Distributed Stochastic Subgradient Projection Algorithms for Convex Optimization

TL;DR

This paper analyzes how stochastic errors in subgradient information affect the convergence of distributed convex optimization algorithms in multi-agent networks, providing theoretical guarantees under various error conditions.

Contribution

It extends existing distributed subgradient methods by incorporating stochastic errors, establishing convergence results in mean, probability, and mean square.

Findings

Convergence in mean with bounded stochastic errors.

Almost sure convergence when errors diminish sufficiently fast.

Performance bounds for algorithms with both diminishing and non-diminishing stepsizes.

Abstract

We consider a distributed multi-agent network system where the goal is to minimize a sum of convex objective functions of the agents subject to a common convex constraint set. Each agent maintains an iterate sequence and communicates the iterates to its neighbors. Then, each agent combines weighted averages of the received iterates with its own iterate, and adjusts the iterate by using subgradient information (known with stochastic errors) of its own function and by projecting onto the constraint set. The goal of this paper is to explore the effects of stochastic subgradient errors on the convergence of the algorithm. We first consider the behavior of the algorithm in mean, and then the convergence with probability 1 and in mean square. We consider general stochastic errors that have uniformly bounded second moments and obtain bounds on the limiting performance of the algorithm in mean for diminishing and non-diminishing stepsizes. When the means of the errors diminish, we prove that there is mean consensus between the agents and mean convergence to the optimum function value for diminishing stepsizes. When the mean errors diminish sufficiently fast, we strengthen the results to consensus and convergence of the iterates to an optimal solution with probability 1 and in mean square.