Continuous-time Discounted Mirror-Descent Dynamics in Monotone Concave Games

TL;DR

This paper introduces new continuous-time mirror descent dynamics for concave monotone games, demonstrating convergence properties under various regularizer conditions, including in less monotone settings.

Contribution

It proposes two classes of discounted mirror descent dynamics based on different regularizers, extending convergence results to hypo-monotone game scenarios.

Findings

Convergence in monotone concave games with pseudo-gradient.

Convergence in hypo-monotone games with strongly convex regularizer.

New dynamics extend applicability to less monotone game settings.

Abstract

In this paper, we consider concave continuous-kernel games characterized by monotonicity properties and propose discounted mirror descent-type dynamics. We introduce two classes of dynamics whereby the associated mirror map is constructed based on a strongly convex or a Legendre regularizer. Depending on the properties of the regularizer we show that these new dynamics can converge asymptotically in concave games with monotone (negative) pseudo-gradient. Furthermore, we show that when the regularizer enjoys strong convexity, the resulting dynamics can converge even in games with hypo-monotone (negative) pseudo-gradient, which corresponds to a shortage of monotonicity.