Learning in games with continuous action sets and unknown payoff functions

TL;DR

This paper analyzes the convergence properties of no-regret learning algorithms, specifically dual averaging, in continuous action games with noisy gradient feedback, introducing variational stability and providing convergence guarantees.

Contribution

It introduces the concept of variational stability and proves convergence of dual averaging in continuous games with noisy gradient feedback.

Findings

Stable equilibria are locally attracting with high probability.

Globally stable equilibria are globally attracting with probability 1.

Provides explicit convergence speed estimates.

Abstract

This paper examines the convergence of no-regret learning in games with continuous action sets. For concreteness, we focus on learning via "dual averaging", a widely used class of no-regret learning schemes where players take small steps along their individual payoff gradients and then "mirror" the output back to their action sets. In terms of feedback, we assume that players can only estimate their payoff gradients up to a zero-mean error with bounded variance. To study the convergence of the induced sequence of play, we introduce the notion of variational stability, and we show that stable equilibria are locally attracting with high probability whereas globally stable equilibria are globally attracting with probability 1. We also discuss some applications to mixed-strategy learning in finite games, and we provide explicit estimates of the method's convergence speed.