  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.
