Distributed Stochastic Subgradient Projection Algorithms for Convex Optimization

• Published in: Journal of optimization theory and applications Vol. 147; no. 3; pp. 516 - 545
• Main Authors: Sundhar Ram, S., Nedić, A., Veeravalli, V. V.
• Format: Journal Article
• Language: English
• Published: Boston Springer US 01.12.2010; Springer; Springer Nature B.V
• Subjects: Algorithms; Applications of Mathematics; Applied sciences; Artificial intelligence; Calculus of Variations and Optimal Control; Optimization; Computer science; control theory; systems; Computer systems and distributed systems. User interface; Convex analysis; Engineering; Exact sciences and technology; Mathematical programming; Mathematics; Mathematics and Statistics; Operational research and scientific management; Operational research. Management science; Operations Research/Decision Theory; Optimization; Software; Studies; Systems analysis; Theory of Computation; Stochastic approximation; Subgradient methods; Convex optimization; Distributed algorithm; Second moment; Convex set; Randomized algorithm; Convex programming; Convex function; Weighted average; Subgradient optimization; Probabilistic approach; Error function; Interconnected power system; Error bound; Distributed system; Set constraint; Mean estimation; Consensus; Value function; Multiagent system; Weighting; Algorithm performance; Optimal solution; Mean error; Objective function; Artificial intelligence
• ISSN: 0022-3239; 1573-2878
• DOI: 10.1007/s10957-010-9737-7

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.