(t)
+x ⟩ + ⟨y(t), u(t)
+y ⟩ = 0, so the last inequality can be in turn cast as
+⟨x(t,⋆), u(t)
+x ⟩ − ⟨x(t), u(t)
+x ⟩ + ⟨y(t,⋆), u(t)
+y ⟩ − ⟨y(t), u(t)
+y ⟩ ≥ 0,
+21
+
+for any t ∈ [[T]]. As a result, summing over all t ∈ [[T]] we have shown that
+DReg(T)
+x (x(1,⋆), . . . , x(T,⋆))+ DReg(T)
+y
+(y(1,⋆), . . . , y(T,⋆))
+=
+T
+�
+t=1
+⟨x(t,⋆), u(t)
+x ⟩ − ⟨x(t), u(t)
+x ⟩ + ⟨y(t,⋆), u(t)
+y ⟩ − ⟨y(t), u(t)
+y ⟩ ≥ 0.
+Property 3.2. Suppose that Z ∋ z(t,⋆) = (x(t,⋆), y(t,⋆)) is an ϵ(t)-approximate Nash equilibrium of
+the t-th game. Then, for s(T)
+x
+= (x(t,⋆))1≤t≤T and s(T)
+y
+= (y(t,⋆))1≤t≤T ,
+DReg(T)
+x (s(T)
+x ) + DReg(T)
+y
+(s(T)
+y
+) ≥ −2
+T
+�
+t=1
+ϵ(t).
+Proof. Given that (x(t,⋆), y(t,⋆)) ∈ Z is an ϵ(t)-approximate Nash equilibrium of the t-th game, it
+follows that ⟨x(t,⋆), A(t)y(t,⋆)⟩ ≤ ⟨x(t), A(t)y(t,⋆)⟩+ϵ(t)
+x and ⟨x(t,⋆), A(t)y(t,⋆)⟩ ≥ ⟨x(t,⋆), A(t)y(t)⟩−ϵ(t)
+y ,
+for some ϵ(t)
+x , ϵ(t)
+y
+≤ ϵ(t). Thus, we have that ⟨x(t), A(t)y(t,⋆)⟩ ≥ ⟨x(t,⋆), A(t)y(t)⟩ − ϵ(t)
+x − ϵ(t)
+y , or
+equivalently, ⟨x(t,⋆), u(t)
+x ⟩ + ⟨y(t,⋆), u(t)
+y ⟩ ≥ −ϵ(t)
+x − ϵ(t)
+y
+≥ −2ϵ(t). As a result,
+⟨x(t,⋆), u(t)
+x ⟩ − ⟨x(t), u(t)
+x ⟩ + ⟨y(t,⋆), u(t)
+y ⟩ − ⟨y(t), u(t)
+y ⟩ ≥ −2ϵ(t),
+(22)
+for any t ∈ [[T]], and the statement follows by summing (22) over all t ∈ [[T]].
+In fact, as we show below (in Property A.3), Property A.2 is a more general consequence of the
+minimax theorem. In particular, for a nonlinear online learning problem, we define dynamic regret
+with respect to a sequence of comparators (x(1,⋆), . . . , x(T,⋆)) ∈ X T as follows:
+DReg(T)
+x (x(1,⋆), . . . , x(T,⋆)) :=
+T
+�
+t=1
+�
+u(t)
+x (x(t,⋆)) − u(t)
+x (x(t))
+�
+,
+(23)
+where u(1)
+x , . . . , u(T)
+x
+: x �→ R are the continuous utility functions observed by the learner, which
+could be in general nonconcave, and (x(t))1≤t≤T is the sequence of strategies produced by the
+learner; (23) generalizes the notion of dynamic regret (1) in online linear optimization, that is, when
+u(t)
+x : x �→ ⟨x, u(t)
+x ⟩, where u(t)
+x ∈ Rdx, for any time t ∈ [[T]].
+Property A.3. Suppose that f(t) : X × Y → R is a continuous function such that for any t ∈ [[T]],
+min
+x∈X max
+y∈Y f(t)(x, y) = max
+y∈Y min
+x∈X f(t)(x, y).
+Let also x(t,⋆) ∈ arg minx∈X maxy∈Y f(t)(x, y) and y(t,⋆) ∈ arg maxy∈Y minx∈X f(t)(x, y), for any
+t ∈ [[T]]. Then, for s(T)
+x
+= (x(t,⋆))1≤t≤T and s(T)
+y
+= (y(t,⋆))1≤t≤T ,
+DReg(T)
+x (s(T)
+x ) + DReg(T)
+y
+(s(T)
+y
+) ≥ 0.
+22
+
+Proof. By definition of dynamic regret (23), it suffices to show that f(t)(x(t), y(t,⋆)) ≥ f(t)(x(t,⋆), y(t)),
+for any time t ∈ [[T]]. Indeed,
+f(t)(x(t), y(t,⋆)) ≥ min
+x∈X f(t)(x, y(t,⋆))
+(24)
+= max
+y∈Y min
+x∈X f(t)(x, y)
+(25)
+= min
+x∈X max
+y∈Y f(t)(x, y)
+(26)
+= max
+y∈Y f(t)(x(t,⋆), y)
+(27)
+≥ f(t)(x(t,⋆), y(t)),
+(28)
+where (24) and (28) are obvious; (25) and (27) follow from the definition of y(t,⋆) ∈ Y and x(t,⋆) ∈ X,
+respectively; and (26) holds by assumption. This concludes the proof.
+Remark A.4 (MVI property). Property (3) can also be generalized beyond time-varying bilinear
+saddle-point problems to more general time-varying variational inequality (VI) problems as follows.
+Let F (t) : Z → Z be the (single-valued) operator of the VI problem at time t. F (t) is said to
+satisfy the MVI property if there exists a point z(t,⋆) ∈ Z such that ⟨z − z(t,⋆), F (t)(z)⟩ ≥ 0 for
+any z ∈ Z. For example, in the special case of a bilinear saddle-point problem, we have that
+F : z := (x, y) �→ (Ay, −A⊤x), and the MVI property is satisfied by virtue of Von Neumann’s
+minimax theorem. It is direct to see that Property A.2 applies to any time-varying VI with respect
+to the sequence (z(t,⋆))1≤t≤T as long as every operator in the sequence (F (1), . . . , F (T)) satisfies
+the MVI property. (Even more broadly, it suffices if almost all operators in the sequence satisfy
+the MVI property—in that their fraction converges to 1 as T → +∞.) This observation enables
+extending Theorem 3.5 beyond time-varying bilinear saddle-point problems.
+A.1.3
+Variation of the Nash Equilibria
+In our next example, we point out that an arbitrarily small change in the entries of the payoff
+matrix can lead to a substantial deviation in the Nash equilibrium.
+Example A.5. Consider a 2 × 2 (two-player) zero-sum game, where X := ∆2, Y := ∆2, described by
+the payoff matrix
+A :=
+�2δ
+0
+0
+δ
+�
+,
+(29)
+for some δ > 0. Then, it is easy to see that the unique Nash equilibrium of this game is such that
+x⋆, y⋆ := ( 1
+3, 2
+3) ∈ ∆2. Suppose now that the original payoff matrix (29) is perturbed to a new
+matrix
+A′ :=
+�δ
+0
+0
+2δ
+�
+.
+(30)
+The new (unique) Nash equilibrium now reads x⋆, y⋆ := ( 2
+3, 1
+3) ∈ ∆2. We conclude that an arbitrarily
+small deviation in the entries of the payoff matrix can lead to a non-trivial change in the Nash
+equilibrium.
+Next, we leverage the simple observation of the example above to establish Proposition 3.3, the
+statement of which is recalled below.
+23
+
+Proposition 3.3. For any T ≥ 4, there is a sequence of T games such that V(T)
+NE ≥ T
+2 while
+V(T)
+ϵ−NE ≤ δ, for any δ > 0.
+Proof. We consider a sequence of T games such that X, Y := ∆2, and
+A(t) =
+�
+A
+if t
+mod 2 = 1,
+A′
+if t
+mod 2 = 0,
+where A, A′ are the payoff matrices defined in (29) and (30), and are parameterized by δ > 0
+(Example A.5). Then, the exact Nash equilibria read
+x(t,⋆), y(t,⋆) =
+�
+( 1
+3, 2
+3)
+if t
+mod 2 = 1,
+( 2
+3, 1
+3)
+if t
+mod 2 = 0.
