Accelerated Regularized Learning in Finite N-Person Games

Authors: kyriakos  lotidis
angeliki  giannou
panayotis  mertikopoulos
nicholas  bambos
Venue: NeurIPS_cc_2024_Conference
Published: 2024-05-15 07:58:01

Abstract: Motivated by the success of Nesterov's accelerated gradient algorithm for convex minimization problems, we examine whether it is possible to achieve similar performance gains in the context of online learning in games.
To that end, we introduce a family of accelerated learning methods, which we call “follow the accelerated leader” (FTXL), and which incorporates the use of momentum within the general framework of regularized learning - and, in particular, the exponential / multiplicative weights algorithm and its variants.
Drawing inspiration and techniques from the continuous-time analysis of Nesterov's algorithm, we show that FTXL converges locally to strict Nash equilibria at a superlinear rate, achieving in this way an exponential speed-up over vanilla regularized learning methods (which, by comparison, converge to strict equilibria at a geometric, linear rate).
Importantly, the FTXL maintains its superlinear convergence rate in a broad range of feedback structures, from deterministic, full information models to stochastic, realization-based ones, and even bandit, payoff-based information, where players are only able to observe their individual realized payoffs.

Main paper: 
 introduction: 
 One of the most important milestones in convex optimization was Nesterov's accelerated gradient (NAG) algorithm, as proposed by Nesterov [28] in 1983. The groundbreaking achievement of Nesterov's algorithm was that it attained an O(1/𝑇 2 ) rate of convergence in Lipschitz smooth convex minimization problems, thus bridging a decades-old gap between the O(1/𝑇) convergence rate of ordinary gradient descent and the corresponding Ω(1/𝑇 2 ) lower bound for said class [27] . In this way, Nesterov's accelerated gradient algorithm opened the door to acceleration in optimization, leading in turn to a wide range of other, likewise influential schemes -such as FISTA and its variants [2] -and jumpstarting a vigorous field of research that remains extremely active to this day. Somewhat peculiarly, despite the great success that NAG has enjoyed in all fields where optimization plays a major role -and, in particular, machine learning and data science -its use has not percolated to the adjoining field of game theory as a suitable algorithm for learning Nash equilibria. Historically, the reasons for this are easy to explain: despite intense scrutiny by the community and an extensive corpus of literature dedicated to deconstructing the algorithm's guarantees, NAG's update structure remains quite opaque -and, to a certain extent, mysterious. Because of this, Nesterov's algorithm could not be considered as a plausible learning scheme that could be employed by boundedly rational human agents involved in a repeated game. Given that this was the predominant tenet in economic thought at the time, the use of Nesterov's algorithm in a game-theoretic context has not been extensively explored, to the best of our knowledge.On the other hand, as far as applications to machine learning and artificial intelligence are concerned, the focus on human agents is no longer a limiting factor. In most current and emerging applications of game-theoretic learning -from multi-agent reinforcement learning to adversarial models in machine learning -the learning agents are algorithms whose computational capacity is only limited by the device on which they are deployed. In view of this, our paper seeks to answer the following question: Can Nesterov's accelerated gradient scheme be deployed in a game-theoretic setting? And, if so, is it possible to achieve similar performance gains as in convex optimization?Our contributions in the context of related work. The answer to the above questions is not easy to guess. On the one hand, given that game theory and convex optimization are fundamentally different fields, a reasonable guess would be "no" -after all, finding a Nash equilibrium is a PPAD-complete problem [6] , whereas convex minimization problems are solvable in polynomial time [4] . On the other, since in the context of online learning each player would have every incentive to use the most efficient unilateral optimization algorithm at their disposal, the use of NAG methods cannot be easily discarded from an algorithmic viewpoint.Our paper examines if it is possible to obtain even a partially positive answer to the above question concerning the application of Nesterov's accelerated gradients techniques to learning in games. We focus throughout on the class of finite 𝑁-person games where, due to the individual concavity of the players' payoff functions, the convergence landscape of online learning in games is relatively well-understood -at least, compared to non-concave games. In particular, it is known that regularized learning algorithms -such as "follow the regularized leader" (FTRL) and its variants -converge locally to strict Nash equilibria at a geometric rate [15] , and strict equilibria are the only locally stable and attracting limit points of regularized learning in the presence of randomness and/or uncertainty [8, 14, 19] . In this regard, we pose the question of (i) whether regularized learning schemes like FTRL can be accelerated; and (ii) whether the above properties are enhanced by this upgrade.We answer both questions in the positive. First, we introduce an accelerated regularized scheme, in both continuous and discrete time, which we call "follow the accelerated leader" (FTXL). In continuous time, our scheme can be seen as a fusion of the continuous-time analogue of NAG proposed by Su, Boyd, and Candès [31] and the dynamics of regularized learning studied by Mertikopoulos & Sandholm [25] . We show that the resulting dynamics exhibit the same qualitative equilibrium convergence properties as the replicator dynamics of Taylor & Jonker [32] (the most widely studied instance of FTRL in continuous time). However, whereas the replicator dynamics converge to strict Nash equilibria at a linear rate, the FTXL dynamics do so at a superlinear rate.In discrete time, we likewise propose an algorithmic implementation of FTXL which can be applied in various information context: (i) full information, that is, players observe their entire mixed payoff vector; (ii) realization-based feedback, i.e., when players get to learn the "what-if" payoff of actions that they did not choose; and (iii) bandit, payoff-based feedback, where players only observe their realized, in-game payoff, and must rely on statistical estimation techniques to reconstruct their payoff vectors. In all cases, we show that FTXL maintains the exponential speedup described above, and converges to strict Nash equilibria at a superlinear rate (though the subleading term in the algorithm's convergence rate becomes increasingly worse as less information is available). We find this feature of FTXL particularly intriguing as superlinear convergence rates are often associated to methods that are second-order in space, not time; the fact that this is achieved even with bandit feedback is quite surprising in this context.Closest to our work is the continuous-time, second-order replicator equation studied by [20] in the context of evolutionary game theory, and derived through a model of pairwise proportional imitation of "long-term success". The dynamics of [20] correspond to the undamped, continuous-time version of FTXL with entropic regularization, and the equilibrium convergence rate obtained by [20] agrees with our analysis. Other than that, the dynamics of Flåm & Morgan [9] also attempted to exploit a Newtonian structure, but they do not yield favorable convergence properties in a general setting. The inertial dynamics proposed in [21] likewise sought to leverage an inertial structure combined with the Hessian-Riemannian underpinnings of the replicator dynamics, but the resulting replicator equation was not even well-posed (in the sense that its solutions exploded in finite time).More recently, Gao & Pavel [12, 13] considered a second-order, inertial version of the dynamics of mirror descent in continuous games, and examined their convergence in the context of variational stability [26] . Albeit related at a high level (given the link between mirror descent and regularized learning), the dynamics of Gao & Pavel [12, 13] are actually incomparable to our own, and there is no overlap in our techniques or results. Other than that, second-order dynamics in games have also been studied in continuous time within the context of control-theoretic passivity, yielding promising results in circumventing the impossibility results of Hart & Mas-Colell [17] , cf. Gao & Pavel [10, 11] , Mabrok & Shamma [24] , Toonsi & Shamma [33] , and references therein. However, the resulting dynamics are also different, and we do not see a way of obtaining comparable rates in our context.Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? Answer: [Yes] Justification: It can be found in Section 3, Section 4 and the appendix. Guidelines:• The answer NA means that the abstract and introduction do not include the claims made in the paper. • The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers. • The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings. • It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper. 
preliminaries: 
