In general, two-agent decision-making problems can be modeled as a two-player game, and a typical solution is to find a Nash equilibrium in such game. Counterfactual regret minimization (CFR) is a well-known method to find a Nash equilibrium strategy in a two-player zero-sum game with imperfect information. The CFR method adopts a regret matching algorithm iteratively to reduce regret values progressively, enabling the average strategy to approach a Nash equilibrium. Although CFR-based methods have achieved significant success in the field of imperfect information games, there is still scope for improvement in the efficiency of convergence. To address this challenge, we propose a novel CFR-based method named exponential counterfactual regret minimization (ECFR). With ECFR, an exponential weighting technique is used to reweight the instantaneous regret value during the process of iteration. A theoretical proof is provided to guarantees convergence of the ECFR algorithm. The result of an extensive set of experimental tests demostrate that the ECFR algorithm converges faster than the current state-of-the-art CFR-based methods.
Minimizing Weighted Counterfactual Regret with Optimistic Online Mirror Descent
Institute of Automation, Chinese Academy of Sciences（中国科学院自动化研究所）；School of Artificial Intelligence, University of Chinese Academy of Sciences（中国科学院大学人工智能学院）；Tencent AI Lab（腾讯AI实验室）；Tsinghua University（清华大学）；School of Future Technology, University of Chinese Academy of Sciences（中国科学院大学未来技术学院）
University of Science and Technology of China（中国科学技术大学）；New York University（纽约大学）
Counterfactual Regret Minimization (CFR) has achieved many fascinating results in solving large-scale Imperfect Information Games (IIGs). Neural network approximation CFR (neural CFR) is one of the promising techniques that can reduce computation and memory consumption by generalizing decision information between similar states. Current neural CFR algorithms have to approximate cumulative regrets. However, efficient and accurate approximation in a large-scale IIG is still a tough challenge. In this paper, a new CFR variant, Recursive CFR (ReCFR), is proposed. In ReCFR, Recursive Substitute Val…
Counterfactual regret minimization (CFR) is a family of iterative algorithms that are the most popular and, in practice, fastest approach to approximately solving large imperfect-information games. In this paper we introduce novel CFR variants that 1) discount regrets from earlier iterations in various ways (in some cases differently for positive and negative regrets), 2) reweight iterations in various ways to obtain the output strategies, 3) use a non-standard regret minimizer and/or 4) leverage "optimistic regret matching". They lead to dramatically improved performance in many settings. For…
Counterfactual Regret Minimization (CFR) is an efficient no-regret learning algorithm for decision problems modeled as extensive games. CFR's regret bounds depend on the requirement of perfect recall: players always remember information that was revealed to them and the order in which it was revealed. In games without perfect recall, however, CFR's guarantees do not apply. In this paper, we present the first regret bound for CFR when applied to a general class of games with imperfect recall. In addition, we show that CFR applied to any abstraction belonging to our general class results in a re…
Facebook AI Research（Facebook人工智能研究院）；Carnegie Mellon University（卡内基梅隆大学）
Counterfactual Regret Minimization (CFR) is the leading framework for solving large imperfect-information games. It converges to an equilibrium by iteratively traversing the game tree. In order to deal with extremely large games, abstraction is typically applied before running CFR. The abstracted game is solved with tabular CFR, and its solution is mapped back to the full game. This process can be problematic because aspects of abstraction are often manual and domain specific, abstraction algorithms may miss important strategic nuances of the game, and there is a chicken-and-egg problem becaus…
Counterfactual Regret Minimization (CFR) is the most popular iterative algorithm for solving zero-sum imperfect-information games. Regret-Based Pruning (RBP) is an improvement that allows poorly-performing actions to be temporarily pruned, thus speeding up CFR. We introduce Total RBP, a new form of RBP that reduces the space requirements of CFR as actions are pruned. We prove that in zero-sum games it asymptotically prunes any action that is not part of a best response to some Nash equilibrium. This leads to provably faster convergence and lower space requirements. Experiments show that Total …
Harbin Institute of Technology, ShenZhen（哈尔滨工业大学（深圳））
Counterfactual regret minimization (CFR) is a popular method to deal with decision-making problems of two-player zero-sum games with imperfect information. Unlike existing studies that mostly explore for solving larger scale problems or accelerating solution efficiency, we propose a framework, RLCFR, which aims at improving the generalization ability of the CFR method. In the RLCFR, the game strategy is solved by the CFR in a reinforcement learning framework. And the dynamic procedure of iterative interactive strategy updating is modeled as a Markov decision process (MDP). Our method, RLCFR, t…