+As a result, it follows that V(T)
+NE := �T−1
+t=1 ∥z(t+1,⋆) − z(t,⋆)∥2 = 2
+3(T − 1) ≥ T
+2 , for T ≥ 4. In contrast,
+it is clear that V(T)
+ϵ−NE ≤ CδT, which follows by simply considering the sequence of strategies wherein
+both players are always selecting actions uniformly at random; we recall that C > 0 here is the
+value that parameterizes V(T)
+ϵ−NE. Thus, taking δ :=
+δ′
+CT , for some arbitrarily small δ′ > 0, concludes
+the proof.
+A.1.4
+Main Result
+Next, we proceed with the proof of our main result, Theorem 3.5. The key ingredient is Theorem 3.4,
+which bounds the second-order path length of OGD in terms of the considered variation measures.
+We first give the precise statement of Theorem 3.4, and we then proceed with its proof.
+Theorem A.6 (Detailed version of Theorem 3.4). Suppose that both players employ OGD with
+learning rate η ≤
+1
+4L in a time-varying bilinear saddle-point problem, where L := maxt∈[[T]] ∥A(t)∥2.
+Then, for any time horizon T ∈ N,
+T
+�
+t=1
+�
+∥z(t) − ˆz(t)∥2
+2 + ∥z(t) − ˆz(t+1)∥2
+2
+�
+≤ 2D2
+Z + 4η2L2∥Z∥2
+2 + 4DZV(T)
+ϵ−NE + 8η2∥Z∥2
+2V(T)
+A .
+Proof of Theorem 3.4. First, for any t ≥ 2 we have that ∥u(t)
+x − m(t)
+x ∥2
+2 is equal to
+∥A(t)y(t) − A(t−1)y(t−1)∥2
+2 ≤ 2∥A(t)(y(t) − y(t−1))∥2
+2 + 2∥(A(t) − A(t−1))y(t−1)∥2
+2
+(31)
+≤ 2∥A(t)∥2
+2∥y(t) − y(t−1)∥2
+2 + 2∥A(t) − A(t−1)∥2
+2∥y(t−1)∥2
+2
+(32)
+≤ 2L2∥y(t) − y(t−1)∥2
+2 + 2∥Y∥2
+2∥A(t) − A(t−1)∥2
+2,
+(33)
+where (31) uses the triangle inequality for the norm ∥ · ∥2 along with the inequality 2ab ≤ a2 + b2 for
+any a, b ∈ R; (32) follows from the definition of the operator norm; and (33) uses the assumption
+that ∥A(t)∥2 ≤ L and ∥y∥2 ≤ ∥Y∥2 for any y ∈ Y. A similar derivaiton shows that for t ≥ 2,
+∥u(t)
+y − m(t)
+y ∥2
+2 ≤ 2L2∥x(t) − x(t−1)∥2
+2 + 2∥X∥2
+2∥A(t) − A(t−1)∥2
+2.
+(34)
+Further, for t = 1 we have that ∥u(1)
+x
+− m(1)
+x ∥2 = ∥u(1)
+x ∥2 = ∥ − A(1)y(1)∥2 ≤ L∥Y∥2, and
+∥u(1)
+y
+− m(1)
+y ∥2 = ∥u(1)
+y ∥2 = ∥(A(1))⊤x(1)∥2 ≤ L∥X∥2. Next, we will use the following simple
+corollary, which follows similarly to Lemma 3.1.
+24
+
+Corollary A.7. For any sequence s(T)
+z
+:= (z(t,⋆))1≤t≤T , the dynamic regret DReg(T)
+z
+(s(T)
+z
+) :=
+DReg(T)
+x (s(T)
+x ) + DReg(T)
+y
+(s(T)
+y
+) can be bounded by
+D2
+Z
+2η + DZ
+η
+T−1
+�
+t=1
+∥z(t+1,⋆) −z(t,⋆)∥2 +η
+T
+�
+t=1
+∥u(t)
+z −m(t)
+z ∥2
+2 − 1
+2η
+T
+�
+t=1
+�
+∥z(t) − ˆz(t)∥2
+2 + ∥z(t) − ˆz(t+1)∥2
+2
+�
+,
+where m(t)
+z
+:= (m(t)
+x , m(t)
+y ) and u(t)
+z
+:= (u(t)
+x , u(t)
+y ) for any t ∈ [[T]].
+As a result, combining (34) and (33) with Corollary A.7 applied for the dynamic regret
+of both players with respect to the sequence of comparators ((x(t,⋆), y(t,⋆)))1≤t≤T yields that
+DReg(T)
+x (x(1,⋆), . . . , x(T,⋆)) + DReg(T)
+y
+(y(1,⋆), . . . , y(T,⋆)) is upper bounded by
+D2
+Z
+2η + ηL2∥Z∥2
+2 + DZ
+η
+T−1
+�
+t=1
+∥z(t+1,⋆) − z(t,⋆)∥2+2η∥Z∥2
+2V(T)
+A
+− 1
+4η
+T
+�
+t=1
+�
+∥z(t) − ˆz(t)∥2
+2 + ∥z(t) − ˆz(t+1)∥2
+2
+�
+,
+where we used the fact that
+2ηL2
+T
+�
+t=2
+∥z(t) − z(t−1)∥2
+2 − 1
+4η
+T
+�
+t=1
+�
+∥z(t) − ˆz(t)∥2
+2 + ∥z(t) − ˆz(t+1)∥2
+2
+�
+≤
+�
+2ηL2 − 1
+8η
+�
+T
+�
+t=2
+∥z(t) − z(t−1)∥2
+2 ≤ 0,
+for η ≤
+1
+4L. Finally, using the fact that DReg(T)
+x (x(1,⋆), . . . , x(T,⋆)) + DReg(T)
+y
+(y(1,⋆), . . . , y(T,⋆)) ≥
+−2 �T
+t=1 ϵ(t) for a suitable sequence of ϵ(t)-approximate Nash equilibria (Property 3.2)—one that
+attains the variation measure V(T)
+ϵ−NE—yields that
+0 ≤ D2
+Z
+2η + ηL2∥Z∥2
+2 + DZ
+η V(T)
+ϵ−NE + 2η∥Z∥2
+2V(T)
+A
+− 1
+4η
+T
+�
+t=1
+�
+∥z(t) − ˆz(t)∥2
+2 + ∥z(t) − ˆz(t+1)∥2
+2
+�
+,
+where it suffices if the parameter C of V(T)
+ϵ−NE is such that 2 ≤ DZ
+η C. Thus, rearranging the last
+displayed inequality concludes the proof.
+Next, we refine this theorem in time-varying games in which the deviation of the payoff matrices
+is bounded by the deviation of the players’ strategies, in the following formal sense.
+Corollary A.8. Suppose that both players employ OGD with learning rate η ≤ min
+�
+1
+4L,
+1
+8W∥Z∥
+�
+in a
+time-varying bilinear saddle-point problem, where L := maxt∈[[T]] ∥A(t)∥2 and V(T)
+A
+≤ W 2 �T−1
+t=1 ∥z(t+1)−
+z(t)∥2
+2, for some parameter W ∈ R>0. Then, for any time horizon T ∈ N,
+T
+�
+t=1
+�
+∥z(t) − ˆz(t)∥2
+2 + ∥ˆz(t) − ˆz(t+1)∥2
+2
+�
+≤ 4D2
+Z + 8η2L2∥Z∥2
+2 + 8DZV(T)
+NE ,
+where V(T)
+NE is defined in (4).
+25
+
+Proof. Following the proof of Theorem 3.4, we have that for any η ≤
+1
+4L,
+0 ≤ D2
+Z
+2η + ηL2∥Z∥2
+2 + DZ
+η V(T)
+NE + 2η∥Z∥2
+2V(T)
+A
+− 1
+4η
+T
+�
+t=1
+�
+∥z(t) − ˆz(t)∥2
+2 + ∥z(t) − ˆz(t+1)∥2
+2
+�
+.
+Further, for η ≤
+1
+8W∥Z∥2 ,
+2η∥Z∥2
+2V(T)
+A
+− 1
+8η
+T
+�
+t=1
+�
+∥z(t) − ˆz(t)∥2
+2 + ∥z(t) − ˆz(t+1)∥2
+2
+�
+≤
+�
+2η∥Z∥2
+2W 2 −
+1
+16η
+� T−1
+�
+t=1
+∥z(t+1) − z(t)∥2
+2 ≤ 0.
+Thus, we have shown that
+0 ≤ D2
+Z
+2η + ηL2∥Z∥2
+2 + DZ
+η V(T)
+NE − 1
+8η
+T
+�
+t=1
+�
+∥z(t) − ˆz(t)∥2
+2 + ∥z(t) − ˆz(t+1)∥2
+2
+�
+,
+and rearranging concludes the proof.
