Forward-backward algorithm for functions with locally Lipschitz gradient: applications to mean field games

TL;DR

This paper extends the forward-backward splitting algorithm to functions with locally Lipschitz gradients, providing convergence guarantees and applying it to mean field games, with improved computational efficiency.

Contribution

It introduces a generalized algorithm for locally Lipschitz gradients, proving convergence and applying it to mean field games with better computational performance.

Findings

Convergence of the generalized algorithm is proven.

Linear convergence rate under local strong convexity.

Numerical results show faster computation compared to benchmarks.

Abstract

In this paper, we provide a generalization of the forward-backward splitting algorithm for minimizing the sum of a proper convex lower semicontinuous function and a differentiable convex function whose gradient satisfies a locally Lipschitztype condition. We prove the convergence of our method and derive a linear convergence rate when the differentiable function is locally strongly convex. We recover classical results in the case when the gradient of the differentiable function is globally Lipschitz continuous and an already known linear convergence rate when the function is globally strongly convex. We apply the algorithm to approximate equilibria of variational mean field game systems with local couplings. Compared with some benchmark algorithms to solve these problems, our numerical tests show similar performances in terms of the number of iterations but an important gain in the required computational time.