Exact Nonparametric Decentralized Online Optimization

Hrusikesha Pradhan, Amrit Singh Bedi, Alec Koppel, Ketan Rajawat

Abstract: We consider online learning over decentralized networks, where nodes observe unique, possibly correlated, observation stream. We focus on the case where agents learn a regression function that belongs to a reproducing kernel Hilbert space (RKHS). In this setting, a decentralized network aims to learn nonlinear statistical models that are optimal in terms of a global stochastic convex functional that aggregates data across the network, with only access to a local data stream. We incentivize coordination while respecting network heterogeneity through the introduction of nonlinear proximity constraints. To solve it, we propose applying a functional variant of stochastic primal-dual (Arrow-Hurwicz) method which yields a decentralized algorithm. To handle the fact that the RKHS parameterization has complexity comparable to the iteration index, we project the primal iterates onto Hilbert subspaces that are greedily constructed from the observation sequence of each node. The resulting proximal stochastic variant of Arrow-Hurwicz is shown to converge in expectation, both in terms of primal sub-optimality and constraint violation to a neighborhood that depends on a given constant step-size selection. Experiments on a correlated random field estimation problem validate our theoretical results.