+Thus, in such time-varying games it is the first-order variation term, V(T)
+NE , that will drive our
+convergence bounds.
+Now before proving Theorem 3.5, we state the connection between the equilibrium gap and the
+deviation of the players’ strategies
+�
+∥z(t) − ˆz(t)∥2 + ∥z(t) − ˆz(t+1)∥2
+�
+. In particular, the following
+claim can be extracted by [Ana+22b, Claim A.14]. (We caution that we use a slightly different
+indexing for the secondary sequence (ˆx(t)
+i ) in the definition of OMD (13) compared to [Ana+22b].)
+Claim A.9. Suppose that the sequences (x(t)
+i )1≤t≤T and (ˆx(t)
+i )1≤t≤T+1 are produced by OMD under
+a G-smooth regularizer 1-strongly convex with respect to a norm ∥ · ∥. Then, for any time t ∈ [[T]]
+and any xi ∈ Xi,
+⟨x(t)
+i , u(t)
+i ⟩ ≥ ⟨xi, u(t)
+i ⟩ − G
+η ∥ˆx(t+1)
+i
+− ˆx(t)
+i ∥ − ∥u(t)
+i ∥∗∥x(t)
+i
+− ˆx(t+1)
+i
+∥.
+We are now ready to prove Theorem 3.5, the precise version of which is stated below.
+Theorem A.10 (Detailed version of Theorem 3.5). Suppose that both players employ OGD with
+learning rate η =
+1
+4L in a time-varying bilinear saddle-point problem, where L := maxt∈[[T]] ∥A(t)∥2.
+Then,
+T
+�
+t=1
+�
+EqGap(t)(z(t))
+�2
+≤ 2L2(4 + ∥Z∥2)2 �
+2D2
+Z + 4η2L2∥Z∥2
+2 + 4DZV(T)
+ϵ−NE + 8η2∥Z∥2
+2V(T)
+A
+�
+,
+where (z(t))1≤t≤T is the sequence of joint strategy profiles produced by OGD.
+26
+
+Proof. Let us first fix a time t ∈ [[T]]. For convenience, we denote by BR(t)
+x (x(t)) := maxx∈X {⟨x, u(t)
+x ⟩}−
+⟨x(t), u(t)
+x ⟩, the best response gap of Player’s x strategy x(t) ∈ X, and similarly for BR(t)
+y (y)(t). By
+definition, it holds that EqGap(t) := max{BR(t)
+x (x(t)), BR(t)
+y (y(t))}. By Claim A.9, we have that
+BR(t)
+x (x(t)) ≤ 1
+η∥ˆx(t+1) − ˆx(t)∥2 + ∥u(t)
+x ∥2∥x(t) − ˆx(t+1)∥2
+(35)
+≤ 4L∥ˆx(t+1) − ˆx(t)∥2 + L∥Y∥2∥x(t) − ˆx(t+1)∥2
+(36)
+≤ L (4 + ∥Z∥2)
+�
+∥x(t) − ˆx(t)∥2 + ∥x(t) − ˆx(t+1)∥2
+�
+,
+(37)
+where (35) follows from Claim A.9 for G = 1 (since the squared Euclidean regularizer φx : x �→ 1
+2∥x∥2
+2)
+is 1-smooth; (36) uses the fact that η :=
+1
+4L and ∥u(t)
+x ∥2 = ∥ − A(t)y(t)∥2 ≤ L∥Y∥; and (37) follows
+from the triangle inequality. A similar derivation shows that
+BR(t)
+y (y(t)) ≤ L(4 + ∥Z∥)
+�
+∥y(t) − ˆy(t)∥2 + ∥y(t) − ˆy(t+1)∥2
+�
+.
+(38)
+Thus,
+T
+�
+t=1
+�
+EqGap(t)(z(t))
+�2
+=
+T
+�
+t=1
+�
+max{BR(t)
+x (x(t)), BR(t)
+y (y(t))}
+�2
+≤
+T
+�
+t=1
+��
+BR(t)
+x (x(t))
+�2
++
+�
+BR(t)
+y (y(t))
+�2�
+≤ 2L2(4 + ∥Z∥2)2
+T
+�
+t=1
+�
+∥z(t) − ˆz(t)∥2
+2 + ∥z(t) − ˆz(t+1)∥2
+2
+�
+,
+(39)
+where the last bound uses (37) and (38). Combining (39) with Theorem A.6 concludes the proof.
+A.1.5
+Variation-Dependent Regret Bounds
+Here we state an important implication of Theorem 3.5 for deriving variation-dependent regret
+bounds in time-varying bilinear saddle-point problems; cf. [Zha+22].
+Corollary A.11 (Detailed version of Corollary 3.7). In the setup of Theorem 3.4, it holds that
+Reg(T)
+x
+≤ D2
+X
+η
++ 8ηL2D2
+Z + ηL2∥Y∥2
+2 + 16η3L4∥Z∥2
+2 + 16ηL2DZV(T)
+NE + (2η∥Y∥2
+2 + 32η3L2∥Z∥2
+2)V(T)
+A ,
+and
+Reg(T)
+y
+≤ D2
+Y
+η
++ 8ηL2D2
+Z + ηL2∥X∥2
+2 + 16η3L4∥Z∥2
+2 + 16ηL2DZV(T)
+NE + (2η∥X∥2
+2 + 32η3L2∥Z∥2
+2)V(T)
+A .
+Proof. First, applying Lemma 3.1 under x(1,⋆) = · · · = x(T,⋆), we have
+Reg(T)
+x
+≤ D2
+X
+η
++ ηL2∥Y∥2
+2 + 2ηL2
+T
+�
+t=2
+∥y(t) − y(t−1)∥2
+2 + 2η∥Y∥2
+2
+T
+�
+t=2
+∥A(t) − A(t−1)∥2
+2,
+(40)
+27
+
+and similarly,
+Reg(T)
+y
+≤ D2
+Y
+η
++ ηL2∥X∥2
+2 + 2ηL2
+T
+�
+t=2
+∥x(t) − x(t−1)∥2
+2 + 2η∥X∥2
+2
+T
+�
+t=2
+∥A(t) − A(t−1)∥2
+2.
+(41)
+Now, by Theorem 3.4 we have
+T
+�
+t=1
+�
+∥z(t) − ˆz(t)∥2
+2 + ∥z(t) − ˆz(t+1)∥2
+2
+�
+≤ 2D2
+Z + 4η2L2∥Z∥2
+2 + 4DZV(T)
+NE + 8η2∥Z∥2
+2V(T)
+A .
+(42)
+Further,
+T
+�
+t=1
+�
+∥z(t) − ˆz(t)∥2
+2 + ∥z(t) − ˆz(t+1)∥2
+2
+�
+≥
+T
+�
+t=1
+�
+∥x(t) − ˆx(t)∥2
+2 + ∥x(t) − ˆx(t+1)∥2
+2
+�
+≥ 1
+2
+T
+�
+t=2
+∥x(t) − x(t−1)∥2
+2.
+Combining this bound with (42) and (41) gives the claimed regret bound on Reg(T)
+y
+, and a similar
+derivation also gives the claimed bound on Reg(T)
+x .
+A.1.6
+Meta-Learning
+We next provide the implication of Theorem 3.5 in the meta-learning setting. We first make a
+remark regarding the effect of the prediction of OGD to Theorem 3.5, and how that relates to an
+assumption present in [Har+22].
+Remark A.12 (Improved predictions). Throughout Section 3.1, we have considered the standard
+prediction m(t)
+x
+:= u(t−1)
+x
+= −A(t−1)y(t−1) for t ≥ 2, and similarly for Player y. It is easy to see
+that using the predictions
+m(t)
+x := −A(t)y(t−1) and m(t)
+y
+:= (A(t))⊤x(t−1)
+(43)
+for t ≥ 1 (where z(0) := ˆz(1)) entirely removes the dependency on V(T)
+A
+on all our convergence
+bounds. While such a prediction cannot be implemented in the standard online learning model,
+there are settings in which we might know the sequence of matrices in advance; the meta-learning
+setting offers such examples, and indeed, Harris et al. [Har+22] use the improved prediction of (43).