 In this section, we outline some notions and definitions required for our analysis. Specifically, we introduce the framework of finite 𝑁-player games, we discuss the solution concept of a Nash equilibrium, and we present the main ideas of regularized learning in games.In this work, we focus exclusively with finite games in normal form. Such games consist of a finite set of players N = {1, . . . , 𝑁 }, each of whom has a finite set of actions -or pure strategies -𝛼 𝑖 ∈ A 𝑖 and a payoff function 𝑢 𝑖 : A → ℝ, where A 𝑖 ∈N A 𝑖 denotes the set of all possible action profiles 𝛼 = (𝛼 1 , . . . , 𝛼 𝑁 ). To keep track of all this, a finite game with the above primitives will be denoted as Γ ≡ Γ(N , A, 𝑢).In addition to pure strategies, players may also randomize their choices by employing mixed strategies, that is, by choosing probability distributions 𝑥 𝑖 ∈ X 𝑖 Δ(A 𝑖 ) over their pure strategies, where Δ(A 𝑖 ) denotes the probability simplex over A 𝑖 . Now, given a strategy profile 𝑥 = (𝑥 1 , . . . , 𝑥 𝑁 ) ∈ X 𝑖 ∈N X 𝑖 , we will use the standard shorthand 𝑥 = (𝑥 𝑖 ; 𝑥 -𝑖 ) to highlight the mixed strategy 𝑥 𝑖 of player 𝑖 against the mixed strategy profile 𝑥 -𝑖 ∈ X -𝑖 𝑗≠𝑖 X 𝑗 of all other players. We also define: 1. The mixed payoff of player 𝑖 under 𝑥 asEQUATION2. The mixed payoff vector of player 𝑖 under 𝑥 asEQUATIONIn words, 𝑣 𝑖 (𝑥) collects the expected rewards 𝑣 𝑖 𝛼 𝑖 (𝑥) 𝑢 𝑖 (𝛼 𝑖 ; 𝑥 -𝑖 ) of each action 𝛼 𝑖 ∈ A 𝑖 of player 𝑖 ∈ N against the mixed strategy profile 𝑥 -𝑖 of all other players. Finally, we write 𝑣(𝑥) = (𝑣 1 (𝑥), . . . , 𝑣 𝑁 (𝑥)) for the concatenation of the players' mixed payoff vectors.In terms of solution concepts, we will say that 𝑥 * is a Nash equilibrium (NE) if no player can benefit by unilaterally deviating from their strategy, that is𝑢 𝑖 (𝑥 * ) ≥ 𝑢 𝑖 (𝑥 𝑖 ; 𝑥 * -𝑖 ) for all 𝑥 𝑖 ∈ X 𝑖 and all 𝑖 ∈ N . (NE)Moreover, we say that 𝑥 * is a strict Nash equilibrium if (NE) holds as a strict inequality for all 𝑥 𝑖 ≠ 𝑥 * 𝑖 , 𝑖 ∈ N , i.e., if any deviation from 𝑥 * 𝑖 results in a strictly worse payoff for the deviating player 𝑖 ∈ N . It is straightforward to verify that a strict equilibrium 𝑥 * ∈ X is also pure in the sense that each player assigns positive probability only to a single pure strategy 𝛼 * 𝑖 ∈ A 𝑖 . Finally, we denote the support of a strategy 𝑥 as the set of actions with non-zero probability mass, i.e., supp(𝑥) = {𝛼 ∈ A : 𝑥 𝛼 > 0}.In the general context of finite games, the most widely used learning scheme is the family of algorithms and dynamics known as "follow the regularized leader" (FTRL). In a nutshell, the main idea behind FTRL is that each player 𝑖 ∈ N plays a "regularized" best response to their cumulative payoff over time, leading to the continuous-time dynamics𝑦 𝑖 (𝑡) = 𝑣 𝑖 (𝑥(𝑡)) 𝑥 𝑖 (𝑡) = 𝑄 𝑖 (𝑦 𝑖 (𝑡)) (FTRL-D)whereEQUATIONdenotes the regularized best response -or mirror -map of player 𝑖 ∈ N , and ℎ 𝑖 : X 𝑖 → ℝ is a strongly convex function known as the method's regularizer. Accordingly, in discrete time, this leads to the algorithm 𝑦 𝑖,𝑛+1 = 𝑦 𝑖,𝑛 + 𝛾 v𝑖,𝑛 𝑥 𝑖,𝑛 = 𝑄 𝑖 (𝑦 𝑖,𝑛 ) (FTRL) where 𝛾 > 0 is a hyperparameter known as the algorithm's learning rate (or step-size) and v𝑖,𝑛 is a black-box "payoff signal" that carries information about 𝑣 𝑖 (𝑥 𝑛 ). In the simplest case, when players have full information about the game being played and the actions taken by their opponents, we have v𝑖,𝑛 = 𝑣 𝑖 (𝑥 𝑛 ); in more information-depleted environments (such as learning with payoff-based, bandit feedback), v𝑖,𝑛 is a reconstruction of 𝑣 𝑖 (𝑥 𝑛 ) based on whatever information is at hand.For concreteness, we close this section with the prototypical example of FTRL methods, the exponential / multiplicative weights (EW) algorithm. Going back to [1, 23, 34] , this method is generated by the negentropy regularizer ℎ 𝑖 (𝑥 𝑖 ) = 𝛼 𝑖 ∈A 𝑖 𝑥 𝑖 𝛼 𝑖 log 𝑥 𝑖 𝛼 𝑖 , which yields the EW update rule𝑦 𝑖,𝑛+1 = 𝑦 𝑖,𝑛 + 𝛾 v𝑖,𝑛 𝑥 𝑖,𝑛 = Λ 𝑖 (𝑦 𝑖,𝑛 ) exp(𝑦 𝑖,𝑛 ) ∥exp(𝑦 𝑖,𝑛 ) ∥ 1 (EW)and, in the continuous-time limit 𝛾 → 0, the exponential weights dynamics𝑦 𝑖 (𝑡) = 𝑣 𝑖 (𝑥(𝑡)) 𝑥 𝑖 (𝑡) = Λ 𝑖 (𝑦 𝑖 (𝑡)) . (EWD)In the above, Λ 𝑖 denotes the regularized best response induced by the method's entropic regularizer, which is known colloquially as a logit best response -or, even more simply, as the logit map. To make the notation more compact in the sequel, we will write 𝑄 = (𝑄 𝑖 ) 𝑖 ∈N and Λ = (Λ 𝑖 ) 𝑖 ∈N for the ensemble of the players' regularized / logit best response maps. Remark 1. To streamline our presentation, in the main part of the paper, quantitative results will be stated for the special case of the EW setup above. In Appendix A, we discuss more general decomposable regularizers of the formℎ 𝑖 (𝑥 𝑖 ) = 𝛼 𝑖 ∈A 𝑖 𝜃 𝑖 (𝑥 𝑖 ) where 𝜃 𝑖 : [0, 1] → ℝ is continuous on [0, 1],and has 𝜃 ′′ (𝑥) > 0 for all 𝑥 ∈ (0, 1] and lim 𝑥→0 + 𝜃 ′ (𝑥) = -∞. Although this set of assumptions can be relaxed, it leads to the clearest presentation of our results, so it will suffice for us. Remark 2. Throughout the paper, we will interchangeably use 𝑔(𝑡) and 𝑑𝑔/𝑑𝑡 to denote the time derivative of 𝑔(𝑡). This dual notation allows us to adopt whichever form is most convenient in the given context. Moreover, for a process 𝑔, we will use the notation 𝑔(𝑡) for 𝑡 ≥ 0 if it evolves in continuous time, and 𝑔 𝑛 for 𝑛 ∈ ℕ if it evolves in discrete time steps, omitting the time-index when it is clear from context.Question: Does the paper discuss the limitations of the work performed by the authors? Answer: [Yes] Justification: It can be found in Section 1.Guidelines:• The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper. • The authors are encouraged to create a separate "Limitations" section in their paper.• The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). 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combining acceleration with regularization: first insights and results: 
 In this section, we proceed to illustrate how Nesterov's accelerated gradient (NAG) method can be combined with FTRL. To keep things as simple as possible, we focus on the continuous-time limit, so we do not have to worry about the choice of hyperparameters, the construction of black-box models for the players' payoff vectors, etc.3.1. Nesterov's accelerated gradient algorithm. We begin by discussing Nesterov's accelerated gradient algorithm as presented in Nesterov's seminal paper [28] in the context of unconstrained smooth convex minimization. Specifically, given a Lipschitz smooth convex function 𝑓 : ℝ 𝑑 → ℝ, the algorithm unfolds iteratively as𝑥 𝑛+1 = 𝑤 𝑛 -𝛾∇ 𝑓 (𝑤 𝑛 ) 𝑤 𝑛+1 = 𝑥 𝑛+1 + 𝑛 𝑛 + 3 (𝑥 𝑛+1 -𝑥 𝑛 ) (NAG)where 𝑤 1 = 𝑥 1 is initialized arbitrarily and 𝛾 > 0 is a step-size parameter (typically chosen as 𝛾 ← 1/𝐿 where 𝐿 is the Lipschitz smoothness modulus of 𝑓 ). The specific iterative structure of (NAG) -and, in particular the "3" in the denominator -can appear quite mysterious, but (NAG) otherwise offers remarkable perfomance gains, improving in particular the rate of convergence of gradient methods from O(1/𝑇) to O(1/𝑇 2 ) [28] , and matching in this way the corresponding Ω(1/𝑇 2 ) lower bound for the minimization of smooth convex functions [27] .This groundbreaking result has since become the cornerstone of a vast and diverse literature expanding on the properties of (NAG) and trying to gain a deeper understanding of the "how" and "why" of its update structure. One perspective that has gained significant traction in this regard is the continuoustime approach of Su et al. [30, 31] ; combining the two equations in (NAG) intoEQUATIONthey modeled (NAG) as a heavy ball with vanishing friction system of the form𝑑 2 𝑥 𝑑𝑡 2 = -∇ 𝑓 (𝑥) - 3 𝑡 𝑑𝑥 𝑑𝑡 (HBVF)The choice of terminology alludes to the fact that (HBVF) describes the dynamics of a heavy ball descending the landscape of 𝑓 under the potential field 𝐹 (𝑥) = -∇ 𝑓 (𝑥) with a vanishing kinetic friction coefficient (the 3/𝑡 factor in front of the momentum term 𝑑𝑥/𝑑𝑡). In this interpretation, the mass of the ball accelerates the system, the friction term dissipates energy to enable convergence, and the vanishing friction coefficient quenches the impact of friction over time in order to avoid decelerating the system too much (so the system is, in a sense, "critically underdamped").As was shown by Su et al. [31] , an explicit Euler discretization of (HBVF) yields (NAG) with exactly the right momentum coefficient 𝑛/(𝑛 + 3); moreover, the rate of convergence of the continuous-time dynamics (HBVF) is the same as that of the discrete-time algorithm (NAG), and the energy function and Lyapunov analysis used to derive the former can also be used to derive the latter. For all these reasons, (HBVF) is universally considered as the de facto continuous-time analogue of (NAG), and we will treat it as such in the sequel.To move from unconstrained convex minimization problems to finite 𝑁-person games -a constrained, non-convex, multi-agent, multi-objective setting -it will be more transparent to start with the continuous-time formulation (HBVF). Indeed, applying the logic behind (HBVF) to the (unconstrained) state variables 𝑦 of (FTRL-D), we obtain the "follow the accelerated leader" dynamics𝑑 2 𝑦 𝑑𝑡 2 = 𝑣(𝑄(𝑦)) - 𝑟 𝑡 𝑑𝑦 𝑑𝑡 (FTXL-D)where the dynamics' driving force 𝐹 (𝑦) = 𝑣(𝑄(𝑦)) is now given by the payoff field of the game, and the factor 𝑟/𝑡, 𝑟 ≥ 0, plays again the role of a vanishing friction coefficient. To avoid confusion, we highlight that in the case of regularized learning, the algorithm's variable that determines the evolution of the system in an autonomous way is the "score variable" 𝑦, not the "strategy variable" 𝑥 (which is an ancillary variable obtained from 𝑦 via the regularized choice map 𝑄).In contrast to (EWD), the accelerated dynamics (FTXL-D) are second-order in time, a fact with fundamental ramifications, not only from a conceptual, but also from an operational viewpoint. Focusing on the latter, we first note that (FTXL-D) requires two sets of initial conditions, 𝑦(0) and 𝑦(0), the latter having no analogue in the first-order setting of (FTRL-D). In general, the evolution of the system depends on both 𝑦(0) and 𝑦(0), but since this would introduce an artificial bias toward a certain direction, we will take 𝑦(0) = 0, in tune with standard practice for (NAG) [31] .We also note that (FTXL-D) can be mapped to an equivalent autonomous first-order system with double the variables: specifically, letting 𝑝 = 𝑦 denote the players' (payoff ) momentum, (FTXL-D) can be rewritten as 𝑑𝑦 𝑑𝑡EQUATIONwith 𝑦(0) initialized arbitrarily and 𝑝(0) = 𝑦(0). In turn, (5) yields 𝑝(𝑡) = 𝑡 -𝑟 ∫ 𝑡 0 𝑡 𝑟 𝑣(𝑄(𝑦(𝜏))) 𝑑𝜏, so 𝑝(𝑡) can be seen as a weighted aggregate of the players' payoffs up to time 𝑡: if 𝑟 = 0 (the undamped regime), all information enters 𝑝(𝑡) with the same weight; if 𝑟 > 0, past information is discounted relative to more recent observations; and, in the overdamped limit 𝑟 → ∞, all weight is assigned to the current point in time, emulating in this way the first-order system (FTRL-D).. From an operational standpoint, the main question of interest is to specify the equilibrium convergence properties of (FTXL-D) -and, later in the paper, its discrete-time analogue. To establish a baseline, the principal equilibrium properties of its first-order counterpart can be summarized as follows: (i) strict Nash equilibria are locally stable and attracting under (FTRL-D) [19, 25] ;1 (ii) the dynamics do not admit any other such points (that is, stable and attracting) [8] ; and (iii) quantitively, in the case of (EWD), the dynamics converge locally to strict Nash equilibria at a geometric rate of the form ∥𝑥(𝑡) -𝑥 * ∥ = O(exp(-𝑐𝑡)) for some 𝑐 > 0 [25] .Our first result below shows that the accelerated dynamics (FTXL-D) exhibit an exponential speed-up relative to (FTRL-D), and the players' orbits converge to strict Nash equilibria at a superlinear rate: Theorem 1. Let 𝑥 * be a strict Nash equilibrium of Γ, and let 𝑥(𝑡) = 𝑄(𝑦(𝑡)) be a solution orbit of (FTXL-D). If 𝑥(0) is sufficiently close to 𝑥 * , then 𝑥(𝑡) converges to 𝑥 * ; in particular, if (FTXL-D) is run with logit best responses (that is, 𝑄 ← Λ), we haveEQUATIONwhere 𝐶 > 0 is a constant that depends only on the initialization of (FTXL-D) and𝑐 = 1 2 min 𝑖 ∈N min 𝛽 𝑖 ∉supp( 𝑥 * 𝑖 ) [𝑢 𝑖 (𝑥 * 𝑖 ; 𝑥 * -𝑖 ) -𝑢 𝑖 (𝛽 𝑖 ; 𝑥 * -𝑖 )] > 0 (7)is the minimum payoff difference at equilibrium.Theorem 1 (which we prove in Appendix B) is representative of the analysis to come, so some remarks are in order. First, we should note that the explicit rate estimate (6) is derived for the special case of logit best responses, which underlie all exponential / multiplicative weights algorithms. To the best of our knowledge, the only comparable result in the literature is the similar rate provided in [20] for the case 𝑟 = 0. In the case of a general