First-order methods (FOMs) are arguably the most scalable algorithms for equilibrium computation in large extensive-form games. To operationalize these methods, a distance-generating function, acting as a regularizer for the strategy space, must be chosen. The ratio between the strong convexity modulus and the diameter of the regularizer is a key parameter in the analysis of FOMs. A natural question is then: what is the optimal distance-generating function for extensive-form decision spaces? In this paper, we make a number of contributions, ultimately establishing that the weight-one dilated entropy (DilEnt) distance-generating function is optimal up to logarithmic factors. The DilEnt regularizer is notable due to its iterate-equivalence with Kernelized OMWU (KOMWU) -- the algorithm with state-of-the-art dependence on the game tree size in extensive-form games -- when used in conjunction with the online mirror descent (OMD) algorithm. However, the standard analysis for OMD is unable to establish such a result; the only current analysis is by appealing to the iterate equivalence to KOMWU. We close this gap by introducing a pair of primal-dual treeplex norms, which we contend form the natural analytic viewpoint for studying the strong convexity of DilEnt. Using these norm pairs, we recover the diameter-to-strong-convexity ratio that predicts the same performance as KOMWU. Along with a new regret lower bound for online learning in sequence-form strategy spaces, we show that this ratio is nearly optimal. Finally, we showcase our analytic techniques by refining the analysis of Clairvoyant OMD when paired with DilEnt, establishing an $\mathcal{O}(n \log |\mathcal{V}| \log T/T)$ approximation rate to coarse correlated equilibrium in $n$-player games, where $|\mathcal{V}|$ is the number of reduced normal-form strategies of the players, establishing the new state of the art.
We study the performance of optimistic regret-minimization algorithms for both minimizing regret in, and computing Nash equilibria of, zero-sum extensive-form games. In order to apply these algorithms to extensive-form games, a distance-generating function is needed. We study the use of the dilated entropy and dilated Euclidean distance functions. For the dilated Euclidean distance function we prove the first explicit bounds on the strong-convexity parameter for general treeplexes. Furthermore, we show that the use of dilated distance-generating functions enable us to decompose the mirror desc…
Solving Hierarchical Information-Sharing Dec-POMDPs: An Extensive-Form Game Approach
Université de Lyon（里昂大学）；INSA Lyon（里昂国立应用科学学院）；Inria（法国国家信息与自动化研究所）；Université de Lorraine（洛林大学）；CNRS（法国国家科学研究中心）；Bernoulli Institute（伯努利研究所）；University of Groningen（格罗宁根大学）
Department of Operations Research and Financial Engineering, Princeton University（普林斯顿大学运筹学与金融工程系）
In this paper, zero-sum mean-field type games (ZSMFTG) with linear dynamics and quadratic cost are studied under infinite-horizon discounted utility function. ZSMFTG are a class of games in which two decision makers whose utilities sum to zero, compete to influence a large population of indistinguishable agents. In particular, the case in which the transition and utility functions depend on the state, the action of the controllers, and the mean of the state and the actions, is investigated. The optimality conditions of the game are analysed for both open-loop and closed-loop controls, and expl…
Tree-form sequential decision making (TFSDM) extends classical one-shot decision making by modeling tree-form interactions between an agent and a potentially adversarial environment. It captures the online decision-making problems that each player faces in an extensive-form game, as well as Markov decision processes and partially-observable Markov decision processes where the agent conditions on observed history. Over the past decade, there has been considerable effort into designing online optimization methods for TFSDM. Virtually all of that work has been in the full-feedback setting, where …
Learning to play optimally against any mixture over a diverse set of strategies is of important practical interests in competitive games. In this paper, we propose simplex-NeuPL that satisfies two desiderata simultaneously: i) learning a population of strategically diverse basis policies, represented by a single conditional network; ii) using the same network, learn best-responses to any mixture over the simplex of basis policies. We show that the resulting conditional policies incorporate prior information about their opponents effectively, enabling near optimal returns against arbitrary mixt…
Federated learning is a distributed learning paradigm where multiple agents, each only with access to local data, jointly learn a global model. There has recently been an explosion of research aiming not only to improve the accuracy rates of federated learning, but also provide certain guarantees around social good properties such as total error. One branch of this research has taken a game-theoretic approach, and in particular, prior work has viewed federated learning as a hedonic game, where error-minimizing players arrange themselves into federating coalitions. This past work proves the exi…
University of Southern California（南加州大学）；Columbia University（哥伦比亚大学）
Regret-based algorithms are highly efficient at finding approximate Nash equilibria in sequential games such as poker games. However, most regret-based algorithms, including counterfactual regret minimization (CFR) and its variants, rely on iterate averaging to achieve convergence. Inspired by recent advances on last-iterate convergence of optimistic algorithms in zero-sum normal-form games, we study this phenomenon in sequential games, and provide a comprehensive study of last-iterate convergence for zero-sum extensive-form games with perfect recall (EFGs), using various optimistic regret-min…