+Proposition A.13 (Meta-learning). Suppose that both players employ OGD with learning rate η =
+1
+4L,
+where L := maxh∈[[H]] ∥A(h)∥2, and the prediction of (43) in a meta-learning bilinear saddle-point
+problem with H ∈ N games, each repeated for m ∈ N consecutive iterations. Then, for an average
+game,
+�
+P
+Hϵ2 + P ′V(H)
+NE
+Hϵ2
+�
+(44)
+iterations suffice to reach an ϵ-approximate Nash equilibrium, where P := 4L2(4 + ∥Z∥2)2D2
+Z,
+P ′ := 8L2(4 + ∥Z∥)2DZ, and
+V(H)
+NE :=
+inf
+z(h,⋆)∈Z(h,⋆),∀h∈[[H]]
+H−1
+�
+h=1
+∥z(h+1,⋆) − z(h,⋆)∥2.
+28
+
+The proof is a direct application of Theorem A.10, where we remark that the term depending
+on V(T)
+A
+and the term 4η2L2∥Z∥2
+2 from Theorem A.10 are eliminated because of the improved
+prediction of Remark A.12. The first term in the iteration complexity bound (44) vanishes in
+the meta-learning regime—as the number of games increases H ≫ 1—while the second term is
+proportional to V(H)
+NE
+H , a natural similarity measure; (44) always recovers the m−1/2 rate, but offers
+significant gains if the games as similar, in the sense that V(H)
+NE
+H
+≪ 1. It is worth noting that, unlike
+the similarity measure derived in [Har+22], V(H)
+NE
+H
+depends on the order of the games. We further
+remark that Proposition A.13 can be readily extended even if each game in the meta-learning
+sequence is not repeated for the same number of iterations.
+A.1.7
+General Variational Inequalities
+Although our main focus in this paper is on the convergence of learning algorithms in time-varying
+games, our techniques could also be of interest for solving (static) general variational inequality
+(VI) problems.
+In particular, let F : Z → Z be a single-valued operator. Solving general VIs is well-known to
+be computationally intractable, and so instead focus has been on identifying broad subclasses that
+elude those intractability barriers (see our overview in Section 1.2). Our framework in Section 3.1
+motivates introducing the following measure of complexity for a VI problem:
+C(F) :=
+inf
+z(1,⋆),...,z(T,⋆)∈Z
+T−1
+�
+t=1
+∥z(t+1,⋆) − z(t,⋆)∥2,
+(45)
+subject to
+DReg(T)
+z
+(z(1,⋆), . . . , z(T,⋆)) ≥ 0 ⇐⇒
+T
+�
+t=1
+⟨z(t) − z(t,⋆), F(z(t))⟩ ≥ 0.
+(46)
+In words, (45) expresses the infimum first-order variation that a sequence of comparators must have
+in order to guarantee nonnegative dynamic regret (46); it is evident that (46) always admits a feasible
+sequence, namely s(T)
+z
+:= (z(t))1≤t≤T . We note that, in accordance to our results in Section 3.1, one
+can also consider an approximate version of the complexity measure (45), which could behave much
+more favorably (recall Proposition 3.3).
+Now in a (static) bilinear saddle-point problem, it holds that C(F) = 0 given that there exists a
+static comparator that guarantees nonnegativity of the dynamic regret. More broadly, our techniques
+imply O(poly(1/ϵ)) iteration-complexity bounds for any VI problem such that C(F) ≤ CT 1−ω, for a
+time-independent parameter C > 0 and ω ∈ (0, 1]:
+Proposition A.14. Consider a variational inequality problem described with the operator F : Z → Z
+such that F is L-Lipschitz continuous, in the sense that ∥F(z) − F(z′)∥2 ≤ L∥z − z′∥2, and
+C(F) ≤ CT 1−ω for C > 0 and ω ∈ (0, 1]. Then, OGD with learning rate η =
+1
+4L reaches an ϵ-strong
+solution z⋆ ∈ Z in O(ϵ−2/ω) iterations; that is, ⟨z − z⋆, F(z⋆)⟩ ≥ −ϵ for any z ∈ Z.
+It is worth comparing (45) with another natural complexity measure, namely infz⋆∈Z
+�T
+t=1⟨z(t)−
+z⋆, F(z(t))⟩; the latter measures how negative (external) regret can be, and has already proven
+useful in certain settings that go bilinear saddle-point problems [YM22], although unlike (45), it
+29
+
+does not appear to be useful in characterizing time-varying bilinear saddle-point problems. In this
+context, O(poly(1/ϵ)) iteration-complexity bounds can also be established whenever
+• infz⋆∈Z
+�T
+t=1⟨z(t) − z⋆, F(z(t))⟩ ≥ −CT 1−ω for a time-invariant C > 0, or
+• infz⋆∈Z
+�T
+t=1⟨z(t) − z⋆, F(z(t))⟩ ≥ −C �T−1
+t=1 ∥z(t+1) − z(t)∥2
+2, for a sufficiently small C > 0.
+Following [YM22], identifying VIs that satisfy those relaxed conditions but not the MVI property
+is an interesting direction. In particular, it is important to understand if those relaxations can shed
+led light into the convergence properties of OGD in Shapley’s two-player zero-sum stochastic games.
+A.2
+Proofs from Section 3.2
+In this subsection, we provide the proofs from Section 3.2, leading to our main result in Theorem 3.8.
+Let us first introduce some additional notation. We let f(t) : X × Y → R be a continuously
+differentiable function for any t ∈ [[T]]. We recall that in Section 3.2 it is assumed that the objective
+function changes after m ∈ N (consecutive) repetitions, which is akin to the meta-learning setting.
+Analogously to our setup for bilinear saddle-point problems (Section 3.1), it is assumed that Player
+x is endeavoring to minimizing the objective function, while Player y is trying to maximize it. We
+will denote by Reg(T)
+L,x(x⋆) := �T
+t=1⟨x(t) − x⋆, −u(t)
+x ⟩ and Reg(T)
+L,y(y⋆) := �T
+t=1⟨y⋆ − y(t), u(t)
+y ⟩, where
+u(t)
+x
+:= −∇xf(x(t), y(t)) and u(t)
+y
+:= ∇yf(x(t), y(t)) for any t ∈ [[T]]; similar notation is used for
+DReg(T)
+L,x, DReg(T)
+L,y.
+Furthermore, we let s(T)
+z
+= ((x(t,⋆), y(t,⋆)))1≤t≤T , so that x(t,⋆) = x(h,⋆) and y(t,⋆) = y(h,⋆) for
+any t ∈ [[T]] such that ⌊(t − 1)/m⌋ = h ∈ [[H]]. The first important step in our analysis is that,
+following the proof of Lemma 3.1,
+DReg(T)
+L,x(s(T)
+x ) ≤ 1
+2η
+H
+�
+h=1
+�
+∥ˆx(h,1) − x(h,⋆)∥2
+2 − ∥ˆx(h,m+1) − x(h,⋆)∥2
+2
+�
++ η
+T
+�
+t=1
+∥u(t)
+x − m(t)
+x ∥2
+2
+− 1
+2η
+T
+�
+t=1
+�
+∥x(t) − ˆx(t)∥2
+2 + ∥x(t) − ˆx(t+1)∥2
+2
+�
+,
+(47)
+where ˆx(h,k) := ˆx((h−1)×m)+k) for any (h, k) ∈ [[H]] × [[m]], ˆx(h,m+1) := ˆx(h+1,1) for h ∈ [[H − 1]], and
+ˆx(H,m+1) := ˆx(T+1). Similarly,
+DReg(T)
+L,y(s(T)
+y
+) ≤ 1
+2η
+H
+�
+h=1
+�
+∥ˆy(h,1) − y(h,⋆)∥2
+2 − ∥ˆy(h,m+1) − y(h,⋆)∥2
+2
+�
++ η
+T
+�
+t=1
+∥u(t)
+y − m(t)
+y ∥2
+2
+− 1
+2η
+T
+�
+t=1
+�
+∥y(t) − ˆy(t)∥2
+2 + ∥y(t) − ˆy(t+1)∥2
+2
+�
+.
+(48)
+Next, we will use the following key observation, which lower bounds the sum of the players’
+(external) regrets under strong convexity-concavity.
+Lemma A.15. Suppose that f : X × Y → R is a µ-strongly convex-concave function with respect to
+∥ · ∥2. Then, for any Nash equilibrium z⋆ = (x⋆, y⋆) ∈ Z,
+Reg(m)
+L,x(x⋆) + Reg(m)
+L,y (y⋆) ≥ µ
+2
+m
+�
+t=1
+∥z(t) − z⋆∥2
+2.