regularizer, an analogous speed-up is observed, but the exact expressions are more involved, so we defer them to Appendix B. A second important point concerns whether the rate estimate (6) is tight or not. Finally, the neighborhood of initial conditions around 𝑥 * is determined by the minimum payoff difference at equilibrium and is roughly O(𝑐) in diameter; we defer the relevant details of this discussion to Appendix B.To answer this question -and, at the same time get a glimpse of the proof strategy for Theorem 1 -it will be instructive to consider a single-player game with two actions. Albeit simple, this toy example is not simplistic, as it provides an incisive look into the problem, and will be used to motivate our design choices in the sequel. EQUATIONAs we show in Appendix B, this non-autonomous differential equation can be solved exactly to yield 𝑧(𝑡) = 𝑧(0) + 𝑡 2 /[2(𝑟 + 1)], and henceEQUATION)Since 𝑐 = 𝑢(A) -𝑢(B) = 1, the rate ( 9) coincides with that of Theorem 1 up to a factor of 1/2. This factor is an artifact of the analysis and, in fact, it can be tightened to (1 -𝜀) for arbitrarily small 𝜀 > 0; we did not provide this more precise expression to lighten notation. By contrast, the factor 2(𝑟 + 1) in ( 6) cannot be lifted; this has important ramifications which we discuss below. ♦The first conclusion that can be drawn from Example 3.1 is that the rate estimate of Theorem 1 is tight and cannot be improved in general. In addition, and in stark contrast to (NAG), Example 3.1 shows that the optimal value for the friction parameter is 𝑟 = 0 (at least from a min-max viewpoint, as this value yields the best possible lower bound for the rate). Of course, this raises the question as to whether this is due to the continuous-time character of the policy;2 however, as we show in detail in Appendix C, this is not the case: the direct handover of (NAG) to Example 3.1 yields the exact same rate (though the proof relies on a significantly more opaque generating function calculation).In view of all this, it becomes apparent that friction only hinders the equilibrium convergence properties of accelerated FTRL schemes in our game-theoretic setting. On that account, we will continue our analysis in the undamped regime 𝑟 = 0.4 Accelerated learning: Analysis and results4.1. The algorithm. To obtain a bona fide, algorithmic implementation of the continuous-time dynamics (FTXL-D), we will proceed with the same explicit, finite-difference scheme leading to the discrete-time algorithm (NAG) from the continuous-time dynamics (HBVF) of Su et al. [31] . Specifically, taking a discretization step 𝛾 > 0 in (FTXL-D) and setting the scheme's friction parameter 𝑟 to zero (which, as we discussed at length in the previous section, is the optimal choice in our setting), a straightforward derivation yields the basic update ruleEQUATIONIn the above, just as in the case of (FTRL), v𝑖,𝑛 ∈ ℝ A 𝑖 denotes a black-box "payoff signal" that carries information about the mixed payoff vector 𝑣 𝑖 (𝑥 𝑛 ) of player 𝑖 at the current strategy profile 𝑥 𝑛 (we provide more details on this below).Alternatively, to obtain an equivalent first-order iterative rule (which is easier to handle and discuss), it will be convenient to introduce the momentum variables 𝑝 𝑛 = (𝑦 𝑛 -𝑦 𝑛-1 )/𝛾. Doing just that, a simple rearrangement of (10) yields the "follow the accelerated leader" scheme𝑦 𝑖,𝑛+1 = 𝑦 𝑖,𝑛 + 𝛾 𝑝 𝑖,𝑛+1 𝑝 𝑖,𝑛+1 = 𝑝 𝑖,𝑛 + 𝛾 v𝑖,𝑛 𝑥 𝑖,𝑛 = 𝑄 𝑖 (𝑦 𝑖,𝑛 ) . (FTXL)The algorithm (FTXL) will be our main object of study in the sequel, and we will examine its convergence properties under three differerent models for v𝑛 :1. Full information, i.e., players get to access their full, mixed payoff vectors:EQUATION2. Realization-based feedback, i.e., after choosing an action profile 𝛼 𝑛 ∼ 𝑥 𝑛 , each player 𝑖 ∈ N observes (or otherwise calculates) the vector of their counterfactual, "what-if" rewards, namelyEQUATION3. Bandit / Payoff-based feedback, i.e., each player only observes their current reward, and must rely on statistical estimation techniques to reconstruct an estimate of 𝑣 𝑖 (𝑥 𝑛 ). For concreteness, we will consider the case where players employ a version of the so-called importance-weighted estimatorEQUATIONwhich we describe in detail later in this section.Of course, this list of information models is not exhaustive, but it is a faithful representation of most scenarios that arise in practice, so it will suffice for our purposes.Now before moving forward with the analysis, it will be useful to keep some high-level remarks in mind. The first is that (FTXL) shares many similarities with (FTRL), but also several notable differences. At the most basic level, (FTRL) and (FTXL) are both "stimulus-response" schemes in the spirit of Erev & Roth [7] , that is, players "respond" with a strategy 𝑥 𝑖,𝑛 = 𝑄 𝑖 (𝑦 𝑖,𝑛 ) to a "stimulus" 𝑦 𝑖,𝑛 generated by the observed payoff signals v𝑖,𝑛 . In this regard, both methods adhere to the online learning setting (and, in particular, to the regularized learning paradigm).However, unlike (FTRL), where players respond to the aggregate of their payoff signals -the process 𝑦 𝑛 in (FTRL) -the accelerated algorithm (FTXL) introduces an additional aggregation layer, which expresses how players "build momentum" based on the same payoff signals -the process 𝑝 𝑛 in (FTXL). Intuitively, we can think of these two processes as the "position" and "momentum" variables of a classical inertial system, not unlike the heavy-ball dynamics of Su et al. [31] . The only conceptual difference is that, instead of rolling along the landscape of a (convex) function, the players now track the "mirrored" payoff field v(𝑦) 𝑣(𝑄(𝑦)). In the rest of this section, we proceed to examine in detail the equilibrium convergence properties of (FTXL) under each of the three models detailed in Eqs. (11a)-(11c) in order. 4.2. Accelerated learning with full information. We begin with the full information model (11a). This is the most straightforward model (due to the absence of randomness and uncertainty) but, admittedly, also the least realistic one. Nevertheless, it will serve as a useful benchmark for the rest, and it will allow us to introduce several important notions.Before we state our result, it is important to note that a finite game can have multiple strict Nash equilibria, so global convergence results are, in general, unattainable; for this reason, we analyze the algorithm's local convergence landscape. In this regard, Theorem 2 below shows that (FTXL) with full information achieves a superlinear local convergence rate to strict Nash equilibria: Theorem 2. Let 𝑥 * be a strict Nash equilibrium of Γ, and let 𝑥 𝑛 = 𝑄(𝑦 𝑛 ) be the sequence of play generated by (FTXL) with full information feedback of the form (11a). If 𝑥 1 is initialized sufficiently close to 𝑥 * , then 𝑥 𝑛 converges to 𝑥 * ; in particular, if (FTXL) is run with logit best responses (that is, 𝑄 ← Λ), we haveEQUATION)where 𝐶 > 0 is a constant that depends only on the initialization of (FTXL) andEQUATIONis the minimum payoff difference at equilibrium.To maintain the flow of our discussion, we defer the proof of Theorem 2 to Appendix C. Instead, we only note here that, just as in the case of (HBVF) and (NAG), Theorem 2 provides essentially the same rate of convergence as its continuous-time counterpart, Theorem 1, modulo a subleading term which has an exponentially small impact on the rate of convergence. In particular, we should stress that the superlinear convergence rate of (FTXL) exhibits an exponential speedup relative to (FTRL), which is known to converge at a geometric rate ∥𝑥 𝑇 -𝑥 * ∥ ∞ = exp(-Θ(𝑇)). This is in direct correspondence to what we observe in continuous time, showing in particular that the continuous-time dynamics (FTXL-D) are a faithful representation of (FTXL).We should also stress here that superlinear convergence rates are typically associated with methods that are second-order in space -like Newton's algorithm -not second-order in time -like (NAG) and (FTXL). We find this observation particularly intriguing as it suggests that accelerated rates can be observed in the context of learning in games without having to pay the excessively high compute cost of second-order methods in optimization.Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof? Answer: [Yes] Justification: It can be found in Section 2, Section 3, Section 4 and the appendix. Guidelines:• The answer NA means that the paper does not include theoretical results.• All the theorems, formulas, and proofs in the paper should be numbered and cross-referenced.• All assumptions should be clearly stated or referenced in the statement of any theorems.• The proofs can either appear in the main paper or the supplemental material, but if they appear in the supplemental material, the authors are encouraged to provide a short proof sketch to provide intuition. • Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material. • Theorems and Lemmas that the proof relies upon should be properly referenced. 
accelerated learning with realization-based feedback.: 
 We now turn to the realization-based model (11b), where players can only assess the rewards of their pure actions in response to the realized actions of all other players. In words, v𝑖,𝑛 = 𝑣 𝑖 (𝛼 𝑛 ) collects the payoffs that player 𝑖 ∈ N would have obtained by playing each of their pure actions 𝛼 𝑖 ∈ A 𝑖 against the action profile 𝛼 -𝑖,𝑛 adopted by the rest of the players.In contrast to the full information model (11a), the realization-based model is stochastic in nature, so our convergence results will likewise be stochastic. Nevertheless, despite the added layer of uncertainty, we show that (FTXL) with realization-based feedback maintains a superlinear convergence rate with high probability: Theorem 3. Let 𝑥 * be a strict Nash equilibrium of Γ, fix some confidence level 𝛿 > 0, and let 𝑥 𝑛 = 𝑄(𝑦 𝑛 ) be the sequence of play generated by (FTXL) with realization-based feedback as per (11b) and a sufficiently small step-size 𝛾 > 0. Then there exists a neighborhood U of 𝑥 * such that In particular, if (FTXL) is run with logit best responses (that is, 𝑄 ← Λ), there exist positive constants 𝐶, 𝑐 > 0 as in Theorem 2 such that on the event {𝑥 𝑛 → 𝑥 * as 𝑛 → ∞}:EQUATIONEQUATION)What is particularly surprising in Theorem 3 is that, (FTXL) maintains the accelerated superlinear rate of Theorem 2 -and, likewise, the exponential speedup relative to (FTRL) -despite the randomness and uncertainty involved in the realization-based model (11b). The salient point enabling this feature of (FTXL) is that v𝑛 can be expressed asEQUATION)where 𝑈 𝑛 ∈ 𝑖 ℝ A 𝑖 is an almost surely bounded conditionally zero-mean stochastic perturbation, that is, 𝔼[𝑈 𝑛 | F 𝑛 ] = 0, where F 𝑛 𝜎(𝑥 1 , . . . , 𝑥 𝑛 ) denotes the history of play up to (and including) time 𝑛. Thanks to the boundedness of ( 16), we are able to derive a series of probabilistic estimates showing that, with high probability (and, in particular, greater than 1 -𝛿), the contribution of the noise in the algorithm's rate becomes subleading, thus allowing the superlinear rate of Theorem 2 to