Organizations: Massachusetts Institute of Technology · OpenAI · University of Maryland, College Park
Abstract
In this paper, we study regret minimization in repeated games with \emph{adaptive} opponents who can respond based on histories of play. The standard metric of \emph{external regret} in online learning is known to fail to capture such adaptivity. To account for players' counterfactual reasoning, we introduce {\tt Repeated Policy Regret (RP-Regret)}, a game-theoretic metric that measures the difference between the \emph{realized} and the \emph{best-in-hindsight} accumulated utility when all players can \emph{respond} to the history of play. Compared to existing regret notions in this setting, ours is native to repeated game playing, enabling stronger comparators and opponents with fewer constraints, while maintaining the possibility of finding better equilibria when all players minimize it. We first identify necessary conditions for obtaining {\tt RP-Regret} sublinear in time, on the variation of the player's comparator strategies in the regret definition and on the memories of both the comparator and opponents' strategies. We then study additional conditions and provable algorithms to minimize {\tt RP-Regret}, which is by definition \emph{non-convex} in the strategy space. To address this challenge, we propose three algorithms: (i) one based on an optimization oracle, as assumed in some prior work in online non-convex learning; (ii) one that minimizes a convex and \emph{linearized} surrogate of {\tt RP-Regret} at each iteration; (iii) one that directly minimizes {\tt RP-Regret} when opponents change strategies slowly. Furthermore, when all players can run algorithms to minimize the {\tt RP-Regret} (or its linearized variant), certain subgame perfect equilibria of the repeated game can be learned. We also provide experiments showing that minimizing our regret notions can lead to more cooperative solutions with higher utility in games such as Stag-Hunt.
This note aims to serve as an entry point to the literature on learning in games, a topic with significant theoretical appeal and a wide range of applications -- from machine learning and data science to economics and beyond. Our presentation is structured around two complementary viewpoints: We first consider a single agent -- the learner -- engaged in a sequential decision process in an unknown, non-stationary, and possibly adversarial environment. We then examine what happens when the environment is shaped by the decisions of several interacting agents, not necessarily aware of each other's actions or goals, and all seeking to improve their individual rewards. In this general context, we examine a family of regularized learning policies based on best-responding to the past history of play, up to a regularization penalty intended to encourage exploration and prevent over-commitment to suboptimal choices. In the single-agent setting, we present some basic regret bounds for regularized learning in adversarial multi-armed bandits; in the multi-agent setting, we describe an ergodic equilibrium convergence result for zero-sum games in the spirit of classical results on fictitious play, as well as a "folk theorem" linking strategic and dynamic notions of stability -- Nash equilibria and attracting points of regularized learning, respectively. We pay special attention to the information available to the players and, through a unified analysis framework, we study both oracle- and payoff-based (bandit) methods. Our goal is to provide a coherent and comprehensible -- albeit, by necessity, not comprehensive -- account of some recent ideas in the field, and to discuss their implications for the study of rationality.
We study sequential decision-making in partially observable environments against strategic, adaptive opponents, modeled as partially observable Markov games (POMGs). The central challenge is to learn latent dynamics from partial observations while facing an adversary whose behavior depends on the learner's strategy, making standard regret notions inadequate. We prove that an epoch-based optimistic maximum-likelihood algorithm achieves O~(T) policy regret for fixed problem parameters, with explicit dependence on the horizon, adversary memory, confidence radius, and the aggregate Eluder dimension of the observable-operator class. The algorithm selects one policy per geometrically growing epoch using confidence sets built cumulatively from past data, which keeps the cost of comparing adversary responses across policies logarithmic in T. We also prove a lower bound matching the T and aggregate-Eluder-dimension dependence, up to problem-dependent and logarithmic factors. Finally, we extend the framework to horizon-adaptive guarantees and adversaries with geometric fading memory.
Uncoupled no-regret dynamics provide a decentralized route to equilibrium, but prior guarantees for individual regret retain a polylogarithmic dependence on the horizon. We remove this dependence for every finite N-player normal-form game under full-information feedback. We introduce \emph{ECHO-OFTRL}: optimistic follow-the-regularized-leader (OFTRL) equipped with an EMA cascade for high-order optimism (ECHO), where EMA denotes exponential moving average. The algorithm is deterministic and fully uncoupled. If mmax denotes the largest action-set size, then, simultaneously for every horizon T≥1, it guarantees that each of the N players in the game incurs regret upper bounded by O(poly(N,logmmax)). Our algorithm leverages a new form of optimism inspired by modern filter design.