+30
+
+Proof. First, by µ-strong convexity of f(x, ·), we have that for any time t ∈ [[m]],
+⟨x(t) − x⋆, ∇xf(x(t), y(t))⟩ ≥ f(x(t), y(t)) − f(x⋆, y(t)) + µ
+2 ∥x(t) − x⋆∥2
+2.
+(49)
+Similarly, by µ-strong concavity of f(·, y), we have that for any time t ∈ [[m]],
+⟨y⋆ − y(t), ∇yf(x(t), y(t))⟩ ≥ f(x(t), y⋆) − f(x(t), y(t)) + µ
+2 ∥y(t) − y⋆∥2
+2.
+(50)
+Further, for any Nash equilibrium (x⋆, y⋆) ∈ Z it holds that f(x(t), y⋆) ≥ f(x(t), y(t)) ≥ f(x⋆, y(t)).
+Combining this fact with (49) and (50) and summing over all t ∈ [[m]] gives the statement.
+In turn, this readily implies the following lower bound for the dynamic regret.
+Lemma A.16. Suppose that f(h) : X × Y → R is a µ-strongly convex-concave function with respect
+to ∥ · ∥2, for any h ∈ [[H]]. Consider a sequence s(T)
+z
+= ((x(t,⋆), y(t,⋆)))1≤t≤T , so that x(t,⋆) = x(h,⋆)
+and y(t,⋆) = y(h,⋆) for any t ∈ [[T]] such that ⌊(t − 1)/m⌋ = h ∈ [[H]]. If (x(h,⋆), y(h,⋆)) ∈ Z is a Nash
+equilibrium of f(h),
+DReg(T)
+L,x(s(T)
+x ) + DReg(T)
+L,y(s(T)
+y
+) ≥ µ
+2
+H
+�
+h=1
+m
+�
+k=1
+∥z(h,k) − z(h,⋆)∥2
+2,
+where z(h,k) := z((h−1)×m)+k) for any (h, k) ∈ [[H]] × [[m]].
+We next combine this with the following monotonicity property of OGD: If z⋆ is a Nash equilibrium,
+∥ˆz(t) − z⋆∥2 is a decreasing function in t [Har+22, Proposition C.10]. This leads to the following
+refinement of Lemma A.16.
+Lemma A.17. Under the assumptions of Lemma A.16, if η ≤
+1
+2µ,
+DReg(T)
+L,x(s(T)
+x ) + DReg(T)
+L,y(s(T)
+y
+) + 1
+4η
+T
+�
+t=1
+∥z(t) − ˆz(t+1)∥2
+2 ≥ µm
+4
+H
+�
+h=1
+∥ˆz(h,m+1) − z(h,⋆)∥2
+2.
+Proof. By Lemma A.16,
+DReg(T)
+L,x(s(T)
+x ) + DReg(T)
+L,y(s(T)
+y
+) + 1
+4η
+T
+�
+t=1
+∥z(t) − ˆz(t+1)∥2
+2 ≥ µ
+2
+H
+�
+h=1
+m
+�
+k=1
+∥z(h,k) − z(h,⋆)∥2
+2
++ 1
+4η
+T
+�
+t=1
+∥z(t) − ˆz(t+1)∥2
+2
+≥ µ
+4
+H
+�
+h=1
+m
+�
+k=1
+∥ˆz(h,k+1) − z(h,⋆)∥2
+2
+(51)
+≥ µm
+4
+H
+�
+h=1
+∥ˆz(h,m+1) − z(h,⋆)∥2
+2,
+(52)
+where (51) uses that
+1
+4η ≥ µ
+2 along with Young’s inequality and triangle inequality, and (52) follows
+from [Har+22, Proposition C.10].
+31
+
+Armed with this important lemma, we are ready to establish our main result (Theorem 3.8), the
+detailed version of which is given below. We first point out that a function f : X × Y → R is said
+to be L-smooth if ∥F(z) − F(z′)∥2 ≤ L∥z − z′∥2, where F(z) := (∇xf(x, y), −∇yf(x, y)).
+Theorem A.18 (Detailed version of Theorem 3.8). Let f(h) : X × Y be a µ-strongly convex-concave
+and L-smooth function, for h ∈ [[H]]. Suppose that both players employ OGD with learning rate
+η = min
+�
+1
+8L, 1
+2µ
+�
+for T repetitions, where T = m × H and m ≥
+2
+ηµ. Then,
+T
+�
+t=1
+�
+∥z(t) − ˆz(t)∥2
+2 + ∥z(t) − ˆz(t+1)∥2
+2
+�
+≤ 4D2
+Z + 8η2∥F(z(1))∥2
+2 + 8S(H)
+NE + 16η2V(H)
+∇f ,
+where S(H)
+NE and V(H)
+∇f are defined in (9) and (11). Thus, �T
+t=1
+�
+EqGap(t)(z(t))
+�2
+= O(1 + S(H)
+NE +
+V(H)
+∇f ).
+Proof. Combining Lemma A.17 with (47) and (48),
+0 ≤ 1
+2η
+H
+�
+h=1
+�
+∥ˆz(h,1) − z(h,⋆)∥2
+2 − 2 + ηµm
+2
+∥ˆz(h,m+1) − z(h,⋆)∥2
+2
+�
++η
+T
+�
+t=1
+∥u(t)
+z − m(t)
+z ∥2
+2 − 1
+4η
+T
+�
+t=1
+�
+∥z(t) − ˆz(t)∥2
+2 + ∥z(t) − ˆz(t+1)∥2
+2
+�
+,
+for a sequence of Nash equilibria (z(h,⋆))1≤h≤H, where we used the notation u(t)
+z
+:= (u(t)
+x , u(t)
+y ) and
+m(t)
+z
+:= (m(t)
+x , m(t)
+y ). Now we bound the first term of the right-hand side above as
+1
+2η
+H
+�
+h=1
+�
+∥ˆz(h,1) − z(h,⋆)∥2
+2 − 2∥ˆz(h,m+1) − z(h,⋆)∥2
+2
+�
+≤
+1
+2η∥ˆz(1,1) − z(1,⋆)∥2
+2 + 1
+2η
+H−1
+�
+h=1
+�
+∥ˆz(h+1,1) − z(h+1,⋆)∥2
+2 − 2∥ˆz(h+1,1) − z(h,⋆)∥2
+2
+�
+,
+where we used the fact that m ≥
+2
+ηµ and ˆz(h,m+1) = ˆz(h+1,1), for h ∈ [[H − 1]]. Hence, continuing
+from above,
+1
+2η
+H
+�
+h=1
+�
+∥ˆz(h,1) − z(h,⋆)∥2
+2 − 2∥ˆz(h,m+1) − z(h,⋆)∥2
+2
+�
+≤ 1
+2η∥ˆz(1,1)−z(1,⋆)∥2
+2+ 1
+η
+H−1
+�
+h=1
+∥z(h+1,⋆)−z(h,⋆)∥2
+2,
+since ∥ˆz(h+1,1) − z(h+1,⋆)∥2
+2 ≤ 2∥ˆz(h+1,1) − z(h,⋆)∥2
+2 + 2∥z(h,⋆) − z(h+1,⋆)∥2
+2, by the triangle inequality
+and Young’s inequality. Moreover, for t ≥ 2,
+∥u(t)
+z −u(t−1)
+z
+∥2
+2 = ∥F (t)(z(t))−F (t−1)(z(t−1))∥2
+2 ≤ 2L2∥z(t)−z(t−1)∥2
+2+2∥F (t)(z(t−1))−F (t−1)(z(t−1))∥2
+2,
+by L-smoothness. As a result,
+T−1
+�
+t=1
+∥u(t+1)
+z
+− u(t)
+z ∥2
+2 ≤ 2L2
+T−1
+�
+t=1
+∥z(t) − z(t−1)∥2
+2 + 2V(T)
+∇f ,
+and the claimed bound on the second-order path length follows. Finally, the second claim of the
+theorem follows from Claim A.9 using convexity-concavity, analogously to Theorem 3.5.
+32
+
+We point out that the improved prediction mechanism described in Remark A.12 can also be used
+in this setting as well, resulting in the elimination of the variation measure (11) from Theorem A.18.
+We conclude this subsection by pointing out an improved variation-dependent regret bound, which
+follows directly from Theorem A.18 (cf. Corollary A.11).
+Corollary A.19. In the setup of Theorem A.18,
+Reg(T)
+L,x, Reg(T)
+L,y ≤ D2
+Z
+η
++ 4ηL2D2
+Z + 32η3L2∥F(z(1)∥2
+2 + 32ηL2S(H)
+NE + (64η3L2 + 2η)V(H)
+∇f .
