Decentralized Online Optimization With Heterogeneous Data Sources

We consider stochastic optimization problems in decentralized settings, where a network of agents aims to learn decision variables which are optimal in terms of a global objective which depends on possibly heterogeneous streaming observations received at each node. Consensus optimization techniques implicitly operate on the hypothesis that each node aims to learn a common parameter vector, which is inappropriate for this context. Motivated by this observation, we formulate a problem where each agent minimizes a global objective while enforcing network proximity constraints that may encode correlation structures among the observations at each node. To solve this problem, we propose a decentralized stochastic saddle point algorithm inspired by Arrow and Hurwicz. 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 the iteration index. Further, the time-average primal vectors converge to the optimal objective while satisfying the network proximity constraints. We apply this method to an online source localization problem and show it outperforms consensus-based schemes.