Non-smooth setting of stochastic decentralized convex optimization problem over time-varying Graphs

Distributed optimization has a rich history. It has demonstrated its effectiveness in many machine learning applications, etc. In this paper we study a subclass of distributed optimization, namely decentralized optimization in a non-smooth setting. Decentralized means that m agents (machines) working in parallel on one problem communicate only with the neighbors agents (machines), i.e. there is no (central) server through which agents communicate. And by non-smooth setting we mean that each agent has a convex stochastic non-smooth function, that is, agents can hold and communicate information only about the value of the objective function, which corresponds to a gradient-free oracle. In this paper, to minimize the global objective function, which consists of the sum of the functions of each agent, we create a gradient-free algorithm by applying a smoothing scheme via l2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l_2$$\end{document} randomization. We also verify in experiments the obtained theoretical convergence results of the gradient-free algorithm proposed in this paper.