+Thus, setting the learning rate optimally implies that Reg(T)
+L,x, Reg(T)
+L,y = O
+��
+S(H)
+NE + V(H)
+∇f
+�
+.
+A.3
+Proofs from Section 3.3
+In this subsection, we provide the proofs from Section 3.3.
+A.3.1
+Potential Games
+We first characterize the behavior of GD in time-varying potential games. Below we give the formal
+definition of an unweighted potential game, represented in normal form.
+Definition A.20 (Potential game). A game admits a potential if there exists a function Φ :
+×
+n
+i=1 Xi → R such that for any Player i ∈ [[n]], any joint strategy profile x−i ∈×i′̸=i Xi′, and any
+pair of strategies xi, x′
+i ∈ Xi,
+Φ(xi, x−i) − Φ(x′
+i, x−i) = ui(xi, x−i) − ui(x′
+i, x−i).
+We also recall that GD is equivalent to OGD under the prediction m(t)
+x
+= 0 for all t. The key
+ingredient in the proof of Theorem 3.10 is the following key bound on the second-order path length
+of the dynamics.
+Proposition A.21. Suppose that each player employs GD with a sufficiently small learning rate
+η > 0 and initialization (x(1)
+1 , . . . , x(1)
+n ) ∈×
+n
+i=1 Xi. Then,
+1
+2η
+T
+�
+t=1
+n
+�
+i=1
+∥x(t+1)
+i
+− x(t)
+i ∥2
+2 ≤
+T
+�
+t=1
+�
+Φ(t)(x(t+1)
+1
+, . . . , x(t+1)
+n
+) − Φ(t)(x(t)
+1 , . . . , x(t)
+n )
+�
+.
+(53)
+This bound can be derived from [Ana+22b, Theorem 4.3]. We note that if Φ(1) = Φ(2) = · · · =
+Φ(T), the right-hand side of (53) telescops, thereby implying that the second-order path-length is
+bounded. More generally, the right-hand side of (53) can be upper bounded by
+2Φmax +
+T−1
+�
+t=1
+�
+Φ(t)(x(t+1)
+1
+, . . . , x(t+1)
+n
+) − Φ(t+1)(x(t+1)
+1
+, . . . , x(t+1)
+n
+)
+�
+≤ 2Φmax + V(T)
+Φ ,
+(54)
+where Φmax is an upper bound on |Φ(t)(·)| for any t ∈ [[T]], and V(T)
+Φ
+is the variation measure of the
+potential functions we introduced in Section 3.3. Furthermore, we know that the Nash equilibrium
+gap in the t-th potential game can be bounded in terms of �n
+i=1 ∥x(t+1)
+i
+− x(t)
+i ∥2 (Claim A.9). As a
+result, combining this property with Proposition A.21 and (54) establishes Theorem 3.10.
+33
+
+A.3.2
+General-Sum Games
+We next turn out attention to general-sum multi-player games using the bilinear formulation
+presented in Section 3.3. To establish Property 3.11, let us first define the regret of any Player
+i ∈ [[n]] as
+Reg(T)
+i
+(¯x⋆
+i ) :=
+T
+�
+t=1
+⟨¯x⋆
+i − ¯x(t)
+i , (A(t)
+i )⊤µ(t)⟩,
+where ¯x⋆
+i ∈ ¯Xi, so that �n
+i=1 Reg(T)
+i
+is easily seen to be equal to the regret of the maximizing
+player in (12). Further, the dynamic regret of the mediator—the minimizing player in (12)—can be
+expressed as
+DReg(T)
+µ (µ(1,⋆), . . . , µ(T,⋆)) :=
+T
+�
+t=1
+⟨µ(t) − µ(t,⋆),
+n
+�
+i=1
+A(t)
+i ¯x(t)
+i ⟩.
+Property 3.11. Suppose that Ξ ∋ µ(t,⋆) is a correlated equilibrium of the game at any time t ∈ [[T]].
+Then,
+DReg(T)
+µ (µ(1,⋆), . . . , µ(T,⋆)) +
+n
+�
+i=1
+Reg(T)
+i
+≥ 0.
+Proof. We have that
+DReg(T)
+µ
++
+n
+�
+i=1
+Reg(T)
+i
+(¯x⋆
+i ) =
+n
+�
+i=1
+T
+�
+t=1
+⟨¯x⋆
+i , (A(t)
+i )⊤µ(t)⟩ −
+T
+�
+t=1
+⟨µ(t,⋆),
+n
+�
+i=1
+A(t)
+i ¯x(t)
+i ⟩.
+Now for any correlated equilibrium µ(t,⋆) of the t-th game we have that ⟨µ(t,⋆), A(t)
+i ¯x(t)
+i ⟩ ≤ 0 for any
+Player i ∈ [[n]], ¯xi ∈ ¯
+Xi, and time t ∈ [[T]], which in turn implies that − �T
+t=1⟨µ(t,⋆), �n
+i=1 A(t)
+i ¯x(t)
+i ⟩ ≥
+0. Moreover, �n
+i=1 max¯x⋆
+i ∈ ¯
+Xi
+�T
+t=1⟨¯x⋆
+i , (A(t)
+i )⊤µ(t)⟩ ≥ 0 given that, by definition, 0 ∈ ¯
+Xi. This
+concludes the proof.
+Next, we provide the main implication of Theorem 3.12 in the meta-learning setting, which
+is similar to the meta-learning guarantee of Proposition A.13 we established earlier in two-player
+zero-sum games. Below, we denote by Ξ(h,⋆) the set of correlated equilibria of the h-th game in the
+meta-learning sequence.
+Corollary A.22 (Meta-learning general-sum). Suppose that each player employ OGD in (12) with a
+suitable learning rate η > 0 and the prediction of (43) in a meta-learning general-sum problem with
+H ∈ N games, each repeated for m ∈ N consecutive iterations. Then, for an average game,
+O
+�
+1
+ϵ2H + V(H)
+CE
+ϵ2H
+�
+(55)
+iterations suffice to reach an ϵ-approximate correlated equilibrium, where
+V(H)
+CE :=
+inf
+µ(h,⋆)∈Ξ(h,⋆) ∥µ(h+1,⋆) − µ(h,⋆)∥2.
+34
+
+In particular, in the meta-learning regime, H ≫ 1, the iteration-complexity bound (55) is
+dominated by the (algorithm-independent) similarity metric of the correlated equilibria
+V(H)
+CE
+H .
+Corollary A.22 establishes significant gains when V(H)
+CE
+H
+≪ 1.
+Finally, we conclude this subsection by providing a variation-dependent regret bound in general-
+sum multi-player games. To do so, we combine Corollary 3.7 with Theorem 3.12, leading to the
+following guarantee.
+Corollary A.23 (Regret in general-sum games). In the setup of Theorem 3.12,
+Reg(T)
+µ , Reg(T)
+i
+= O
+�1
+η + η
+�
+V(T)
+CE + V(T)
+A
+��
+,
+for any Player i ∈ [[n]].
+In particular, if one selects optimally the learning rate, Corollary A.23 implies that the individual
+regret of each player is bounded by O
+��
+V(T)
+CE + V(T)
+A
+�
+. We note again that there are techniques
+that would allow (nearly) recovering such regret guarantees without having to know the variation
+measures in advance [Zha+22].
+A.4
+Proofs from Section 3.4
+Finally, in this subsection we present the proofs omitted from Section 3.4. We begin with Proposi-
+tion 3.13, the statement of which is recalled below. We first recall that a regularizer φx, 1-strongly
+convex with respect to a norm ∥ · ∥, is said to be G-smooth if ∥∇φx(x) − ∇φx(x′)∥∗ ≤ G∥x − x′∥,
+for all x, x′.
+Proposition 3.13. Suppose that both players in a (static) two-player zero-sum game employ OMD
+with a smooth regularizer. Then, DReg(T)
+x , DReg(T)
+y
+= O(
+√
+T).
+Proof. First, using Claim A.9, it follows that the dynamic regret DReg(T)
+x
+of Player x up to time T
+can be bounded as
+T
+�
+t=1
+�
+max
+x(t,⋆)∈X
+�
+⟨x(t,⋆), u(t)
+x ⟩
+�
+− ⟨x(t), u(t)
+x ⟩
+�
+≤
+T
+�
+t=1
+� �G
+η + ∥u(t)
+x ∥∗
+�
+∥x(t) − ˆx(t+1)∥ + G
+η ∥x(t) − ˆx(t)∥
+�
+,
+(56)
+where G > 0 is the smoothness parameter of the regularizer, and η > 0 is the learning rate. We further
+know that �T
+t=1
+�
+∥x(t) − ˆx(t)∥2 + ∥x(t) − ˆx(t+1)���2�
+= O(1) for any instance of OMD in a two-player
+zero-sum game [Ana+22b], which in turn implies that �T
+t=1
+�
+∥x(t) − ˆx(t)∥ + ∥x(t) − ˆx(t+1)∥
+�
+=
+O(
+√
+T) by Cauchy-Schwarz. Thus, combining with (56) we have shown that DReg(T)
+x
+= O(
+√
+T).
