One-Point Gradient-Free Methods for Smooth and Non-Smooth Saddle-Point Problems

Abstract: In this paper, we analyze gradient-free methods with one-point feedback for stochastic saddle point problems $\min_{x}\max_{y} \varphi(x, y)$. For non-smooth and smooth cases, we present analysis in a general geometric setup with arbitrary Bregman divergence. For problems with higher-order smoothness, the analysis is carried out only in the Euclidean case. The estimates we have obtained repeat the best currently known estimates of gradient-free methods with one-point feedback for problems of imagining a convex or strongly convex function. The paper uses three main approaches to recovering the gradient through finite differences: standard with a random direction, as well as its modifications with kernels and residual feedback. We also provide experiments to compare these approaches for the matrix game.