  This paper studies non-smooth problems of convex stochastic optimization.
Using the smoothing technique based on the replacement of the function value at
the considered point by the averaged function value over a ball (in $l_1$-norm
or $l_2$-norm) of small radius with the center in this point, the original
problem is reduced to a smooth problem (whose Lipschitz constant of the
gradient is inversely proportional to the radius of the ball). An important
property of the smoothing used is the possibility to calculate an unbiased
estimation of the gradient of a smoothed function based only on realizations of
the original function. The obtained smooth stochastic optimization problem is
proposed to be solved in a distributed federated learning architecture (the
problem is solved in parallel: nodes make local steps, e.g. stochastic gradient
descent, then they communicate - all with all, then all this is repeated). The
goal of this paper is to build on the current advances in gradient-free
non-smooth optimization and in feild of federated learning, gradient-free
methods for solving non-smooth stochastic optimization problems in federated
learning architecture.
