In this paper we establish efficient and uncoupled learning dynamics so that, when employed by all players in a general-sum multiplayer game, the swap regret of each player after T repetitions of the game is bounded by O(log T ), improving over the prior best bounds of O(log 4 (T )). At the same time, we guarantee optimal O( √ T ) swap regret in the adversarial regime as well. To obtain these results, our primary contribution is to show that when all players follow our dynamics with a time-invariant learning rate, the second-order path lengths of the dynamics up to time T are bounded by O(log T ), a fundamental property which could have further implications beyond near-optimally bounding the (swap) regret. Our proposed learning dynamics combine in a novel way optimistic regularized learning with the use of self-concordant barriers. Further, our analysis is remarkably simple, bypassing the cumbersome framework of higher-order smoothness recently developed by Daskalakis, Fishelson, and Golowich (NeurIPS'21).

Most existing results about last-iterate convergence of learning dynamics are limited to two-player zero-sum games, and only apply under rigid assumptions about what dynamics the players follow. In this paper we provide new results and techniques that apply to broader families of games and learning dynamics. First, we use a regret-based analysis to show that in a class of games that includes constant-sum polymatrix and strategically zero-sum games, dynamics such as optimistic mirror descent (OMD) have bounded second-order path lengths, a property which holds even when players employ different algorithms and prediction mechanisms. This enables us to obtain O(1/ √ T ) rates and optimal O(1) regret bounds. Our analysis also reveals a surprising property: OMD either reaches arbitrarily close to a Nash equilibrium, or it outperforms the robust price of anarchy in efficiency. Moreover, for potential games we establish convergence to an -equilibrium after O(1/ 2 ) iterations for mirror descent under a broad class of regularizers, as well as optimal O(1) regret bounds for OMD variants. Our framework also extends to near-potential games, and unifies known analyses for distributed learning in Fisher's market model. Finally, we analyze the convergence, efficiency, and robustness of optimistic gradient descent (OGD) in general-sum continuous games.

Recently, Daskalakis, Fishelson, and Golowich (DFG) (NeurIPS '21) showed that if all agents in a multi-player general-sum normal-form game employ Optimistic Multiplicative Weights Update (OMWU), the external regret of every player is 𝑂 (polylog(𝑇 )) after 𝑇 repetitions of the game. In this paper we extend their result from external regret to internal and swap regret, thereby establishing uncoupled learning dynamics that converge to an approximate correlated equilibrium at the rate of 𝑂 𝑇 −1 . This substantially improves over the prior best rate of convergence of 𝑂 (𝑇 −3/4 ) due to Chen and Peng (NeurIPS '20), and it is optimal up to polylogarithmic factors.To obtain these results, we develop new techniques for establishing higher-order smoothness for learning dynamics involving fixed point operations. Specifically, we first establish that the no-internalregret learning dynamics of Stoltz and Lugosi (Mach Learn '05) are equivalently simulated by no-external-regret dynamics on a combinatorial space. This allows us to trade the computation of the stationary distribution on a polynomial-sized Markov chain for a (much more well-behaved) linear transformation on an exponentialsized set, enabling us to leverage similar techniques as DGF to near-optimally bound the internal regret.Moreover, we establish an 𝑂 (polylog(𝑇 )) no-swap-regret bound for the classic algorithm of Blum and Mansour (BM) (JMLR '07). We do so by introducing a technique based on the Cauchy Integral Formula that circumvents the more limited combinatorial arguments of DFG. In addition to shedding clarity on the near-optimal regret guarantees of BM, our arguments provide insights into the various ways in which the techniques by DFG can be extended and leveraged in the analysis of more involved learning algorithms.

In the literature on game-theoretic equilibrium finding, focus has mainly been on solving a single game in isolation. In practice, however, strategic interactions-ranging from routing problems to online advertising auctions-evolve dynamically, thereby leading to many similar games to be solved. To address this gap, we introduce meta-learning for equilibrium finding and learning to play games. We establish the first meta-learning guarantees for a variety of fundamental and well-studied classes of games, including two-player zero-sum games, general-sum games, and Stackelberg games. In particular, we obtain rates of convergence to different game-theoretic equilibria that depend on natural notions of similarity between the sequence of games encountered, while at the same time recovering the known single-game guarantees when the sequence of games is arbitrary. Along the way, we prove a number of new results in the single-game regime through a simple and unified framework, which may be of independent interest. Finally, we evaluate our meta-learning algorithms on endgames faced by the poker agent Libratus against top human professionals. The experiments show that games with varying stack sizes can be solved significantly faster using our meta-learning techniques than by solving them separately, often by an order of magnitude.

In this work, we study the metric distortion problem in voting theory under a limited amount of ordinal information. Our primary contribution is threefold. First, we consider mechanisms that perform a sequence of pairwise comparisons between candidates. We show that a popular deterministic mechanism employed in many knockout phases yields distortion O(log m) while eliciting only m − 1 out of the Θ(m2 ) possible pairwise comparisons, where m represents the number of candidates. Our analysis for this mechanism leverages a powerful technical lemma developed by Kempe (AAAI ‘20). We also provide a matching lower bound on its distortion. In contrast, we prove that any mechanism which performs fewer than m−1 pairwise comparisons is destined to have unbounded distortion. Moreover, we study the power of deterministic mechanisms under incomplete rankings. Most notably, when agents provide their k-top preferences we show an upper bound of 6m/k + 1 on the distortion, for any k ∈ {1, 2, . . . , m}. Thus, we substantially improve over the previous bound of 12m/k established by Kempe (AAAI ‘20), and we come closer to matching the best-known lower bound. Finally, we are concerned with the sample complexity required to ensure near-optimal distortion with high probability. Our main contribution is to show that a random sample of Θ(m/ϵ2 ) voters suffices to guarantee distortion 3 + ϵ with high probability, for any sufficiently small ϵ > 0. This result is based on analyzing the sensitivity of the deterministic mechanism introduced by Gkatzelis, Halpern, and Shah (FOCS ‘20). Importantly, all of our sample-complexity bounds are distribution-independent. From an experimental standpoint, we present several empirical findings on real-life voting applications, comparing the scoring systems employed in practice with a mechanism explicitly minimizing (metric) distortion. Interestingly, for our case studies, we find that the winner in the actual competition is typically the candidate who minimizes the distortion.