Doubly Optimal No-Regret Learning in Monotone Games

• URL: http://arxiv.org/abs/2301.13120v2
• Date: Mon, 4 Sep 2023 18:07:17 GMT
• Title: Doubly Optimal No-Regret Learning in Monotone Games
• Authors: Yang Cai, Weiqiang Zheng
• Abstract summary: We propose the first doubly optimal no-regret learning algorithm for smooth monotone games. Our algorithm achieves both (i) the optimal $O(sqrtT)$ regret in the adversarial setting under smooth and convex loss functions and (ii) the optimal $O(frac1T)$ last-iterate convergence rate to a Nash equilibrium.
• Score: 10.760195409078591
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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