emerge. As in the case of Theorem 2, we defer the proof of Theorem 3 to the appendix.The last framework we consider is the bandit model where players only observe their realized rewards, a scalar from which they must reconstruct their entire payoff vector. To do so, a standard technique from the multi-armed bandit literature is the so-called importance weighted estimator (IWE) [5, 22] , defined in our setting asv𝑖𝛼 𝑖 ,𝑛 = 1{𝛼 𝑖,𝑛 = 𝛼 𝑖 } x𝑖𝛼 𝑖,𝑛 𝑢 𝑖 (𝛼 𝑖 ; 𝛼 -𝑖,𝑛 ) (IWE)where x𝑖,𝑛 = (1 -𝜀 𝑛 )𝑥 𝑖,𝑛 + 𝜀 𝑛 unif A 𝑖 is a mixture of 𝑥 𝑖,𝑛 and the uniform distribution on A 𝑖 (a mechanism known in the literature as explicit exploration). Importantly, this estimator is unbiased relative to the perturbed strategy x𝑥 𝑛 , which thus incurs an O(𝜀 𝑛 ) non-zero-sum error to the estimation of 𝑣 𝑖 (𝑥 𝑛 ). This error can be made arbitrarily small by taking 𝜀 𝑛 → 0 but, in doing so, the variance of v𝑖,𝑛 diverges, leading to a bias-variance trade-off that is difficult to tame.Despite these added difficulties, we show below that (FTXL) maintains its superlinear convergence rate even with bandit, payoff-based feedback: Theorem 4. Let 𝑥 * be a strict Nash equilibrium of Γ, fix some confidence level 𝛿 > 0, and let 𝑥 𝑛 = 𝑄(𝑦 𝑛 ) be the sequence of play generated by (FTXL) with bandit feedback of the form (11c), an IWE exploration parameter 𝜀 𝑛 ∝ 1/𝑛 ℓ 𝜀 for some ℓ 𝜀 ∈ (0, 1/2), and a sufficiently small step-size 𝛾 > 0. Then there exists a neighborhood U of 𝑥 * in X such thatEQUATION)In particular, if (FTXL) is run with logit best responses (that is, 𝑄 ← Λ), there exist positive constants 𝐶, 𝑐 > 0 as in Theorem 2 such that on the eventEQUATIONTheorem 4 (which we prove in Appendix D shows that, despite the degradation of the subleading term, (FTXL) retains its superlinear convergence rate even with bandit, payoff-based feedback (for a numerical demonstration, see Fig. 1 above ). We find this feature of (FTXL) particularly important as it shows that the algorithm remains exceptionally robust in the face of randomness and uncertainty, even as we move toward increasingly information-starved environments -from full information, to realization-based observations and, ultimately, to bandit feedback. This has important ramifications from an operational standpoint, which we intend to examine further in future work.4.5. Numerical Experiments. Finally, we provide numerical simulations to validate the performance of (FTXL). To this end, we consider two game paradigms, (i) a 2-player zero-sum game, and (ii) a congestion game.Zero-sum Game. First, we consider a 2-player zero-sum game with actions {𝛼 1 , 𝛼 2 , 𝛼 3 } and {𝛽 1 , 𝛽 2 , 𝛽 3 }, and payoff matrix𝑃 = (2, -2) (1, -1) (2, -2) (-2, 2) (-1, 1) (-2, 2) (-2, 2) (-1, 1) (-2, 2)Here, the rows of 𝑃 correspond to the actions of player 𝐴 and the columns to the actions of player 𝐵, while the first item of each entry of 𝑃 corresponds to the payoff of 𝐴, and the second one to the payoff of 𝐵. Clearly, the action profile (𝛼 1 , 𝛽 2 ) is a strict Nash equilibrium.Congestion Game. As a second example, we consider a congestion game with 𝑁 = 100 and 2 roads, 𝑟 1 and 𝑟 2 , with costs 𝑐 1 = 1.1 and 𝑐 2 = 𝑑/𝑁 where 𝑑 is the number of drivers on 𝑟 2 . In words, 𝑟 1 has a fixed delay equal to 1.1, while 𝑟 2 has a delay proportional to the drivers using it. Note, that the strategy profile where all players are using 𝑟 2 is a strict Nash equilibrium.In Fig. 1 , we assess the convergence of (FTXL) with logit best responses, under realization-based and bandit feedback, and compare it to the standard (EW) with the same level of information. The figures verify that (FTXL) outperforms (EW) regarding the convergence to a strict Nash equilibrium both for the realization-based and the bandit feedback, as expected from the theoretical findings. Specifically, they validate the faster convergence rate of (FTXL) compared to that of the (EW) algorithm. Moreover, we observe that both algorithms perform worse under bandit feedback than under realization-based feedback. This behavior is expected as less information becomes available. More details can be found in Appendix E.Lemma EQUATIONwhere 𝐶 > 0 is a constant that depends only on the initialization of (FTXL-D) and EQUATION𝑧(𝑡) = 𝑣 𝛼 (𝑥(𝑡)) -𝑣 𝛼 * (𝑥(𝑡)) - 𝑟 𝑡 𝑧 𝛼 (𝑡) (B.7)Let 𝑦(0) such that 𝑧 𝛼 (0) = -𝑀 -𝜀, for all 𝛼 ≠ 𝛼 * ∈ A, where 𝜀 > 0 small. We will, first, show that 𝑧(𝑡) < -𝑀 for all 𝑡 ≥ 0. For the sake of contradiction, and denoting 𝑇 0 inf{𝑡 ≥ 0 : 𝑧(𝑡) ≥ -𝑀 }, suppose that 𝑇 0 < ∞. Then, we readily get that for all 𝑡 < 𝑇 0 , it holds𝑣 𝛼 (𝑥(𝑡)) -𝑣 𝛼 * (𝑥(𝑡)) < -𝑐 (B.8)and therefore, for all 𝑡 ≤ 𝑇 0 :𝑧 𝛼 (𝑡)𝑡 𝑟 + 𝑟𝑡 𝑟 -1 𝑧 𝛼 (𝑡) = 𝑡 𝑟 [𝑣 𝛼 (𝑥) -𝑣 𝛼 * (𝑥)] ≤ -𝑐𝑡 𝑟 (B.9)which can be rewritten as: 𝑑 𝑑𝑡( 𝑧 𝛼 (𝑡)𝑡 𝑟 ) ≤ -𝑐𝑡 𝑟 (B.10)Integrating over 𝑡 < 𝑇 0 , we obtain 𝑧 𝛼 (𝑡)𝑡 𝑟 ≤ -𝑐𝑡 𝑟+1 /(𝑟 + 1), which readily implies:𝑧 𝛼 (𝑡) ≤ 𝑧 𝛼 (0) - 𝑐 2(𝑟 + 1) 𝑡 2 < -𝑀 - 𝑐 2(𝑟 + 1) 𝑡 2 (B.11)By sending 𝑡 → 𝑇 0 , we arrive at a contradiction. Therefore 𝑧 𝛼 (𝑡) < -𝑀 for all 𝑡 ≥ 0, and the previous equation implies that for all 𝑡 ≥ 0 :𝑧 𝛼 (𝑡) ≤ 𝑧 𝛼 (0) - 𝑐 2(𝑟 + 1)𝑡 2 (B.12)and invoking Lemma A.1, we get the convergence result. Finally, translating the score-differences to the primal space X , we get:∥𝑥(𝑡) -𝑥 * ∥ ∞ = max 𝑖 ∈N 1 -𝑥 𝑖 𝛼 * 𝑖 (𝑡) (B.13)For the case of logit best responses, i.e., when 𝑄 ← Λ, and assuming that the maximum above is attained for player 𝑖 ∈ N , we obtain Let 𝑦(0) such that 𝑧 𝛼 (0) = -𝑀 -𝜀, for all 𝛼 ≠ 𝛼 * ∈ A, where 𝜀 > 0 small. As in the proof of Theorem 1, we will, first, show that 𝑧(𝑡) < -𝑀 for all 𝑡 ≥ 0. For the sake of contradiction, and denoting 𝑇 0 inf{𝑡 ≥ 0 : 𝑧(𝑡) ≥ -𝑀 }, suppose that 𝑇 0 < ∞. Then, we readily get that for all 𝑡 < 𝑇 0 , it holds 𝑣 𝛼 (𝑥(𝑡)) -𝑣 𝛼 * (𝑥(𝑡)) < -𝑐 (B.23) and therefore, for all 𝑡 ≤ 𝑇 0 :∥𝑥(𝑡) -𝑥 * ∥ ∞ = 𝛼 𝑖 ≠𝛼 * 𝑖 exp(𝑧 𝛼 𝑖 (𝑡)) 1 + 𝛼 𝑖 ≠𝛼 * 𝑖 exp(𝑧 𝛼 𝑖 (𝑡)) (B.𝑧 𝛼 (𝑡)𝑒 𝑟𝑡 + 𝑟𝑒 𝑟𝑡 𝑧 𝛼 (𝑡) = 𝑒 𝑟𝑡 [𝑣 𝛼 (𝑥) -𝑣 𝛼 * (𝑥)] ≤ -𝑐𝑒 𝑟𝑡 (B.24)which can be rewritten as:𝑑 𝑑𝑡 𝑧 𝛼 (𝑡)𝑒 𝑟𝑡 ≤ -𝑐𝑒 𝑟𝑡 (B.25)Integrating over 𝑡 < 𝑇 0 , and using that 𝑧 𝛼 (0) = 0, we obtain 𝑧 𝛼 (𝑡) ≤ -𝑐/𝑟 + 𝑐𝑒 -𝑟𝑡 /𝑟, which implies:𝑧 𝛼 (𝑡) ≤ 𝑧 𝛼 (0) - 𝑐 𝑟 𝑡 - 𝑐 𝑟 2 𝑒 -𝑟𝑡 + 𝑐 𝑟 2 (B.26) = 𝑧 𝛼 (0) - 𝑐 𝑟 2 𝑟𝑡 + 𝑒 -𝑟𝑡 -1 (B.27) < 𝑧 𝛼 (0) (B.28) < -𝑀 (B.29)where we used that 𝑥 + 𝑒 -𝑥 -1 ≥ 0 for all 𝑥 ∈ ℝ with equality if and only if 𝑥 = 0. By sending 𝑡 → 𝑇 0 , we arrive at a contradiction. Therefore 𝑧 𝛼 (𝑡) < -𝑀 for all 𝑡 ≥ 0, and the previous equation implies that for all 𝑡 ≥ 0 :𝑧 𝛼 (𝑡) ≤ 𝑧 𝛼 (0) - 𝑐 𝑟 𝑡 - 𝑐 𝑟 2 𝑒 -𝑟𝑡 + 𝑐 𝑟 2 (B.30)and invoking Lemma A.1 for 𝜃 (𝑥) = 𝑥 log 𝑥, we get the convergence result.In this section, we provide the results for the (FTXL) algorithm with full-information feedback. First, we discuss the rates obtained by the direct discretization of (FTXL-D) with both vanishing and nonvanishing friction, and then provide the proof of Theorem 2, our main result, for the full-information case.C.1. FTXL with vanishing friction. First, we provide the rate of convergence for the discrete version of (FTXL-D) with vanishing friction:𝑝 𝑖,𝑛+1 = 𝑝 𝑖,𝑛 1 - 𝛾𝑟 𝑛 + 𝛾 v𝑖,𝑛 𝑦 𝑖,𝑛+1 = 𝑦 𝑖,𝑛 + 𝛾 𝑝 𝑖,𝑛+1 (C.1)To streamline our presentation, we consider the setup of Example 3.1 that provides a lower bound for the algorithm. Proposition C.1. Consider the single-player game Γ with actions A and B such that 𝑢(A) -𝑢(B) = 1 of Example 3.1, and let 𝑥 𝑛 = Λ(𝑦 𝑛 ) be the sequence of play generated by (C.1). Then, denoting by 𝑥 * = (1, 0) the strict Nash equilibrium, we have:∥𝑥 𝑇 -𝑥 * ∥ ∞ ∼ exp 𝐶 - 𝛾 2 𝑇 2 2(𝛾𝑟 + 1) . (C.2)where 𝐶 > 0 is a constant that depends only on the initialization of the algorithm.Proof. We first define the score-difference𝑤 𝑛 𝑝 B,𝑛 -𝑝 A,𝑛 (C.3)with initial condition 𝑤 1 = 0. Then, unfolding according to the sequence of play, we obtain:𝑤 𝑛+1 = 𝑤 𝑛 1 - 𝛾𝑟 𝑛 + 𝛾(𝑢(B) -𝑢(A)) (C.4) = 𝑤 𝑛 1 - 𝛾𝑟 𝑛 -𝛾 (C.5) = -𝛾 𝑛-1 ∑︁ 𝑘=1 𝑘-1 ℓ=0 1 - 𝛾𝑟 𝑛 -ℓ -𝛾 (C.6)We next define for 𝑛 ∈ ℕ the difference 𝑧 𝑛 𝑦 B,𝑛 -𝑦 A,𝑛 . Thus, unfolding it, we obtain:𝑧 𝑛+1 = 𝑧 𝑛 + 𝛾𝑤 𝑛+1 (C.7) = 𝑧 𝑛 -𝛾 2 1 + 𝑛-1 ∑︁ 𝑘=1 𝑘-1 ℓ=0 1 - 𝛾𝑟 𝑛 -ℓ (C.8) = 𝑧 1 -𝛾 2 𝑛 ∑︁ 𝑚=1 1 + 𝑚-1 ∑︁ 𝑘=1 𝑘-1 ℓ=0 1 - 𝛾𝑟 𝑚 -ℓ (C.9) (C.10)Now, using Lemma C.1, which we provide after this proof, we obtain that𝑧 𝑛+1 = 𝑧 1 -𝛾 2 𝑛 ∑︁ 𝑚=1 1 + 𝑚 -𝛾𝑟 1 + 𝛾𝑟 - 1 1 + 𝛾𝑟 𝑚 ℓ=1 1 - 𝛾𝑟 ℓ (C.11) = 𝑧 1 -𝛾 2 𝑛(𝑛 + 1) 2(1 + 𝛾𝑟) -𝛾 2 𝑛 1 - 𝛾𝑟 1 + 𝛾𝑟 + 𝛾 2 1 + 𝛾𝑟 𝑛 ∑︁ 𝑚=1 𝑚 ℓ=1 1 - 𝛾𝑟 ℓ (C.12) = 𝑧 1 - 𝛾 2 𝑛 2 2(1 + 𝛾𝑟) + Θ(𝑛) (C.13)and invoking Lemma A.1 for 𝜃 (𝑥) = 𝑥 log 𝑥, we get the result. ■The following lemma is a necessary tool for obtaining the exact convergence rate in Proposition C.2. Lemma C.1. For any 𝑚 ∈ ℕ and 𝑎 > 0, we have that𝑚-1 ∑︁ 𝑘=1 𝑘-1 ℓ=0 (1 - 𝑎 𝑚 -ℓ ) = 𝑚 -𝑎 1 + 𝑎 - 1 1 + 𝑎 𝑚 ℓ=1 1 - 𝑎 ℓ (C.14)Proof. First, by expanding the inner product, we can rewrite the expression as𝑚-1 ∑︁ 𝑘=1 𝑘-1 ℓ=0 (1 - 𝑎 𝑚 -ℓ ) = 𝑚-1 ∑︁ 𝑘=1 𝑘-1 ℓ=0 ( 𝑚 -ℓ -𝑎 𝑚 -ℓ ) (C.15) 𝑚-1 ∑︁ 𝑘=1 (𝑚 -𝑎) . . . (𝑚 -𝑘 + 1 -𝑎) 𝑚 . . . (𝑚 -𝑘 + 1) (C.16) = 𝑚-1 ∑︁ 𝑘=1 (𝑚 -𝑎)!(𝑚 -𝑘)! (𝑚 -𝑘 -𝑎)!𝑚! (C.17) = (𝑚 -𝑎)! 𝑚! 𝑚-1 ∑︁ 𝑘=1 (𝑚 -𝑘)! (𝑚 -𝑘 -𝑎)! (C.18)where with a slight abuse of notation we use the factorial notation (𝑚 -𝑎)! to denote the Gamma function evaluated at 𝑚 -𝑎 + 1, i.e., Γ(𝑚 -𝑎 + 1).𝐹 𝑚 (𝑚 -𝑎)! 𝑚! 𝑚 ∑︁ 𝑘=1 (𝑚 -𝑘)! (𝑚 -𝑘 -𝑎)!the difference of two consecutive terms evolves as:𝐹 𝑚+1 -𝐹 𝑚 = (𝑚 + 1 -𝑎)! (𝑚 + 1)! 𝑚+1 ∑︁ 𝑘=1 (𝑚 + 1 -𝑘)! (𝑚 + 1 -𝑘 -𝑎)! - (𝑚 -𝑎)! 𝑚! 𝑚 ∑︁ 𝑘=1 (𝑚 -𝑘)! (𝑚 -𝑘 -𝑎)! (C.19) = 𝑚 + 1 -𝑎 𝑚 + 1 + (𝑚 + 1 -𝑎)! (𝑚 + 1)! 𝑚+1 ∑︁ 𝑘=2 (𝑚 + 1 -𝑘)! (𝑚 + 1 -𝑘 -𝑎)! - (𝑚 -𝑎)! 𝑚! 𝑚 ∑︁ 𝑘=1 (𝑚 -𝑘)! (𝑚 -𝑘 -𝑎)! (C.20) = 𝑚 + 1 -𝑎 𝑚 + 1 + (𝑚 + 1 -𝑎)! (𝑚 + 1)! 𝑚+1 ∑︁ 𝑘=2 (𝑚 + 1 -𝑘)! (𝑚 + 1 -𝑘 -𝑎)! - (𝑚 -𝑎)! 𝑚! 𝑚+1 ∑︁ 𝑘=2 (𝑚 -𝑘 + 1)! (𝑚 -𝑘 + 1 -𝑎)! (C.21) = 𝑚 + 1 -𝑎 𝑚 + 1 + 𝑚+1 ∑︁ 𝑘=2 (𝑚 + 1 -𝑎)!(𝑚 + 1 -𝑘)! -(𝑚 + 1) (𝑚 -𝑎)!(𝑚 -𝑘 + 1)! (𝑚 + 1)!(𝑚 + 1 -𝑎 -𝑘)! (C.22) = 𝑚 + 1 -𝑎 𝑚 + 1 + 𝑚+1 ∑︁ 𝑘=2 (𝑚 -𝑎)!(𝑚 + 1 -𝑘)!(𝑚 + 1 -𝑎 -𝑚 -1) (𝑚 + 1)!(𝑚 + 1 -𝑎 -𝑘)! (C.23) = 𝑚 + 1 -𝑎 𝑚 + 1 -𝑎 𝑚+1 ∑︁ 𝑘=2 (𝑚 -𝑎)!(𝑚 + 1 -𝑘)! (𝑚 + 1)!(𝑚 + 1 -𝑎 -𝑘)! (C.24) = 𝑚 + 1 -𝑎 𝑚 + 1 - 𝑎 𝑚 + 1 -𝑎 𝑚+1 ∑︁ 𝑘=1 (𝑚 + 1 -𝑘)!(𝑚 + 1 -𝑎)! (𝑚 + 1)!(𝑚 + 1 -𝑎 -𝑘)! - 𝑚 + 1 -𝑎 𝑚 + 1 (C.25) = 1 - 𝑎 𝑚 + 1 -𝑎 𝐹 𝑚+1 (C.26)Thus, we readily obtain the recurrence relation𝑚 + 1 𝑚 + 1 -𝑎 𝐹 𝑚+1 = 𝐹 𝑚 + 1 . (C.27)We continue the proof by induction. To this end, we will show that𝐹 𝑚 = 𝑚 -𝑎 1 + 𝑎 + 𝑎 1 + 𝑎 𝑚 ℓ=1 ℓ -𝑎 ℓ . (C.28)For the base case, note that𝐹 1 = (1 -𝑎) = 1 -𝑎 1 + 𝑎 + 𝑎 1 + 𝑎 (1 -𝑎) (C.29)For the inductive step, suppose that (C.28) holds for 𝑚 ∈ ℕ. Then, we have:𝑚 + 1 𝑚 + 1 -𝑎 𝐹 𝑚+1 = 𝑚 -𝑎 1 + 𝑎 + 𝑎 1 + 𝑎 𝑚 ℓ=1 ℓ -𝑎 ℓ + 1 (C.30) = 𝑚 + 1 1 + 𝑎 + 𝑎 1 + 𝑎 𝑚 ℓ=1 ℓ -𝑎 ℓ (C.31)which implies the inductive step𝐹 𝑚+1 = 𝑚 + 1 -𝑎 1 + 𝑎 + 𝑎 1 + 𝑎 𝑚+1 ℓ=1 ℓ -𝑎 ℓ (C.32)and thus (C.28) holds for all 𝑚 ∈ ℕ. To complete the proof notice that𝑚-1 ∑︁ 𝑘=1 𝑘-1 ℓ=0 (1 - 𝑎 𝑚 -ℓ ) = (𝑚 -𝑎)! 𝑚! 