Improved rates for derivative free play in convex games

The influential work of Bravo et al. (2018) shows that derivative free play in strongly monotone games converges at the rate $O\left(\frac{d^2}{T^{1/3}}\right)$, where $T$ is the time horizon and $d$ is the ambient dimension. This note shows that the convergence rate is actually $O\left(\frac{d^2}{T^{1/2}}\right)$, thereby matching the classical guarantees for derivative free methods in optimization. The argument we present is elementary, simply interpreting the method as stochastic gradient play on a slightly perturbed strongly monotone game.