Excess-Risk of Distributed Stochastic Learners

TL;DR

This paper analyzes the excess-risk behavior of diffusion and consensus distributed learners, revealing their advantages over non-cooperative and centralized schemes in terms of convergence rate and asymptotic performance.

Contribution

It provides closed-form expressions for excess-risk evolution and demonstrates diffusion's superior asymptotic convergence and robustness to network topology variations.

Findings

Diffusion strategies improve asymptotic excess-risk convergence rates.

Optimal in-network cooperation can outperform naive centralized processing.

Diffusion outperforms consensus asymptotically, invariant to network topology.

Abstract

This work studies the learning ability of consensus and diffusion distributed learners from continuous streams of data arising from different but related statistical distributions. Four distinctive features for diffusion learners are revealed in relation to other decentralized schemes even under left-stochastic combination policies. First, closed-form expressions for the evolution of their excess-risk are derived for strongly-convex risk functions under a diminishing step-size rule. Second, using these results, it is shown that the diffusion strategy improves the asymptotic convergence rate of the excess-risk relative to non-cooperative schemes. Third, it is shown that when the in-network cooperation rules are designed optimally, the performance of the diffusion implementation can outperform that of naive centralized processing. Finally, the arguments further show that diffusion outperforms consensus strategies asymptotically, and that the asymptotic excess-risk expression is invariant to the particular network topology. The framework adopted in this work studies convergence in the stronger mean-square-error sense, rather than in distribution, and develops tools that enable a close examination of the differences between distributed strategies in terms of asymptotic behavior, as well as in terms of convergence rates.