Doubly Optimal No-Regret Learning in Monotone Games


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.

Proceedings of the 40th International Conference on Machine Learning (ICML)
Click the Cite button above to demo the feature to enable visitors to import publication metadata into their reference management software.
Create your slides in Markdown - click the Slides button to check out the example.

Add the publication’s full text or supplementary notes here. You can use rich formatting such as including code, math, and images.