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We present two easy-to-implement gradient-free/zeroth-order methods tooptimize a stochastic non-smooth function accessible only via a black-box. Themethods are built upon efficient first-order methods in the heavy-tailed case,i.e., when the gradient noise has infinite variance but bounded $(1+\kappa)$-thmoment for some $\kappa \in(0,1]$. The first algorithm is based on thestochastic mirror descent with a particular class of uniformly convex mirrormaps which is robust to heavy-tailed noise. The second algorithm is based onthe stochastic mirror descent and gradient clipping technique. Additionally,for the objective functions satisfying the $r$-growth condition, fasteralgorithms are proposed based on these methods and the restart technique.

Gradient-Free Methods for Non-Smooth Convex Stochastic Optimization with Heavy-Tailed Noise on Convex Compact