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We consider non-smooth saddle point optimization problems. To solve theseproblems, we propose a zeroth-order method under bounded or Lipschitzcontinuous noise, possible adversarial. In contrast to the state-of-the-artalgorithms, our algorithm is optimal in terms of both criteria: oracle callscomplexity and the maximum value of admissible noise. The proposed method issimple and easy to implement as it is built on zeroth-order version of thestochastic mirror descent. The convergence analysis is given in terms of theaverage and probability. We also pay special attention to the duality gap$r$-growth condition $(r\geq 1)$, for which we provide a modification of ouralgorithm using the restart technique. We also comment on infinite noisevariance and upper bounds in the case of Lipschitz noise. The results obtainedin this paper are significant not only for saddle point problems but also forconvex optimization.

Gradient-Free Optimization for Non-Smooth Saddle Point Problems under Adversarial Noise