School of Computer Science（计算机科学学院）；University of Waterloo（滑铁卢大学）；Vector Institute（向量研究所）
Convergence to a saddle point for convex-concave functions has been studied for decades, while recent years has seen a surge of interest in non-convex (zero-sum) smooth games, motivated by their recent wide applications. It remains an intriguing research challenge how local optimal points are defined and which algorithm can converge to such points. An interesting concept is known as the local minimax point, which strongly correlates with the widely-known gradient descent ascent algorithm. This paper aims to provide a comprehensive analysis of local minimax points, such as their relation with o…
Decoding Game: On Minimax Optimality of Heuristic Text Generation Strategies
Department of Electrical and Computer Engineering, Princeton University（普林斯顿大学电气工程与计算机科学系）；Department of Operations Research and Financial Engineering, Princeton University（普林斯顿大学运筹学与金融工程系）
A conceptually appealing approach for learning Extensive-Form Games (EFGs) is to convert them to Normal-Form Games (NFGs). This approach enables us to directly translate state-of-the-art techniques and analyses in NFGs to learning EFGs, but typically suffers from computational intractability due to the exponential blow-up of the game size introduced by the conversion. In this paper, we address this problem in natural and important setups for the \emph{$Φ$-Hedge} algorithm -- A generic algorithm capable of learning a large class of equilibria for NFGs. We show that $Φ$-Hedge can be directly use…
Technische Universität München（慕尼黑工业大学）；Munich Center for Quantum Science and Technology（慕尼黑量子科学与技术中心）
In this work, we study the system of interacting non-cooperative two Q-learning agents, where one agent has the privilege of observing the other's actions. We show that this information asymmetry can lead to a stable outcome of population learning, which generally does not occur in an environment of general independent learners. The resulting post-learning policies are almost optimal in the underlying game sense, i.e., they form a Nash equilibrium. Furthermore, we propose in this work a Q-learning algorithm, requiring predictive observation of two subsequent opponent's actions, yielding an opt…
The Hong Kong University of Science Technology（香港科技大学）；Peking University（北京大学）；University of Virginia（弗吉尼亚大学）
Offline reinforcement learning (RL) aims at learning an optimal strategy using a pre-collected dataset without further interactions with the environment. While various algorithms have been proposed for offline RL in the previous literature, the minimax optimality has only been (nearly) established for tabular Markov decision processes (MDPs). In this paper, we focus on offline RL with linear function approximation and propose a new pessimism-based algorithm for offline linear MDP. At the core of our algorithm is the uncertainty decomposition via a reference function, which is new in the litera…
We study reinforcement learning for two-player zero-sum Markov games with simultaneous moves in the finite-horizon setting, where the transition kernel of the underlying Markov games can be parameterized by a linear function over the current state, both players' actions and the next state. In particular, we assume that we can control both players and aim to find the Nash Equilibrium by minimizing the duality gap. We propose an algorithm Nash-UCRL based on the principle "Optimism-in-Face-of-Uncertainty". Our algorithm only needs to find a Coarse Correlated Equilibrium (CCE), which is computatio…
University College London（伦敦大学学院）；University of Alberta（阿尔伯塔大学）
In a single-agent setting, reinforcement learning (RL) tasks can be cast into an inference problem by introducing a binary random variable o, which stands for the "optimality". In this paper, we redefine the binary random variable o in multi-agent setting and formalize multi-agent reinforcement learning (MARL) as probabilistic inference. We derive a variational lower bound of the likelihood of achieving the optimality and name it as Regularized Opponent Model with Maximum Entropy Objective (ROMMEO). From ROMMEO, we present a novel perspective on opponent modeling and show how it can improve th…
We methodologically address the problem of Q-value overestimation in deep reinforcement learning to handle high-dimensional state spaces efficiently. By adapting concepts from information theory, we introduce an intrinsic penalty signal encouraging reduced Q-value estimates. The resultant algorithm encompasses a wide range of learning outcomes containing deep Q-networks as a special case. Different learning outcomes can be demonstrated by tuning a Lagrange multiplier accordingly. We furthermore propose a novel scheduling scheme for this Lagrange multiplier to ensure efficient and robust learni…
Towards Global Optimality in Cooperative MARL with the Transformation And Distillation Framework
Washington University in St. Louis（华盛顿大学圣路易斯分校）；Institute for Interdisciplinary Information Sciences, Tsinghua University（清华大学交叉信息研究院）；Department of Computer Science and Technology, Tsinghua University（清华大学计算机科学与技术系）
Centrum Wiskunde & Informatica（数学与计算机科学研究中心）；Inria Lille（里尔法国国家信息与自动化研究所）
Pure exploration (aka active testing) is the fundamental task of sequentially gathering information to answer a query about a stochastic environment. Good algorithms make few mistakes and take few samples. Lower bounds (for multi-armed bandit models with arms in an exponential family) reveal that the sample complexity is determined by the solution to an optimisation problem. The existing state of the art algorithms achieve asymptotic optimality by solving a plug-in estimate of that optimisation problem at each step. We interpret the optimisation problem as an unknown game, and propose sampling…