𝑚-1 ∑︁ 𝑘=1 (𝑚 -𝑘)! (𝑚 -𝑘 -𝑎)! (C.33) = 𝐹 𝑚 - 𝑚-1 ℓ=0 1 - 𝑎 𝑚 -ℓ (C.34) = 𝑚 -𝑎 1 + 𝑎 + 𝑎 1 + 𝑎 𝑚 ℓ=1 ℓ -𝑎 ℓ - 𝑚 ℓ=1 1 - 𝑎 ℓ (C.35) = 𝑚 -𝑎 1 + 𝑎 - 1 1 + 𝑎 𝑚 ℓ=1 1 - 𝑎 ℓ (C.36)■ Next, we discuss the cases of non-vanishing and zero friction.C.2. FTXL with non-vanishing friction. We continue this section by considering the case of non-vanishing friction in analogy to the continuous-time case, as per Appendix B. Specifically, we consider the discrete version of (FTXL-D) with non-vanishing friction, as follows:𝑝 𝑖,𝑛+1 = 𝑝 𝑖,𝑛 (1 -𝛾𝑟) + 𝛾 v𝑖,𝑛 𝑦 𝑖,𝑛+1 = 𝑦 𝑖,𝑛 + 𝛾 𝑝 𝑖,𝑛+1(C.37) with 𝛾𝑟 < 1. Below, we provide the rate of convergence for the setup of 𝐸𝑥𝑎𝑚 𝑝𝑙𝑒 3.1, as we did before. Namely, we obtain a linear convergence rate, as the following proposition suggests. Proposition C.2. Consider the single-player game Γ with actions A and B such that 𝑢(A) -𝑢(B) = 1 of Example 3.1, and let 𝑥 𝑛 = Λ(𝑦 𝑛 ) be the sequence of play generated by (C.37). Then, denoting by 𝑥 * = (1, 0) the strict Nash equilibrium, we have:∥𝑥 𝑛 -𝑥 * ∥ ∞ ∼ exp 𝐶 - 𝛾 𝑟 𝑛 . (C.38)where 𝐶 > 0 is a constant that depends on the initialization of the algorithm.Proof. We first define the score-difference𝑤 𝑛 𝑝 B,𝑛 -𝑝 A,𝑛 (C.39)with initial condition 𝑤 1 = 0. Then, unfolding according to the sequence of play, we obtain:𝑤 𝑛+1 = 𝑤 𝑛 (1 -𝛾𝑟) + 𝛾(𝑢(B) -𝑢(A)) (C.40) = 𝑤 𝑛 (1 -𝛾𝑟) -𝛾 (C.41) = . . . = -𝛾 𝑛-1 ∑︁ 𝑘=0 (1 -𝛾𝑟) 𝑘 (C.42) = - 1 -(1 -𝛾𝑟) 𝑛 𝑟 (C.43)We next define for 𝑛 ∈ ℕ the difference 𝑧 𝑛 𝑦 B,𝑛 -𝑦 A,𝑛 . Thus, unfolding it, we obtain: 𝑧 𝑛+1 = 𝑧 𝑛 + 𝛾𝑤 𝑛+1 (C.44) = 𝑧 𝑛 -𝛾 1 -(1 -𝛾𝑟) 𝑛 𝑟 (C.45) = 𝑧 1 -𝛾 𝑛 ∑︁ 𝑚=1 1 -(1 -𝛾𝑟) 𝑚 𝑟 (C.46) = 𝑧 1 - 𝛾 𝑟 𝑛 -(1 -𝛾𝑟) 1 -(1 -𝛾𝑟) 𝑛 𝛾𝑟 (C.47) = 𝑧 1 - 𝛾 𝑟 𝑛 + O(∥𝑥 𝑇 -𝑥 * ∥ ∞ ≤ exp 𝐶 -𝑐𝛾 2 𝑇 (𝑇 -1) 2 = exp -Θ(𝑇 2 ) (12)where 𝐶 > 0 is a constant that depends only on the initialization of (FTXL) and where 𝑒 𝛼 , 𝑒 * 𝛼 are the standard basis vectors corresponding to 𝛼, 𝛼 * ∈ A. Let initial conditions 𝑦 1 such that 𝑦 𝛼,1 -𝑦 𝛼 * ,1 = -𝑀 -𝜀, for all 𝛼 ≠ 𝛼 * ∈ A, where 𝜀 > 0 small, and 𝑝 1 = 0. We will first show by induction that 𝑧 𝛼,𝑛 < -𝑀 for all 𝑛 ∈ ℕ. To this end, unfolding the recursion, we obtain:EQUATION𝑤 𝛼,𝑛+1 = 𝑤 𝛼,𝑛 + 𝛾⟨v 𝑛 , 𝑒 𝛼 -𝑒 * 𝛼 ⟩ = 𝑤 𝛼,𝑛 + 𝛾⟨𝑣(𝑥 𝑛 ), 𝑒 𝛼 -𝑒 * 𝛼 ⟩ = 𝛾 𝑛 ∑︁ 𝑘=1 ⟨𝑣(𝑥 𝑘 ), 𝑒 𝛼 -𝑒 * 𝛼 ⟩ (C.51)where we used that 𝑤 1 = 0. Now, for the sake of induction, suppose that 𝑧 𝛼,𝑘 < -𝑀 for all 𝑘 = 1, . . . , 𝑛 (C.52) which implies that ⟨𝑣(𝑥 𝑘 ), 𝑒 𝛼 -𝑒 * 𝛼 ⟩ < -𝑐. With this in hand, we will prove that 𝑧 𝛼,𝑛+1 < -𝑀, as well. Specifically, we have:𝑧 𝛼,𝑛+1 = 𝑧 𝛼,𝑛 + 𝛾𝑤 𝛼,𝑛+1 = 𝑧 𝛼,𝑛 + 𝛾 2 𝑛 ∑︁ 𝑘=1 ⟨𝑣(𝑥 𝑘 ), 𝑒 𝛼 -𝑒 * 𝛼 ⟩ ≤ 𝑧 𝛼,𝑛 -𝑐𝛾 2 𝑛 ≤ 𝑧 𝛼,1 -𝑐𝛾 2 𝑛 ∑︁ ℓ=1 ℓ < -𝑀 (C.53)where we used the inductive hypothesis and the initial condition. Therefore, we conclude by induction that 𝑧 𝛼,𝑛 < -𝑀 for all 𝑛 ∈ ℕ. Thus, we readily obtain that after 𝑇 time-step:𝑧 𝑇 ≤ 𝑧 𝛼,1 -𝑐𝛾 2 𝑇 -1 ∑︁ ℓ=1 ℓ ≤ 𝑧 𝛼,1 -𝑐𝛾 2 𝑇 (𝑇 -1) 2 (C.54)and invoking Lemma A.1 for 𝜃 (𝑥) = 𝑥 log 𝑥, we get the result. ■In this appendix, we provide the proofs of Theorem 3 and Theorem 4 that correspond to the convergence of (FTXL) with realization-based and bandit feedback, respectively. For this, we need the following lemma, which provides a maximal bound on a martingale process. Namely, we have: Lemma D. for some 𝑞 > 2. Then, for 𝜇 ∈ (0, 1) and 𝑛 ∈ ℕ:ℙ sup 𝑘 ≤𝑛 |𝑀 𝑘 | > 𝑐 𝑛 ∑︁ 𝑘=1 𝛾 𝑘 𝜇 ≤ 𝐴 𝑞 𝑛 𝑘=1 𝛾 𝑞/2+1 𝑘 𝜎 𝑞 𝑘 𝑛 𝑘=1 𝛾 𝑘 1+𝑞 ( 𝜇-1/2) (D.1)where 𝐴 𝑞 is a constant depending only on 𝑐 and 𝑞.Proof. Fix some 𝜇 ∈ (0, 1). By Doob's maximal inequality [16, Corollary 2.1], we have:ℙ sup 𝑘 ≤𝑛 |𝑀 𝑘 | > 𝑐 𝑛 ∑︁ 𝑘=1 𝛾 𝑘 𝜇 ≤ 𝔼[|𝑀 𝑛 | 𝑞 ] 𝑐 𝑞 𝑛 𝑘=1 𝛾 𝑘 𝑞𝜇 (D.2)Now, applying the Burkholder-Davis-Gundy inequality [16, Theorem 2.10], we get that𝔼[|𝑀 𝑛 | 𝑞 ] ≤ 𝐴 𝑞 𝔼       𝑛 ∑︁ 𝑘=1 𝛾 2 𝑘 ∥𝜉 𝑘 ∥ 2 * 𝑞/2      (D.3)where 𝐴 𝑞 is a constant depending only on 𝑐 and 𝑞. Now, we will invoke the generalized Hölder's inequality [3] :𝑛 ∑︁ 𝑘=1 𝑎 𝑘 𝑏 𝑘 𝜌 ≤ 𝑛 ∑︁ 𝑘=1 𝑎 𝜆𝜌 𝜌-1 𝑘 𝜌-1 𝑛 ∑︁ 𝑘=1 𝑎 (1-𝜆)𝜌 𝑘 𝑏 𝜌 𝑘 (D.4)for 𝑎 𝑘 , 𝑏 𝑘 ≥ 0, 𝜌 > 1 and 𝜆 ∈ [0, 1). Thus, setting 𝑎 𝑘 = 𝛾 2 𝑘 , 𝑏 𝑘 = ∥𝜉 𝑘 ∥ 2 * , 𝜌 = 𝑞/2 and 𝜆 = 1/2 -1/𝑞, (D.2), combined with (D.3), becomes:ℙ sup 𝑘 ≤𝑛 |𝑀 𝑘 | > 𝑐 𝑛 ∑︁ 𝑘=1 𝛾 𝑘 𝜇 ≤ 𝐴 𝑞 𝑛 𝑘=1 𝛾 𝑘 𝑞/2-1 𝑛 𝑘=1 𝛾 𝑞/2+1 𝑘 𝔼[∥𝜉 𝑘 ∥ 𝑞 * ] 𝑛 𝑘=1 𝛾 𝑘 𝑞𝜇 ≤ 𝐴 𝑞 𝑛 𝑘=1 𝛾 𝑞/2+1 𝑘 𝜎 𝑞 𝑘 𝑛 𝑘=1 𝛾 𝑘 1+𝑞 ( 𝜇-1/2) (D.5)and the proof is complete. ■ With this tool in hand, we move forward to the convergence of (FTXL) under realization-based feedback. Theorem 3. Let 𝑥 * be a strict Nash equilibrium of Γ, fix some confidence level 𝛿 > 0, and let 𝑥 𝑛 = 𝑄(𝑦 𝑛 ) be the sequence of play generated by (FTXL) with realization-based feedback as per (11b) and a sufficiently small step-size 𝛾 > 0. Then there exists a neighborhood U of 𝑥 * such thatEQUATIONIn particular, if (FTXL) is run with logit best responses (that is, 𝑄 ← Λ), there exist positive constants 𝐶, 𝑐 > 0 as in Theorem 2 such that on the event {𝑥 𝑛 → 𝑥 * as 𝑛 → ∞}:EQUATIONProof. First of all, since 𝑥 * is a strict Nash equilibrium, by Lemma B. where 𝑒 𝛼 , 𝑒 * 𝛼 are the standard basis vectors corresponding to 𝛼, 𝛼 * ∈ A. Then, unfolding the recursion, we obtain:𝑤 𝛼,𝑛+1 = 𝑤 𝛼,𝑛 + 𝛾⟨v 𝑛 , 𝑒 𝛼 -𝑒 * 𝛼 ⟩ = 𝑤 𝛼,𝑛 + 𝛾⟨𝑣(𝑥 𝑛 ), 𝑒 𝛼 -𝑒 * 𝛼 ⟩ + 𝛾⟨𝑈 𝑛 , 𝑒 𝛼 -𝑒 * 𝛼 ⟩ = 𝛾 𝑛 ∑︁ 𝑘=1 ⟨𝑣(𝑥 𝑘 ), 𝑒 𝛼 -𝑒 * 𝛼 ⟩ + 𝛾 𝑛 ∑︁ 𝑘=1 ⟨𝑈 𝑘 , 𝑒 𝛼 -𝑒 * 𝛼 ⟩ (D.8)where we used that 𝑤 1 = 0. Now, define the stochastic process {𝑀 𝑛 } 𝑛∈ℕ as𝑀 𝑛 𝛾 𝑛 ∑︁ 𝑘=1 ⟨𝑈 𝑘 , 𝑒 𝛼 -𝑒 * 𝛼 ⟩ (D.9)which is a martingale, since 𝔼[𝑈 𝑛 | F 𝑛 ] = 0. Moreover, note that∥𝑈 𝑛 ∥ * = ∥𝑣(𝛼 𝑛 ) -𝑣(𝑥 𝑛 ) ∥ * ≤ 2 max 𝛼∈A ∥𝑣(𝛼) ∥ * (D.10)and, thus, we readily obtain that𝔼[∥𝑈 𝑛 ∥ 𝑞 * | F 𝑛 ] ≤ 𝜎 𝑞 for 𝜎 = 2 max 𝛼∈A ∥𝑣(𝛼) ∥ * and all 𝑞 ∈ [1, ∞]. By Lemma D.1 for 𝛾 𝑛 = 𝛾, 𝜎 𝑛 = 𝜎, 𝜉 𝑛 = ⟨𝑈 𝑛 , 𝑒 𝛼 -𝑒 *𝛼 ⟩, 𝑐 as in Theorem 2, and 𝜇 ∈ (0, 1), 𝑞 > 2 whose values will be determined next, there exists 𝐴 𝑞 > 0 such that:𝛿 𝑛 ℙ sup 𝑘 ≤𝑛 |𝑀 𝑘 | > 𝑐(𝛾𝑛) 𝜇 ≤ 𝐴 𝑞 𝜎 𝑞 𝑛𝛾 𝑞/2+1 (𝛾𝑛) 1+𝑞 ( 𝜇-1/2) ≤ 𝐴 𝑞 𝜎 𝑞 𝛾 𝑞 (1-𝜇) 𝑛 𝑞 ( 𝜇-1/2) (D.11)Now, we need to guarantee that there exist 𝜇 ∈ (0, 1), 𝑞 > 2, such that∞ ∑︁ 𝑛=1 𝛿 𝑛 < ∞ (D.12)For this, we simply need 𝑞(𝜇 -1/2) > 1, or equivalently, 𝜇 > 1/2 + 1/𝑞, which implies that 𝜇 ∈ (1/2, 1). Therefore, for 𝛾 small enough, we get ∞ 𝑛=1 𝛿 𝑛 < 𝛿, and therefore:ℙ ∞ 𝑛=1 sup 𝑘 ≤𝑛 |𝑀 𝑘 | ≤ 𝑐(𝛾𝑛) 𝜇 = 1 -ℙ ∞ 𝑛=1 sup 𝑘 ≤𝑛 |𝑀 𝑘 | > 𝑐(𝛾𝑛) 𝜇 ≥ 1 - ∞ ∑︁ 𝑛=1 𝛿 𝑛 ≥ 1 -𝛿 (D.13)From now on, we denote the good event ∞ 𝑛=1 sup 𝑘 ≤𝑛 |𝑀 𝑘 | ≤ 𝑐(𝛾𝑛) 𝜇 by 𝐸. Then, with probability at least 1 -𝛿:𝑤 𝛼,𝑛+1 ≤ 𝛾 𝑛 ∑︁ 𝑘=1 ⟨𝑣(𝑥 𝑘 ), 𝑒 𝛼 -𝑒 * 𝛼 ⟩ + 𝑐(𝛾𝑛) 𝜇 for all 𝑛 ∈ ℕ. (D.14)Furthermore, we have that for 𝑛 > 𝑁 0 ⌈1/𝛾⌉, we readily get that 𝛾𝑛 > (𝛾𝑛) 𝜇 . Therefore, setting𝑅 𝑐𝛾 𝑁 0 -1 ∑︁ 𝑘=1 ((𝛾𝑘) 𝜇 -𝛾𝑘) (D.15)we obtain:-𝑐𝛾 𝑛 ∑︁ 𝑘=1 (𝛾𝑘 -(𝛾𝑘) 𝜇 ) ≤ 𝑅 (D.16)for all 𝑛 ∈ ℕ. Then, initializing 𝑦 1 such that 𝑧 𝛼,1 < -𝑀 -𝑅, we will show that 𝑧 𝛼,𝑛 < -𝑀 for all 𝑛 ∈ ℕ with probability at least 1 -𝛿. For this, suppose that 𝐸 is realized, and assume that 𝑧 𝛼,𝑘 < -𝑀 for all 𝑘 = 1, . . . , 𝑛 (D.17)We will show that 𝑧 𝛼,𝑛+1 < -𝑀, as well. For this, we have:𝑧 𝛼,𝑛+1 = 𝑧 𝛼,𝑛 + 𝛾𝑤 𝛼,𝑛+1 ≤ 𝑧 𝛼,𝑛 + 𝛾 𝛾 𝑛 ∑︁ 𝑘=1 ⟨𝑣(𝑥 𝑘 ), 𝑒 𝛼 -𝑒 * 𝛼 ⟩ + 𝑐(𝛾𝑛) 𝜇 ≤ 𝑧 𝛼,𝑛 -𝑐𝛾(𝛾𝑛 -(𝛾𝑛) 𝜇 ) ≤ 𝑧 𝛼,1 -𝑐𝛾 𝑛 ∑︁ 𝑘=1 (𝛾𝑘 -(𝛾𝑘) 𝜇 ) ≤ -𝑀 -𝑅 -𝑐𝛾 𝑛 ∑︁ 𝑘=1 (𝛾𝑘 -(𝛾𝑘) 𝜇 ) < -𝑀 (D.18)Therefore, we conclude by induction that 𝑧 𝛼,𝑛 < -𝑀 for all 𝑛 ∈ ℕ. Thus, we readily obtain that with probability at least 1 -𝛿 it holds:𝑧 𝛼,𝑇 ≤ 𝑧 𝛼,1 -𝑐𝛾 𝑇 -1 ∑︁ 𝑘=1 (𝛾𝑘 -(𝛾𝑘) 𝜇 ) (D.19) ≤ 𝑧 𝛼,1 -𝑐𝛾 2 𝑇 (𝑇 -1) 2 + 𝑐𝛾 1+𝜇 ∫ 𝑇 0 𝑡 𝜇 𝑑𝑡 (D.20) ≤ 𝑧 𝛼,1 -𝑐𝛾 2 𝑇 (𝑇 -1) 2 + 𝑐𝛾 1+𝜇 𝑇 𝜇+1 𝜇 + 1 (D.21) (D.22)for all 𝑇 ∈ ℕ. Setting 𝜇 = 2/3 and invoking Lemma A.1 for 𝜃 (𝑥) = 𝑥 log 𝑥, we get the result. ■ Finally, we prove the convergence of (FTXL) under realization-based feedback. Theorem 4. Let 𝑥 * be a strict Nash equilibrium of Γ, fix some confidence level 𝛿 > 0, and let 𝑥 𝑛 = 𝑄(𝑦 𝑛 ) be the sequence of play generated by (FTXL) with bandit feedback of the form (11c), an IWE exploration parameter 𝜀 𝑛 ∝ 1/𝑛 ℓ 𝜀 for some ℓ 𝜀 ∈ (0, 1/2), and a sufficiently small step-size 𝛾 > 0. Then there exists a neighborhood U of 𝑥 * in X such thatEQUATIONIn particular, if (FTXL) is run with logit best responses (that is, 𝑄 ← Λ), there exist positive constants 𝐶, 𝑐 > 0 as in Theorem 2 such that on the eventEQUATIONProof. First of all, since 𝑥 * is a strict Nash equilibrium, by Lemma B. where 𝑒 𝛼 , 𝑒 * 𝛼 are the standard basis vectors corresponding to 𝛼, 𝛼 * ∈ A. For notational convenience, we focus on player 𝑖 and drop the player-specific indices altogether. Now, decomposing the IWE v𝑛 , we obtain v𝑛 = 𝑣(𝑥 𝑛 ) + 𝑈 𝑛 + 𝑏 𝑛 (D. 25) where 𝑈 𝑛 v𝑛 -𝑣 𝑖 ( x𝑛 ) is a zero-mean noise, and 𝑏 𝑖,𝑛 𝑣 𝑖 ( x𝑛 ) -𝑣 𝑖 (𝑥 𝑛 ). Then, unfolding the recursion, we obtain: with 𝑛 𝑘=1 𝑘 𝑞ℓ 𝜀 = Θ(𝑛 1+𝑞ℓ 𝜀 ). Therefore,𝑤 𝛼,𝑛+1 = 𝑤 𝛼,𝑛 + 𝛾⟨v 𝑛 , 𝑒 𝛼 -𝑒 * 𝛼 ⟩ = 𝑤 𝛼,𝑛 + 𝛾⟨𝑣(𝑥 𝑛 ), 𝑒 𝛼 -𝑒 * 𝛼 ⟩ + 𝛾⟨𝑈 𝑛 , 𝑒 𝛼 -𝑒 * 𝛼 ⟩ + 𝛾⟨𝑏 𝑛 , 𝑒 𝛼 -𝑒 * 𝛼 ⟩ ≤ 𝑤 𝛼,𝛿 𝑛 ≤ 𝐴 ′ 𝑞 𝛾 𝑞 (1-𝜇) 𝜀 -𝑞 𝑛 1+𝑞ℓ 𝜀 𝑛 1+𝑞 ( 𝜇-1/2) ≤ 𝐴 ′ 𝑞 𝛾 𝑞 (1-𝜇) 𝜀 -𝑞 𝑛 𝑞 ( 𝜇-1/2-ℓ 𝜀 ) (D.31)Now, we need to guarantee that there exist 𝜇 ∈ (0, 1), 𝑞 > 2, such that∞ ∑︁ 𝑛=1 𝛿 𝑛 < ∞ (D.32)For this, we need to ensure that 𝑞(𝜇 -1/2ℓ 𝜀 ) > 1, or, equivalently,ℓ 𝜀 < 𝜇 -1/2 -1/𝑞 (D.33)which we will do later. Then, we will get for 𝛾 small enough:ℙ ∞ 𝑛=1 sup 𝑘 ≤𝑛 |𝑀 𝑘 | ≤ 𝑐 2 (𝛾𝑛) 𝜇 = 1 -ℙ ∞ 𝑛=1 sup 𝑘 ≤𝑛 |𝑀 𝑘 | > 𝑐 2 (𝛾𝑛) 𝜇 ≥ 1 - ∞ ∑︁ 𝑛=1 𝛿 𝑛 ≥ 1 -𝛿 (D.34)Regarding the term 2𝛾𝐵 𝑛 𝑘=1 𝜀 𝑘 in (D.26), we have that:2𝛾𝐵 𝑛 ∑︁ 𝑘=1 𝜀 𝑘 = 2𝐵𝛾𝜀 𝑛 ∑︁ 𝑘=1 𝑘 -ℓ 𝜀 ≤ 𝐵 ′ 𝛾𝜀𝑛 1-ℓ 𝜀 (D.35)where we used that 𝑛 𝑘=1 𝑘 -ℓ 𝜀 = Θ(𝑛 1-ℓ 𝜀 ). Thus, for 1ℓ 𝜀 < 𝜇 (D.36)we have for 𝜀, 𝛾 > 0 small enough: EQUATIONIn this section, we provide numerical simulations to validate and explore the performance of (FTXL).To this end, we consider two game paradigms, (i) a zero-sum game, and (ii) a congestion game.Zero-sum Game. First, we consider a 2-player zero-sum game with actions {𝛼 1 , 𝛼 2 , 𝛼 3 } and {𝛽 1 , 𝛽 2 , 𝛽 3 }, and payoff matrix𝑃 = (2, -2) (1, -1) (2, -2) (-2, 2) (-1, 1) (-2, 2) (-2, 2) (-1, 1) (-2, 2)Here, the rows of 𝑃 correspond to the actions of player 𝐴 and the columns to the actions of player 𝐵, while the first item of each entry of 𝑃 corresponds to the payoff of 𝐴, and the second one to the payoff of 𝐵. Clearly, the action profile (𝛼 1 , 𝛽 2 ) is a strict Nash equilibrium.Congestion Game. As a second example, we consider a congestion game with 𝑁 = 100 and 2 roads, 𝑟 1 and 𝑟 2 , with costs 𝑐 1 = 1.1 and 𝑐 2 = 𝑑/𝑁 where 𝑑 is the number of drivers on 𝑟 2 . In words, 𝑟 1 has a fixed delay equal to 1.1, while 𝑟 2 has a delay proportional to the drivers using it. Note, that the strategy profile where all players are using 𝑟 2 is a strict Nash equilibrium.In Fig. 1 , we assess the convergence of (FTXL) with logit best responses, under realization-based and bandit feedback, and compare it to the standard (EW) with the same level of information. For each feedback mode, we conducted 100 separate trials, each with 𝑇 = 10 3 steps, and calculated the average norm ∥𝑥 𝑛 -𝑥 * ∥ 1 as a function of the iteration counter 𝑛 = 1, 2, ..., 𝑇. The solid lines represent the average distance from equilibrium for each method, while the shaded areas enclose the range of ±1 standard deviation from the mean across the different trials. All the plots are displayed in logarithmic scale. For the zero-sum game, all runs were initialized with 𝑦 1 = 0, and we used constant step-size 𝛾 = 10 -2 , and exploration parameter 𝜀 = 10 -1 , where applicable. For the congestion game, the initial state 𝑦 1 for each run was drawn uniformly at random in [-1, 1] 2 , and we used constant step-size 𝛾 = 10 -2 , and exploration parameter 𝜀 𝑛 = 1/𝑛 1/4 , where applicable.The experiments have been implemented using Python 3.11.5 on a M1 MacBook Air with 16GB of RAM.NeurIPS Paper ChecklistQuestion: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)? 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 Appendix:  


