Learning Distributed Equilibria in Linear-Quadratic Stochastic Differential Games: An $\alpha$-Potential Approach

TL;DR

This paper demonstrates that independent policy-gradient methods can efficiently find equilibria in large-scale linear-quadratic stochastic differential games by leveraging an $ ext{alpha}$-potential structure, with proven convergence guarantees.

Contribution

It introduces an $ ext{alpha}$-potential framework for analyzing distributed policy-gradient learning in LQ stochastic differential games, providing convergence results for symmetric and asymmetric interactions.

Findings

Global linear convergence of PG methods to equilibrium

Complexity scales linearly with population size

Numerical experiments confirm theoretical results

Abstract

We analyze independent policy-gradient (PG) learning in $N$-player linear-quadratic (LQ) stochastic differential games. Each player employs a distributed policy that depends only on its own state and updates the policy independently using the gradient of its own objective. We establish global linear convergence of these methods to an equilibrium by showing that the LQ game admits an $\alpha$-potential structure, with $\alpha$ determined by the degree of pairwise interaction asymmetry. For pairwise-symmetric interactions, we construct an affine distributed equilibrium by minimizing the potential function and show that independent PG methods converge globally to this equilibrium, with complexity scaling linearly in the population size and logarithmically in the desired accuracy. For asymmetric interactions, we prove that independent projected PG algorithms converge linearly to an approximate equilibrium, with suboptimality proportional to the degree of asymmetry. Numerical experiments confirm the theoretical results across both symmetric and asymmetric interaction networks.