Distributed Zeroth Order Optimization Over Random Networks: A Kiefer-Wolfowitz Stochastic Approximation Approach

TL;DR

This paper introduces a distributed zeroth order optimization algorithm for randomly varying networks using a Kiefer-Wolfowitz stochastic approximation approach, achieving convergence rates comparable to centralized methods.

Contribution

It develops a novel distributed zeroth order optimization method for random networks, addressing the lack of gradient information and establishing its convergence rate.

Findings

Achieves an $O(1/k^{1/2})$ convergence rate under standard assumptions.

Handles noisy function evaluations without gradients or Hessians.

Works effectively over randomly varying network topologies.

Abstract

We study a standard distributed optimization framework where $N$ networked nodes collaboratively minimize the sum of their local convex costs. The main body of existing work considers the described problem when the underling network is either static or deterministically varying, and the distributed optimization algorithm is of first or second order, i.e., it involves the local costs' gradients and possibly the local Hessians. In this paper, we consider the currently understudied but highly relevant scenarios when: 1) only noisy function values' estimates are available (no gradients nor Hessians can be evaluated); and 2) the underlying network is randomly varying (according to an independent, identically distributed process). For the described random networks-zeroth order optimization setting, we develop a distributed stochastic approximation method of the Kiefer-Wolfowitz type. Furthermore, under standard smoothness and strong convexity assumptions on the local costs, we establish the $O(1/k^{1/2})$ mean square convergence rate for the method -- the rate that matches that of the method's centralized counterpart under equivalent conditions.