Learning in games with continuous action sets and unknown payoff functions

• Published in: Mathematical programming Vol. 173; no. 1-2; pp. 465 - 507
• Main Authors: Mertikopoulos, Panayotis, Zhou, Zhengyuan
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.01.2019; Springer Nature B.V
• Subjects: Algorithms; Calculus of Variations and Optimal Control; Optimization; Combinatorics; Convergence; Distance learning; Equilibrium; Error analysis; Feedback; Full Length Paper; Game theory; Games; Learning; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Mathematics of Computing; Numerical Analysis; Optimization; Theoretical; Variational stability; Continuous games; Dual averaging; Primary 91A26; Fenchel coupling; Secondary 90C33; Nash equilibrium; 90C15; 68Q32
• DOI: 10.1007/s10107-018-1254-8
• ISSN: 0025-5610

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.