Game-theoretical control with continuous action sets

TL;DR

This paper introduces a convergent actor-critic reinforcement learning algorithm for potential games with continuous actions, extending finite-dimensional theories to infinite-dimensional spaces for distributed control.

Contribution

It develops an infinite-dimensional mean-field analysis and proves convergence of a new learning algorithm in continuous action potential games.

Findings

The algorithm converges to equilibrium in potential games with continuous controls.

The analysis extends stochastic approximation theory to infinite-dimensional Banach spaces.

Players do not need to track other agents' controls during learning.

Abstract

Motivated by the recent applications of game-theoretical learning techniques to the design of distributed control systems, we study a class of control problems that can be formulated as potential games with continuous action sets, and we propose an actor-critic reinforcement learning algorithm that provably converges to equilibrium in this class of problems. The method employed is to analyse the learning process under study through a mean-field dynamical system that evolves in an infinite-dimensional function space (the space of probability distributions over the players' continuous controls). To do so, we extend the theory of finite-dimensional two-timescale stochastic approximation to an infinite-dimensional, Banach space setting, and we prove that the continuous dynamics of the process converge to equilibrium in the case of potential games. These results combine to give a provably-convergent learning algorithm in which players do not need to keep track of the controls selected by the other agents.