Proximity Without Consensus in Online Multi-Agent Optimization

TL;DR

This paper introduces a decentralized stochastic optimization method that relaxes agreement constraints, allowing for data heterogeneity across agents, and proves convergence properties with applications in sensor networks and source localization.

Contribution

It proposes a novel decentralized saddle point algorithm for online multi-agent optimization with proximity constraints, extending beyond traditional consensus frameworks.

Findings

Convergence of time-average primal vectors to the optimal solution.

Neighborhood bounds on suboptimality and constraint violation under constant step-size.

Empirical validation in sensor network estimation and source localization.

Abstract

We consider stochastic optimization problems in multi-agent settings, where a network of agents aims to learn parameters which are optimal in terms of a global objective, while giving preference to locally observed streaming information. To do so, we depart from the canonical decentralized optimization framework where agreement constraints are enforced, and instead formulate a problem where each agent minimizes a global objective while enforcing network proximity constraints. This formulation includes online consensus optimization as a special case, but allows for the more general hypothesis that there is data heterogeneity across the network. To solve this problem, we propose using a stochastic saddle point algorithm inspired by Arrow and Hurwicz. This method yields a decentralized algorithm for processing observations sequentially received at each node of the network. Using Lagrange multipliers to penalize the discrepancy between them, only neighboring nodes exchange model information. We establish that under a constant step-size regime the time-average suboptimality and constraint violation are contained in a neighborhood whose radius vanishes with increasing number of iterations. As a consequence, we prove that the time-average primal vectors converge to the optimal objective while satisfying the network proximity constraints. We apply this method to the problem of sequentially estimating a correlated random field in a sensor network, as well as an online source localization problem, both of which demonstrate the empirical validity of the aforementioned convergence results.