A Unified View of Large-scale Zero-sum Equilibrium Computation

The task of computing approximate Nash equilibria in large zero-sum extensive-form games has received a tremendous amount of attention due mainly to the Annual Computer Poker Competition. Immediately after its inception, two competing and seemingly different approaches emerged---one an application of no-regret online learning, the other a sophisticated gradient method applied to a convex-concave saddle-point formulation. Since then, both approaches have grown in relative isolation with advancements on one side not effecting the other. In this paper, we rectify this by dissecting and, in a sense, unify the two views.Comment: AAAI Workshop on Computer Poker and Imperfect Informatio

Local and adaptive mirror descents in extensive-form games

We study how to learn $\epsilon$-optimal strategies in zero-sum imperfect information games (IIG) with trajectory feedback. In this setting, players update their policies sequentially based on their observations over a fixed number of episodes, denoted by $T$. Existing procedures suffer from high variance due to the use of importance sampling over sequences of actions (Steinberger et al., 2020; McAleer et al., 2022). To reduce this variance, we consider a fixed sampling approach, where players still update their policies over time, but with observations obtained through a given fixed sampling policy. Our approach is based on an adaptive Online Mirror Descent (OMD) algorithm that applies OMD locally to each information set, using individually decreasing learning rates and a regularized loss. We show that this approach guarantees a convergence rate of $\tilde{\mathcal{O}}(T^{-1/2})$ with high probability and has a near-optimal dependence on the game parameters when applied with the best theoretical choices of learning rates and sampling policies. To achieve these results, we generalize the notion of OMD stabilization, allowing for time-varying regularization with convex increments

Online Sequential Decision-Making with Unknown Delays

In the field of online sequential decision-making, we address the problem with delays utilizing the framework of online convex optimization (OCO), where the feedback of a decision can arrive with an unknown delay. Unlike previous research that is limited to Euclidean norm and gradient information, we propose three families of delayed algorithms based on approximate solutions to handle different types of received feedback. Our proposed algorithms are versatile and applicable to universal norms. Specifically, we introduce a family of Follow the Delayed Regularized Leader algorithms for feedback with full information on the loss function, a family of Delayed Mirror Descent algorithms for feedback with gradient information on the loss function and a family of Simplified Delayed Mirror Descent algorithms for feedback with the value information of the loss function's gradients at corresponding decision points. For each type of algorithm, we provide corresponding regret bounds under cases of general convexity and relative strong convexity, respectively. We also demonstrate the efficiency of each algorithm under different norms through concrete examples. Furthermore, our theoretical results are consistent with the current best bounds when degenerated to standard settings

Efficient Last-iterate Convergence Algorithms in Solving Games

No-regret algorithms are popular for learning Nash equilibrium (NE) in two-player zero-sum normal-form games (NFGs) and extensive-form games (EFGs). Many recent works consider the last-iterate convergence no-regret algorithms. Among them, the two most famous algorithms are Optimistic Gradient Descent Ascent (OGDA) and Optimistic Multiplicative Weight Update (OMWU). However, OGDA has high per-iteration complexity. OMWU exhibits a lower per-iteration complexity but poorer empirical performance, and its convergence holds only when NE is unique. Recent works propose a Reward Transformation (RT) framework for MWU, which removes the uniqueness condition and achieves competitive performance with OMWU. Unfortunately, RT-based algorithms perform worse than OGDA under the same number of iterations, and their convergence guarantee is based on the continuous-time feedback assumption, which does not hold in most scenarios. To address these issues, we provide a closer analysis of the RT framework, which holds for both continuous and discrete-time feedback. We demonstrate that the essence of the RT framework is to transform the problem of learning NE in the original game into a series of strongly convex-concave optimization problems (SCCPs). We show that the bottleneck of RT-based algorithms is the speed of solving SCCPs. To improve the their empirical performance, we design a novel transformation method to enable the SCCPs can be solved by Regret Matching+ (RM+), a no-regret algorithm with better empirical performance, resulting in Reward Transformation RM+ (RTRM+). RTRM+ enjoys last-iterate convergence under the discrete-time feedback setting. Using the counterfactual regret decomposition framework, we propose Reward Transformation CFR+ (RTCFR+) to extend RTRM+ to EFGs. Experimental results show that our algorithms significantly outperform existing last-iterate convergence algorithms and RM+ (CFR+)