A unified stochastic approximation framework for learning in games

TL;DR

This paper introduces a comprehensive stochastic approximation framework that unifies and extends the analysis of various learning algorithms in games, providing new convergence insights for both continuous and finite settings.

Contribution

It offers a unified analysis template for multiple learning algorithms in games, including new convergence results and the concept of coherence for finite-time convergence.

Findings

Established criteria for convergence to Nash equilibria.

Introduced the notion of coherence for finite-time convergence.

Applicable to payoff-based and bandit learning methods.

Abstract

We develop a flexible stochastic approximation framework for analyzing the long-run behavior of learning in games (both continuous and finite). The proposed analysis template incorporates a wide array of popular learning algorithms, including gradient-based methods, the exponential/multiplicative weights algorithm for learning in finite games, optimistic and bandit variants of the above, etc. In addition to providing an integrated view of these algorithms, our framework further allows us to obtain several new convergence results, both asymptotic and in finite time, in both continuous and finite games. Specifically, we provide a range of criteria for identifying classes of Nash equilibria and sets of action profiles that are attracting with high probability, and we also introduce the notion of coherence, a game-theoretic property that includes strict and sharp equilibria, and which leads to convergence in finite time. Importantly, our analysis applies to both oracle-based and bandit, payoff-based methods - that is, when players only observe their realized payoffs.