Generalized Bandit Regret Minimizer Framework in Imperfect Information Extensive-Form Game

TL;DR

This paper introduces a generalized framework for regret minimization in imperfect information games with bandit feedback, enabling more efficient learning of approximate Nash equilibria.

Contribution

It proposes a modular theoretical framework for bandit regret minimization, analyzes existing methods as special cases, and introduces a novel, more efficient algorithm SIX-OMD.

Findings

SIX-OMD significantly improves convergence rates over previous methods.

The framework unifies analysis of various bandit regret minimization algorithms.

SIX-OMD is computationally efficient, requiring only current and average strategy updates.

Abstract

Regret minimization methods are a powerful tool for learning approximate Nash equilibrium (NE) in two-player zero-sum imperfect information extensive-form games (IIEGs). We consider the problem in the interactive bandit-feedback setting where we don't know the dynamics of the IIEG. In general, only the interactive trajectory and the reached terminal node value $v(z^t)$ are revealed. To learn NE, the regret minimizer is required to estimate the full-feedback loss gradient $\ell^t$ by $v(z^t)$ and minimize the regret. In this paper, we propose a generalized framework for this learning setting. It presents a theoretical framework for the design and the modular analysis of the bandit regret minimization methods. We demonstrate that the most recent bandit regret minimization methods can be analyzed as a particular case of our framework. Following this framework, we describe a novel method SIX-OMD to learn approximate NE. It is model-free and extremely improves the best existing convergence rate from the order of $O(\sqrt{X B/T}+\sqrt{Y C/T})$ to $O(\sqrt{ M_{\mathcal{X}}/T} +\sqrt{ M_{\mathcal{Y}}/T})$. Moreover, SIX-OMD is computationally efficient as it needs to perform the current strategy and average strategy updates only along the sampled trajectory.