Decision: This paper considers the problem of learning in games. The main result is to provide a general family of algorithms called *follow the accelerated leader (FTXL)* which incorporates acceleration into the FTRL framework, and achieves a speedup over vanilla FTRL under various feedback structures.


Reviewers generally agreed that this paper makes a solid and clear contribution by integrating Nesterov-style acceleration into learning in games, and is likely to inspire further work in this direction, meriting acceptance. They also found the paper to be clear and well written.


A minor comment for the final version: The authors use the term "quadratic rate" to refer to exp(-T^2) convergence (or \sqrt{\log(1/eps)} iteration complexity to achieve precision \eps). This is different from the typical use of the term in optimization (e.g., in the context of Newton's method), where quadratic convergence typically refers to exp(-exp(T)) or \log\log(1/eps) convergence. I recommend using another term to avoid confusion here.

Reviews: review: paper_summary: paper_summary: This paper primarily focuses on introducing a Nesterov's accelerated gradient (NAG) algorithm for online learning in games.
Initially, the author shows that a continuous-time version of NAG converges to a strict Nash equilibrium at a rate that is quadratic.
This rate of convergence is notably faster than that of standard FTRL algorithms.
Following this, the paper demonstrates that the convergence rate of NAG is preserved even in a bandit feedback setting.
The experimental results confirm that the proposed algorithm converges to an equilibrium more quickly than a FTRL algorithm.
strength_weakness: strength_weakness: My primary concern is that the theoretical results are only applicable to games with a strict Nash equilibrium.
This is somewhat problematic as even in a simple game like rock-paper-scissors, a strict Nash equilibrium does not exist.
Moreover, the convergence in games that do not have a strict equilibrium has been the subject of extensive research.
Therefore, I'm wondering how efficient NAG algorithms are in two-player zero-sum games and monotone games.

