Rate analysis of dual averaging for nonconvex distributed optimization

TL;DR

This paper analyzes the convergence of dual averaging algorithms in nonconvex distributed optimization, establishing rate results and validating the suboptimality measure as a stationarity indicator.

Contribution

It extends the convergence analysis of dual averaging to nonconvex distributed problems, providing rate guarantees and a practical stationarity measure.

Findings

Squared norm of suboptimality measure converges at rate O(1/t)

Distributed dual averaging converges to stationary points at rate O(1/t)

Numerical example confirms theoretical convergence rates

Abstract

This work studies nonconvex distributed constrained optimization over stochastic communication networks. We revisit the distributed dual averaging algorithm, which is known to converge for convex problems. We start from the centralized case, for which the change of two consecutive updates is taken as the suboptimality measure. We validate the use of such a measure by showing that it is closely related to stationarity. This equips us with a handle to study the convergence of dual averaging in nonconvex optimization. We prove that the squared norm of this suboptimality measure converges at rate $\mathcal{O}(1/t)$. Then, for the distributed setup we show convergence to the stationary point at rate $\mathcal{O}(1/t)$. Finally, a numerical example is given to illustrate our theoretical results.