Doubly Optimal No-Regret Learning in Monotone Games

TL;DR

This paper introduces the AOG algorithm, achieving the first doubly optimal no-regret learning with both optimal regret and convergence rates in smooth monotone games, surpassing previous methods.

Contribution

The paper presents the accelerated optimistic gradient (AOG) algorithm, the first to attain both optimal regret and convergence rates in smooth monotone games.

Findings

Achieves $O(rac{1}{T})$ last-iterate convergence rate to Nash equilibrium.

Attains $O( oot{T}ar{)}$ regret in adversarial settings.

Reduces individual dynamic regret to $O( ext{log } T)$.

Abstract

We consider online learning in multi-player smooth monotone games. Existing algorithms have limitations such as (1) being only applicable to strongly monotone games; (2) lacking the no-regret guarantee; (3) having only asymptotic or slow $O(\frac{1}{\sqrt{T}})$ last-iterate convergence rate to a Nash equilibrium. While the $O(\frac{1}{\sqrt{T}})$ rate is tight for a large class of algorithms including the well-studied extragradient algorithm and optimistic gradient algorithm, it is not optimal for all gradient-based algorithms. We propose the accelerated optimistic gradient (AOG) algorithm, the first doubly optimal no-regret learning algorithm for smooth monotone games. Namely, our algorithm achieves both (i) the optimal $O(\sqrt{T})$ regret in the adversarial setting under smooth and convex loss functions and (ii) the optimal $O(\frac{1}{T})$ last-iterate convergence rate to a Nash equilibrium in multi-player smooth monotone games. As a byproduct of the accelerated last-iterate convergence rate, we further show that each player suffers only an $O(\log T)$ individual worst-case dynamic regret, providing an exponential improvement over the previous state-of-the-art $O(\sqrt{T})$ bound.