+Similar reasoning yields that DReg(T)
+y
+= O(
+√
+T), concluding the proof.
+In contrast, we next show that such a result is precluded in general-sum games. In particular,
+we note that the following computational-hardness result holds beyond the online learning setting.
+35
+
+It should be stressed that without imposing computational or memory restrictions there are trivial
+online algorithms that guarantee even O(1) dynamic regret by first exploring the payoff matrices
+and then computing a Nash equilibrium; we suspect that under the memory limitations imposed in
+our work, as in [DDK11], there could be unconditional information-theoretic lower bounds, but that
+is left for future work.
+Proposition 3.15. Unless PPAD ⊆ P, any polynomial-time algorithm incurs �n
+i=1 DReg(T)
+i
+= Ω(T),
+even if n = 2, where Ω(·) here hides polynomial factors.
+Proof. We will use the fact that computing a Nash equilibrium in two-player (normal-form) games
+to a sufficiently small accuracy ϵ := 1/poly is PPAD-hard [CDT09]. Indeed, suppose that there exist
+polynomial-time algorithms that always guarantee that �n
+i=1 DReg(T)
+i
+≤ ϵT, where n := 2. Then,
+this implies that there exists a time t ∈ [[T]] such that
+max
+x(t,⋆)
+1
+∈X1
+⟨x(t,⋆)
+1
+, u(t)
+1 ⟩ − ⟨x(t)
+1 , u(t)
+1 ⟩ +
+max
+x(t,⋆)
+2
+∈X2
+⟨x(t,⋆)
+2
+, u(t)
+2 ⟩ − ⟨x(t)
+2 , u(t)
+2 ⟩ ≤ ϵ,
+which in turn implies that (x(t)
+1 , x(t)
+2 ) is an ϵ-approximate Nash equilibrium. Further, such a time
+t ∈ [[T]] can be identified in polynomial time. But this would imply that PPAD ⊆ P, concluding the
+proof.
+Finally, we provide the proof of Theorem 3.16, the detailed version of which is provided below.
+Theorem A.24 (Detailed version of Theorem 3.16). Consider an n-player game such that ∥∇xiui(z)−
+∇xiui(z′)∥2 ≤ L∥z − z′∥2, where z, z′ ∈ ×
+n
+i=1 Xi, for any Player i ∈ [[n]]. Then, if all players
+employ OGD with learning rate η > 0 it holds that
+1. �n
+i=1 K-DReg(T)
+i
+= O(K√nL) for η = Θ
+�
+1
+L√n
+�
+;
+2. K-DReg(T)
+i
+= O(K3/4T 1/4n1/4√
+L), for any Player i ∈ [[n]], for η = Θ
+�
+K1/4
+n1/4L1/2
+�
+.
+Proof. First, applying Lemma 3.1 subject to the constraint that �T−1
+t=1 1{x(t+1,⋆) ̸= x(t,⋆)} ≤ K − 1
+gives that for any Player i ∈ [[n]],
+K-DReg(T)
+i
+≤ D2
+Xi
+2η (2K − 1) + η∥u(1)
+i ∥2
+2 + η
+T−1
+�
+t=1
+∥u(t+1)
+i
+− u(t)
+i ∥2
+2 − 1
+4η
+T−1
+�
+t=1
+∥x(t+1)
+i
+− x(t)
+i ∥2
+2.
+(57)
+Further, by L-smoothness we have that
+∥u(t+1)
+i
+− u(t)
+i ∥2
+2 = ∥∇xiui(z(t+1)) − ∇xiui(z(t))∥2
+2 ≤ L2
+n
+�
+i=1
+∥x(t+1)
+i
+− x(t)
+i ∥2
+2,
+for any t ∈ [[T − 1]], where (x(t)
+1 , . . . , x(t)
+n ) = z(t) ∈×
+n
+i=1 Xi is the joint strategy profile at time
+t. Thus, summing (57) over all i ∈ [[n]] and taking η ≤
+1
+2L√n implies that �n
+i=1 K-DReg(T)
+i
+≤
+2K−1
+2η
+�n
+i=1 D2
+Xi + η �n
+i=1 ∥u(1)
+i ∥2
+2, yielding the first part of the statement. The second part follows
+directly from (57) using the stability property of OGD: ∥x(t+1)
+i
+− x(t)
+i ∥2 = O(η), for any time
+t ∈ [[T − 1]].
+36
+
+B
+Experimental Examples
+Finally, although the focus of this paper is theoretical, in this section we provide some illustrative
+experimental examples. In particular, Appendix B.1 contains experiments on time-varying potential
+games, while Appendix B.2 focuses on time-varying (two-player) zero-sum games. For simplicity, we
+will be assuming that each game is represented in normal form.
+B.1
+Time-Varying Potential Games
+Here we consider time-varying 2-player identical-interest games. We point out that such games
+are potential games (recall Definition A.20), and as such they are indeed amenable to our theory
+in Section 3.3.
+In our first experiment, we first sampled two matrices A, P ∈ Rdx×dy, where dx = dy = 1000.
+Then, we defined each payoff matrix as A(t) := A(t−1) + Pt−α for t ≥ 1, where A(0) := A. Here,
+α > 0 is a parameter that controls the variation of the payoff matrices. In this time-varying setup,
+we let each player employ (online) GD with learning rate η := 0.1. The results obtained under
+different random initializations of matrices A and P are illustrated in Figure 1.
+Next, we operate in the same time-varying setup but each player is now employing multiplicative
+weights update (MWU), instead of gradient descent, with η := 0.1. As shown in Figure 2, while the
+cumulative equilibrium gap is much larger compared to using GD (Figure 1), the dynamics still
+appear to be approaching equilibria, although our theory does not cover MWU. We suspect that
+theoretical results such as Theorem 3.10 should hold for MWU as well, but that has been left for
+future work.
+In our third experiment for identical-interest games, we again first sampled two matrices
+A, P ∈ Rdx×dy, where dx = dy = 1000. Then, we defined A(t) := A(t−1) + ϵP for t ≥ 1, where
+A(0) := A. Here, ϵ > 0 is the parameter intended to capture the variation of the payoff matrices.
+The results obtained under different random initializations of A and P are illustrated in Figure 3.
+As an aside, it is worth pointing out that this particular setting can be thought of as a game in
+which the variation in the payoff matrices is controlled by another learning agent. In particular, our
+theoretical results are helpful for characterizing the convergence properties of two-timescale learning
+algorithms, in which the deviation of the game is controlled by a player constrained to be updating
+its strategies with a much smaller learning rate.
+B.2
+Time-Varying Zero-Sum Games
+We next conduct experiments on time-varying bilinear saddle-point problems when players are
+employing OGD. Such problems have been studied extensively in Section 3.1 from a theoretical
+standpoint.
+First, we sampled two matrices A, P ∈ Rdx×dy, where dx = dy = 10; here we consider lower-
+dimensional payoff matrices compared to the experiments in Appendix B.1 for convenience in the
+graphical illustrations. Then, we defined each payoff matrix as A(t) := A(t−1) + Pt−α for t ≥ 1,
+where A(1) := A. The results obtained under different random initializations are illustrated in
+Figure 4.