Furthermore, Theorems 1, 2, 3, and 4 hinge on the presumption that the initial strategy is in close proximity to an equilibrium.
Could you provide some insight into how close the initial point needs to be to the equilibrium?
questions: questions: My major concerns and questions are stated in Weaknesses.
I also have the following additional questions:
* Why is the update rule of $y_{i,n+1}=y_{i,n}+\gamma p_{i,n+1}$ in (FTXL) derived? Looking at the original NAG's update rule in (NAG), it seems more natural that $y_{i,n+1}=y_{i,n}+\gamma (p_{i,n+1}-p_{i,n})$.
* Is it possible to recover FTRL algorithms by setting $r$ appropriately in FTXL?
limitations: limitations: I see no negative societal impacts that need to be addressed.
paper_summary: paper_summary: The paper proposes a momentum-based follow-the-regularized-leader (FTRL) type algorithm for finding the Nash equilibrium in finite games. The paper first investigates the continuous second order ODE and a concrete game, and then devises the appropriate FTRL scheme. The convergence rate for the proposed algorithm in all three information settings of reward is quadratic (O( exp(-T^2))).
strength_weakness: strength_weakness: See question.
questions: questions: Minor comments
1. Are lower bounds of learning a finite game with the three kinds feedbacks known? If so, it would be great to provide a survey.
2. In line 305 the phrasing "second-order in space" is a bit confusing. In my opinion, it would be better to say "using Hessian information of relevant functions".

