Distributed Stochastic Approximation for Constrained and Unconstrained Optimization

TL;DR

This paper studies a distributed stochastic approximation algorithm for multi-agent optimization, proving convergence to local minima and Kuhn-Tucker points, with stability conditions and a central limit theorem, exemplified in wireless power allocation.

Contribution

It introduces a convergence analysis for a distributed Robbins-Monro algorithm applicable to constrained and unconstrained problems, including stability and fluctuation results.

Findings

Agents reach consensus on estimates.

Algorithm converges to Kuhn-Tucker points.

Asymptotic fluctuations follow a central limit theorem.

Abstract

In this paper, we analyze the convergence of a distributed Robbins-Monro algorithm for both constrained and unconstrained optimization in multi-agent systems. The algorithm searches for local minima of a (nonconvex) objective function which is supposed to coincide with a sum of local utility functions of the agents. The algorithm under study consists of two steps: a local stochastic gradient descent at each agent and a gossip step that drives the network of agents to a consensus. It is proved that i) an agreement is achieved between agents on the value of the estimate, ii) the algorithm converges to the set of Kuhn-Tucker points of the optimization problem. The proof relies on recent results about differential inclusions. In the context of unconstrained optimization, intelligible sufficient conditions are provided in order to ensure the stability of the algorithm. In the latter case, we also provide a central limit theorem which governs the asymptotic fluctuations of the estimate. We illustrate our results in the case of distributed power allocation for ad-hoc wireless networks.