Function approximation is a powerful approach for structuring large decision problems that has facilitated great achievements in the areas of reinforcement learning and game playing. Regression counterfactual regret minimization (RCFR) is a simple algorithm for approximately solving imperfect information games with normalized rectified linear unit (ReLU) parameterized policies. In contrast, the more conventional softmax parameterization is standard in the field of reinforcement learning and yields a regret bound with a better dependence on the number of actions. We derive approximation error-a…
Learning strategies for imperfect information games from samples of interaction is a challenging problem. A common method for this setting, Monte Carlo Counterfactual Regret Minimization (MCCFR), can have slow long-term convergence rates due to high variance. In this paper, we introduce a variance reduction technique (VR-MCCFR) that applies to any sampling variant of MCCFR. Using this technique, per-iteration estimated values and updates are reformulated as a function of sampled values and state-action baselines, similar to their use in policy gradient reinforcement learning. The new formulati…
The CFR framework has been a powerful tool for solving large-scale extensive-form games in practice. However, the theoretical rate at which past CFR-based algorithms converge to the Nash equilibrium is on the order of $O(T^{-1/2})$, where $T$ is the number of iterations. In contrast, first-order methods can be used to achieve a $O(T^{-1})$ dependence on iterations, yet these methods have been less successful in practice. In this work we present the first CFR variant that breaks the square-root dependence on iterations. By combining and extending recent advances on predictive and stable regret …
Counterfactual Regret Minimization and variants (e.g. Public Chance Sampling CFR and Pure CFR) have been known as the best approaches for creating approximate Nash equilibrium solutions for imperfect information games such as poker. This paper introduces CFR$^+$, a new algorithm that typically outperforms the previously known algorithms by an order of magnitude or more in terms of computation time while also potentially requiring less memory.
Counterfactual Regret Minimization (CFR) is the most successful algorithm for finding approximate Nash equilibria in imperfect information games. However, CFR's reliance on full game-tree traversals limits its scalability. For this reason, the game's state- and action-space is often abstracted (i.e. simplified) for CFR, and the resulting strategy is then translated back to the full game, which requires extensive expert-knowledge and often converges to highly exploitable policies. A recently proposed method, Deep CFR, applies deep learning directly to CFR, allowing the agent to intrinsically ab…
Online learning algorithms that minimize regret provide strong guarantees in situations that involve repeatedly making decisions in an uncertain environment, e.g. a driver deciding what route to drive to work every day. While regret minimization has been extensively studied in repeated games, we study regret minimization for a richer class of games called bounded memory games. In each round of a two-player bounded memory-m game, both players simultaneously play an action, observe an outcome and receive a reward. The reward may depend on the last m outcomes as well as the actions of the players…
Modeling strategic conflict from a game theoretical perspective involves dealing with epistemic uncertainty. Payoff uncertainty models are typically restricted to simple probability models due to computational restrictions. Recent breakthroughs Artificial Intelligence (AI) research applied to Poker have resulted in novel approximation approaches such as counterfactual regret minimization, that can successfully deal with large-scale imperfect games. By drawing from these ideas, this work addresses the problem of arbitrary continuous payoff distributions. We propose a method, Harsanyi-Counterfac…
A recent emerging trend in the literature on learning in games has been concerned with providing faster learning dynamics for correlated and coarse correlated equilibria in normal-form games. Much less is known about the significantly more challenging setting of extensive-form games, which can capture both sequential and simultaneous moves, as well as imperfect information. In this paper we establish faster no-regret learning dynamics for \textit{extensive-form correlated equilibria (EFCE)} in multiplayer general-sum imperfect-information extensive-form games. When all players follow our accel…
Online game playing algorithms produce high-quality strategies with a fraction of memory and computation required by their offline alternatives. Continual Resolving (CR) is a recent theoretically sound approach to online game playing that has been used to outperform human professionals in poker. However, parts of the algorithm were specific to poker, which enjoys many properties not shared by other imperfect information games. We present a domain-independent formulation of CR applicable to any two-player zero-sum extensive-form games that works with an abstract resolving algorithm. We further …
In this tutorial article, we give an overview of new challenges and representative results on distributed no-regret learning in multi-agent systems modeled as repeated unknown games. Four emerging game characteristics---dynamicity, incomplete and imperfect feedback, bounded rationality, and heterogeneity---that challenge canonical game models are explored. For each of the four characteristics, we illuminate its implications and ramifications in game modeling, notions of regret, feasible game outcomes, and the design and analysis of distributed learning algorithms.
