Improved Rates for Derivative Free Gradient Play in Strongly Monotone Games

TL;DR

This paper improves the complexity bounds for derivative-free gradient play in strongly monotone games from $O(d^2/ ext{accuracy}^3)$ to $O(d^2/ ext{accuracy}^2)$, aligning it with unconstrained optimization results.

Contribution

It provides a tighter complexity bound for derivative-free gradient play in strongly monotone games, showing the method is more efficient than previously thought.

Findings

Complexity bound improved from $O(d^2/ ext{accuracy}^3)$ to $O(d^2/ ext{accuracy}^2)$.

Method interpreted as stochastic gradient play on a perturbed game.

Results match known bounds for unconstrained optimization.

Abstract

The influential work of Bravo et al. 2018 shows that derivative free play in strongly monotone games has complexity $O(d^2/\varepsilon^3)$, where $\varepsilon$ is the target accuracy on the expected squared distance to the solution. This note shows that the efficiency estimate is actually $O(d^2/\varepsilon^2)$, which reduces to the known efficiency guarantee for the method in unconstrained optimization. The argument we present simple interprets the method as stochastic gradient play on a slightly perturbed strongly monotone game.