Improved Rates for Derivative Free Gradient Play in Strongly Monotone Games

The influential work of Bravo et al. 2018 shows that derivative free play in strongly monotone games has complexity O(d2/varepsilon3), where varepsilon is the target accuracy on the expected squared distance to the solution. This note shows that the efficiency estimate is actually O(d2/varepsilon2), 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.