Convergence Analysis of Gradient-Based Learning with Non-Uniform Learning Rates in Non-Cooperative Multi-Agent Settings

TL;DR

This paper analyzes the convergence of gradient-based multi-agent learning algorithms with non-uniform learning rates in non-cooperative games, providing finite-time guarantees and exploring how non-uniform rates affect convergence dynamics.

Contribution

It introduces a novel analysis of how non-uniform learning rates influence convergence and stability in multi-agent gradient learning, supported by finite-time guarantees and numerical illustrations.

Findings

Non-uniform learning rates act like preconditioning, affecting convergence speed.

Finite-time convergence guarantees are established for deterministic settings.

High-probability convergence to a neighborhood is shown in stochastic settings.

Abstract

Considering a class of gradient-based multi-agent learning algorithms in non-cooperative settings, we provide local convergence guarantees to a neighborhood of a stable local Nash equilibrium. In particular, we consider continuous games where agents learn in (i) deterministic settings with oracle access to their gradient and (ii) stochastic settings with an unbiased estimator of their gradient. Utilizing the minimum and maximum singular values of the game Jacobian, we provide finite-time convergence guarantees in the deterministic case. On the other hand, in the stochastic case, we provide concentration bounds guaranteeing that with high probability agents will converge to a neighborhood of a stable local Nash equilibrium in finite time. Different than other works in this vein, we also study the effects of non-uniform learning rates on the learning dynamics and convergence rates. We find that much like preconditioning in optimization, non-uniform learning rates cause a distortion in the vector field which can, in turn, change the rate of convergence and the shape of the region of attraction. The analysis is supported by numerical examples that illustrate different aspects of the theory. We conclude with discussion of the results and open questions.