+37
+
+0
+50
+100
+150
+200
+0
+2
+4
+�t
+τ=1(EG(τ))2
+0
+50
+100
+150
+200
+0.0
+0.2
+0.4
+0.6
+EG(t)
+α = 0.5
+α = 0.2
+α = 0.1
+0
+50
+100
+150
+200
+0
+2
+4
+6
+max(Reg(t)
+x , Reg(t)
+y )
+0
+50
+100
+150
+200
+0
+1
+2
+3
+�t
+τ=1(EG(τ))2
+0
+50
+100
+150
+200
+0.0
+0.2
+0.4
+0.6
+EG(t)
+α = 0.5
+α = 0.2
+α = 0.1
+0
+50
+100
+150
+200
+0
+2
+4
+6
+max(Reg(t)
+x , Reg(t)
+y )
+0
+50
+100
+150
+200
+0
+1
+2
+3
+4
+�t
+τ=1(EG(τ))2
+0
+50
+100
+150
+200
+0.0
+0.2
+0.4
+0.6
+0.8
+EG(t)
+α = 0.5
+α = 0.2
+α = 0.1
+0
+50
+100
+150
+200
+0
+2
+4
+6
+max(Reg(t)
+x , Reg(t)
+y )
+0
+50
+100
+150
+200
+Iteration (t)
+0
+2
+4
+�t
+τ=1(EG(τ))2
+0
+50
+100
+150
+200
+Iteration (t)
+0.0
+0.2
+0.4
+0.6
+0.8
+EG(t)
+α = 0.5
+α = 0.2
+α = 0.1
+0
+50
+100
+150
+200
+Iteration (t)
+0
+2
+4
+6
+max(Reg(t)
+x , Reg(t)
+y )
+Figure 1: The equilibrium gap and the players’ regrets in 2-player time-varying identical-interest
+games when both players are employing (online) GD with learning rate η := 0.1 for T := 200 iterations.
+Each row corresponds to a different random initialization of the matrices A, P ∈ Rdx×dy, which in
+turn induces a different time-varying game. Further, each figure contains trajectories corresponding
+to three different values of α ∈ {0.1, 0.2, 0.5}, but under the same initialization of A and P. As
+expected, smaller values of α generally increase the equilibrium gap since the variation of the games
+is more significant. Nevertheless, for all games we observe that the players are gradually approaching
+equilibria.
+38
+
+0
+50
+100
+150
+200
+0
+100
+200
+�t
+τ=1(EG(τ))2
+0
+50
+100
+150
+200
+0
+2
+4
+EG(t)
+α = 0.5
+α = 0.2
+α = 0.1
+0
+20
+40
+60
+80
+100
+0
+2
+4
+6
+max(Reg(t)
+x , Reg(t)
+y )
+0
+50
+100
+150
+200
+0
+100
+200
+300
+�t
+τ=1(EG(τ))2
+0
+50
+100
+150
+200
+0
+2
+4
+EG(t)
+α = 0.5
+α = 0.2
+α = 0.1
+0
+20
+40
+60
+80
+100
+0
+2
+4
+6
+max(Reg(t)
+x , Reg(t)
+y )
+0
+50
+100
+150
+200
+0
+100
+200
+�t
+τ=1(EG(τ))2
+0
+50
+100
+150
+200
+0
+2
+4
+EG(t)
+α = 0.5
+α = 0.2
+α = 0.1
+0
+20
+40
+60
+80
+100
+0
+2
+4
+6
+max(Reg(t)
+x , Reg(t)
+y )
+0
+50
+100
+150
+200
+Iterations
+0
+100
+200
+300
+�t
+τ=1(EG(τ))2
+0
+50
+100
+150
+200
+Iterations
+0
+2
+4
+EG(t)
+α = 0.5
+α = 0.2
+α = 0.1
+0
+20
+40
+60
+80
+100
+Iterations
+0
+2
+4
+6
+max(Reg(t)
+x , Reg(t)
+y )
+Figure 2: The equilibrium gap and the players’ regrets in 2-player time-varying identical-interest
+games when both players are employing (online) GD with learning rate η := 0.1 for T := 200 iterations.
+Each row corresponds to a different random initialization of the matrices A, P ∈ Rdx×dy, which in
+turn induces a different time-varying game. Further, each figure contains trajectories corresponding
+to three different values of α ∈ {0.1, 0.2, 0.5}, but under the same initialization of A and P. The
+MWU dynamics still appear to be approaching equilibria, although the cumulative gap is much larger
+compared to GD (Figure 1).
+39
+
+0
+100
+200
+300
+400
+500
+0.0
+0.5
+1.0
+�t
+τ=1(EG(τ))2
+0
+100
+200
+300
+400
+500
+0.0
+0.1
+0.2
+EG(t)
+ϵ = 0.001
+ϵ = 0.01
+ϵ = 0.1
+0
+50
+100
+150
+200
+0
+2
+4
+6
+max(Reg(t)
+x , Reg(t)
+y )
+0
+100
+200
+300
+400
+500
+0.0
+0.5
+1.0
+�t
+τ=1(EG(τ))2
+0
+100
+200
+300
+400
+500
+0.0
+0.1
+0.2
+0.3
+EG(t)
+ϵ = 0.001
+ϵ = 0.01
+ϵ = 0.1
+0
+50
+100
+150
+200
+0
+2
+4
+6
+max(Reg(t)
+x , Reg(t)
+y )
+0
+100
+200
+300
+400
+500
+0
+1
+2
+�t
+τ=1(EG(τ))2
+0
+100
+200
+300
+400
+500
+0.0
+0.1
+0.2
+0.3
+EG(t)
+ϵ = 0.001
+ϵ = 0.01
+ϵ = 0.1
+0
+50
+100
+150
+200
+0
+2
+4
+6
+max(Reg(t)
+x , Reg(t)
+y )
+0
+100
+200
+300
+400
+500
+Iterations
+0
+1
+2
+3
+�t
+τ=1(EG(τ))2
+0
+100
+200
+300
+400
+500
+Iterations
+0.0
+0.1
+0.2
+EG(t)
+ϵ = 0.001
+ϵ = 0.01
+ϵ = 0.1
+0
+50
+100
+150
+200
+Iterations
+0
+2
+4
+6
+max(Reg(t)
+x , Reg(t)
+y )
+Figure 3: The equilibrium gap and the players’ regrets in 2-player time-varying identical-interest
+games when both players are employing (online) GD with learning rate η := 0.1 for T := 500 iterations.
+Each row corresponds to a different random initialization of the matrices A, P ∈ Rdx×dy, which in
+turn induces a different time-varying game. Further, each figure contains trajectories from three
+different values of ϵ ∈ {0.1, 0.01, 0.001}, but under the same initialization of A and P. As expected,
+larger values of ϵ generally increase the equilibrium gap since the variation of the games is more
+significant. Yet, even for the larger value ϵ = 0.1, the dynamics are still appear to be approaching
+Nash equilibria.
+40
+
+0
+200
+400
+600
+800
+1000
+0
+10000
+20000
+30000
+�t
+τ=1(EG(τ))2
+0
+200
+400
+600
+800
+1000
+0.0
+0.2
+0.4
+0.6
+0.8
+EG(t)
+α = 2
+α = 1
+α = 0.7
+0
+200
+400
+600
+800
+1000
+0
+5
+10
+15
+20
+max(Reg(t)
+x , Reg(t)
+y )
+0
+200
+400
+600
+800
+1000
+0
+2500
+5000
+7500
+10000
+�t
+τ=1(EG(τ))2
+0
+200
+400
+600
+800
+1000
+0.0
+0.2
+0.4
+0.6
+0.8
+EG(t)
+α = 2
+α = 1
+α = 0.7
+0
+200
+400
+600
+800
+1000
+0
+10
+20
+30
+40
+max(Reg(t)
+x , Reg(t)
+y )
+0
+200
+400
+600
+800
+1000
+0
+20000
+40000
+�t
+τ=1(EG(τ))2
+0
+200
+400
+600
+800
+1000
+0.0
+0.2
+0.4
+0.6
+0.8
+EG(t)
+α = 2
+α = 1
+α = 0.7
+0
+200
+400
+600
+800
+1000
+0
+10
+20
+30
+max(Reg(t)
+x , Reg(t)
+y )
+0
+200
+400
+600
+800
+1000
+Iteration (t)
+0
+2500
+5000
+7500
+10000
+�t
+τ=1(EG(τ))2
+0
+200
+400
+600
+800
+1000
+Iteration (t)
+0.0
+0.2
+0.4
+0.6
+EG(t)
+α = 2
+α = 1
+α = 0.7
+0
+200
+400
+600
+800
+1000
+Iteration (t)
+0
+10
+20
+max(Reg(t)
+x , Reg(t)
+y )
+Figure 4: The equilibrium gap and the players’ regrets in 2-player time-varying zero-sum games
+when both players are employing OGD with learning rate η := 0.01 and T := 1000 iterations. Each
+row corresponds to a different random initialization of the matrices A, P ∈ Rdx×dy, which in turn
+induces a different time-varying game. Further, each figure contains trajectories from three different
+values of α ∈ {0.7, 1, 2}, but under the same initialization of A and P. The OGD dynamics appear
+to be approaching equilibria, albeit with a much slower rate compared to the ones observed earlier
+for potential games (Figure 1).
+41
+