Blackwell approachability is a framework for reasoning about repeated games with vector-valued payoffs. We introduce predictive Blackwell approachability, where an estimate of the next payoff vector is given, and the decision maker tries to achieve better performance based on the accuracy of that estimator. In order to derive algorithms that achieve predictive Blackwell approachability, we start by showing a powerful connection between four well-known algorithms. Follow-the-regularized-leader (FTRL) and online mirror descent (OMD) are the most prevalent regret minimizers in online convex optim…
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…
No-regret learning has emerged as a powerful tool for solving extensive-form games. This was facilitated by the counterfactual-regret minimization (CFR) framework, which relies on the instantiation of regret minimizers for simplexes at each information set of the game. We use an instantiation of the CFR framework to develop algorithms for solving behaviorally-constrained (and, as a special case, perturbed in the Selten sense) extensive-form games, which allows us to compute approximate Nash equilibrium refinements. Nash equilibrium refinements are motivated by a major deficiency in Nash equili…
Monte-Carlo counterfactual regret minimization (MCCFR) is the state-of-the-art algorithm for solving sequential games that are too large for full tree traversals. It works by using gradient estimates that can be computed via sampling. However, stochastic methods for sequential games have not been investigated extensively beyond MCCFR. In this paper we develop a new framework for developing stochastic regret minimization methods. This framework allows us to use any regret-minimization algorithm, coupled with any gradient estimator. The MCCFR algorithm can be analyzed as a special case of our fr…
A dominant approach to solving large imperfect-information games is Counterfactural Regret Minimization (CFR). In CFR, many regret minimization problems are combined to solve the game. For very large games, abstraction is typically needed to render CFR tractable. Abstractions are often manually tuned, possibly removing important strategic differences in the full game and harming performance. Function approximation provides a natural solution to finding good abstractions to approximate the full game. A common approach to incorporating function approximation is to learn the inputs needed for a r…
We introduce DREAM, a deep reinforcement learning algorithm that finds optimal strategies in imperfect-information games with multiple agents. Formally, DREAM converges to a Nash Equilibrium in two-player zero-sum games and to an extensive-form coarse correlated equilibrium in all other games. Our primary innovation is an effective algorithm that, in contrast to other regret-based deep learning algorithms, does not require access to a perfect simulator of the game to achieve good performance. We show that DREAM empirically achieves state-of-the-art performance among model-free algorithms in po…
OSCC: Certified Observation-Safe Coupling Optimization for Gradient-Noise Control in Imperfect-Information Learning
School of Artificial Intelligence, University of Chinese Academy of Sciences（中国科学院大学人工智能学院）；Institute of Information Engineering, Chinese Academy of Sciences（中国科学院信息工程研究所）
We develop new parameter-free and scale-free algorithms for solving convex-concave saddle-point problems. Our results are based on a new simple regret minimizer, the Conic Blackwell Algorithm$^+$ (CBA$^+$), which attains $O(1/\sqrt{T})$ average regret. Intuitively, our approach generalizes to other decision sets of interest ideas from the Counterfactual Regret minimization (CFR$^+$) algorithm, which has very strong practical performance for solving sequential games on simplexes. We show how to implement CBA$^+$ for the simplex, $\ell_{p}$ norm balls, and ellipsoidal confidence regions in the s…
In two-player zero-sum games, if both players minimize their average external regret, then the average of the strategy profiles converges to a Nash equilibrium. For n-player general-sum games, however, theoretical guarantees for regret minimization are less understood. Nonetheless, Counterfactual Regret Minimization (CFR), a popular regret minimization algorithm for extensive-form games, has generated winning three-player Texas Hold'em agents in the Annual Computer Poker Competition (ACPC). In this paper, we provide the first set of theoretical properties for regret minimization algorithms in …
Extensive-form games (EFGs) model finite sequential interactions between players. The amount of memory required to represent these games is the main bottleneck of algorithms for computing optimal strategies and the size of these strategies is often impractical for real-world applications. A common approach to tackle the memory bottleneck is to use information abstraction that removes parts of information available to players thus reducing the number of decision points in the game. However, existing information-abstraction techniques are either specific for a particular domain, they do not prov…
We compare performance of the genetic algorithm and the counterfactual regret minimization algorithm in computing the near-equilibrium strategies in the simplified poker games. We focus on the von Neumann poker and the simplified version of the Texas Hold'Em poker, and test outputs of the considered algorithms against analytical expressions defining the Nash equilibrium strategies. We comment on the performance of the studied algorithms against opponents deviating from equilibrium.
Regret minimization is a powerful tool for solving large-scale extensive-form games. State-of-the-art methods rely on minimizing regret locally at each decision point. In this work we derive a new framework for regret minimization on sequential decision problems and extensive-form games with general compact convex sets at each decision point and general convex losses, as opposed to prior work which has been for simplex decision points and linear losses. We call our framework laminar regret decomposition. It generalizes the CFR algorithm to this more general setting. Furthermore, our framework …