Typo 
1. At Eq FTRL, near line 131, the second equation should be x_i,n = Q_i (y_i, n).
limitations: limitations: The paper provides a extensive discussion of momentum method for finite game solving, and does not have major limitations in my opnion.
paper_summary: paper_summary: The submission studies the convergence of finite N-person games. Combining the idea of (NAG) and (FTRL), the submission proposes a continuous-time dynamic (FTXL-D) and its discrete-time scheme (FTXL). The novelty of integrating momentum and regularization into an algorithm allows the construction of quadratic rate methods.
strength_weakness: strength_weakness: - A smoother transition toward (FTXL-D) in lines 182–186 is desired. In a game, $x$ is the response and $y$ is an aggregated payoff. For (HBVF), the interpretation is for a moving object. 
  - What is the correspondence of $x$ and $y$ of a game in (HBVF)? Do we correspond $x$ (position) in (HBVF) to $y$ (payoff) in a game? If so, what corresponds to the response ($x$ in a game) in (HBVF)? Is it the control $w$ (my naive guess)? 
  - These analogs, with the aid of a clearer transition, would help us to better understand the first paragraph of Section 3.2. Maybe the response x in a game and the position x in (HBVF) introduces a bit of confusion, too? 
- There are two sets of notations, one for the continuous-time regime and the other for the discrete-time regime. A small table to distinguish them might help the readability.
- Although in general, the submission is easy to read, a minor aspect of improving readability could be explaining the meaning $\dot{y}$ (lines 192 and 196) and $\ddot{z}$ (line 486), which could ease the reading barrier for people who are more comfortable with discrete-time analysis.
questions: questions: - In line 245, the friction parameter is set to zero ($r=0$) according to the discussion of Example 3. This setting, by direct substitution, would cancel out the desired effect of the momentum in (FTXL-D) in line 186 or (4) in line 197. However, in (FTXL) (line 258), the momentum appears to play an important role in the leftmost equality $y_{n+1} = y_n + \gamma p_n$. Did I misunderstand something?
- In Figure 1, why does FTXL perform poorly in the early rounds? Is it due to the constants in the bound or is there another reason?
limitations: limitations: Yes.
paper_summary: paper_summary: The paper studies the extension of Nesterov’s accelerated gradient algorithm to the solution of N-player games, named "follow the accelerated leader" (FTXL). The method is studied in both continuous and discrete time, and under various information oracles. The convergence of the algorithm is super-linear when started from an initialization close enough to the NE. Simulations are conducted which confirms the accelerated convergence numerically.
strength_weakness: strength_weakness: The convergence of FTXL in all settings requires sufficiently close initialization or has to do with the neighborhood $\mathcal{U}$. The exact definition of "sufficiently close" or this neighborhood is not given in the main paper. I believe this should be added in the statement of the theorems for clarity. The authors should also discuss how realistic it is that the initial point satisfies the requirement (and/or how likely the iterates of FTXL or FTRL may run into this "good neighborhood").

It seems that no assumption on the convexity of the payoff function is made. With a general non-convex payoff function, I wonder what is the problem structure and technical apparatus that allow the authors to derive the quadratic convergence (or even convergence at all). This is currently not sufficiently discussed in the main paper. While I am happy to study the detailed analysis, I would appreciate a sketch of the proof which highlights the main techniques.
questions: questions: Please see Weaknesses section.
limitations: limitations: Paper of theoretical nature. No negative social impact is foreseeable.

Comments: title: Thank you
Small question
comment: Thank you again for your time and input.

Kind regards,

The authors
Thank you again for your time, input, and positive evaluation!

Kind regards,

The authors
Thank you for your response. I have no further questions, and I will keep my score.
Dear AC,

> Can the authors comment on whether there are any connections between their FTXL method and the "linear coupling" method of Allen-Zhu and Orecchia [1], which also aims to combine mirror descent/FTRL and acceleration? The technique I mention is not analyzed in the context of games, but I am curious of there are connections nonetheless.

Excellent question, thanks for bringing it up! It's a complex issue, so we're giving two answers, one shorter and one (much) longer.

---

### **Short answer**

The linear coupling approach of [1] is related to the momentum-based approach of Su, Boyd and Candès which forms the basis of (FTXL), but the two are not equivalent. In essence, they are both different approaches to acceleration – and, in turn, both different than the original "estimate sequences" approach of Nesterov – and we do not see a way of directly applying the techniques of [1] to our setting.

---

### **Long answer**

Because [1] is not taking a momentum-based approach (see e.g., footnote 5 in p. 4 of the arxiv version of the paper), it is difficult to accurately translate the coupling approach of [1] to our setting and provide a direct comparison between the two methods.

One of the main reasons for this is that [1] is essentially using two step-sizes: the first is taken equal to the inverse Lipschitz modulus of the function being minimized and is used to take a gradient step; the second step-size sequence is much more aggressive, and it is used to generate an ancillary, exploration sequence which "scouts ahead". These two sequences are then "coupled" with a mixing coefficient which plays a role "similar" - but *not equivalent* - to the friction coefficient in the (HBVF) formulation of (NAG) by Su, Boyd, and Candès.

The above is the best high-level description and analogy we can make between the coupling approach of [1] and the momentum-driven analysis of Su, Boyd, and Candès and/or momentum analysis in Nesterov's 2004 textbook (which is itself quite different from Nesterov's original 1983 paper).

At a low level (and omitting certain technical details and distinctions that are not central to this discussion), the linear coupling approach of [1] applied to our setting would correspond to the update scheme:

\begin{align}
x_t &= Q(y_t)
    \\\\
w_t &= \lambda_t z_t + (1-\lambda_t) x_t
    \\\\
y_{t+1} &= y_t + (1-\lambda_t) \eta_t \hat v_t
    \\\\
z_{t+1} &= \lambda_t z_t + (1-\lambda_t) x_{t+1}
\end{align}

with $\hat v_t$ obtained by querying a first-order oracle at $w_t$ – that is, $\hat v_t$ is an estimate, possibly imperfect, of $v(w_t)$.

The first and third lines of this update scheme are similar to the corresponding update structure of (FTXL). However, whereas (FTXL) builds momentum by the aggregation of gradient information via the momentum variables $p_t$, the linear coupling method above achieves acceleration through the coupling of the sequences $w_t$, $z_t$ and $x_t$, and by taking an increasing step-size sequence $\eta_t$ that grows roughly as $\Theta(t)$, and a mixing coefficient $\lambda_t$ that evolves as $\lambda_t = 1 - 1/(L\eta_t)$, where $L$ is the Lipschitz modulus of $v$.

Beyond this comparison, we cannot provide a term-by-term correspondence between the momentum-based and coupling-based approaches, because the two methors are not equivalent (even though they give the same value convergence rates). In particular, we do not see a way of linking the parameters $\eta_t$ and $\lambda_t$ of the coupling approach to the friction and step-size parameters of the momentum approach.

In the context of convex minimization problems, the coupling-based approach of [1] is more amenable to a regret-based analysis – this is the "unification" that [1] referred to – while the momentum-based approach of Su, Boyd, and Candès facilitates a Lyapunov-based analysis. Ultimately, these are all different - though, of course, equally valid - approaches to acceleration. From a game-theoretic standpoint, the momentum-based approach seems to be more fruitful and easier to implement, but studying the linear coupling approach of [1] could also be very relevant.

---

We will be happy to include a version of the above discussion at the first possible revision opportunity. In the meantime, please let us know if you have any follow-up questions on the above, and thanks again for your question and for handling our submission!

Kind regards,

The authors
Thank you for your time and your support!

Kind regards,

The authors
Hi,

Can the authors comment on whether there are any connections between their FTXL method and the "linear coupling" method of Allen-Zhu and Orecchia [1], which also aims to combine mirror descent/FTRL and acceleration? The technique I mention is not analyzed in the context of games, but I am curious of there are connections nonetheless.

Thank you!

[1] Linear Coupling: An Ultimate Unification of Gradient and Mirror Descent
Thank you for your response. I have no further questions.
I thank the authors for the clarification and would like to maintain my score. I support the acceptance of the paper.