Asymptotic Properties of $\mathcal{S}$-$\mathcal{AB}$ Method with Diminishing Stepsize

TL;DR

This paper studies the asymptotic behavior of the stochastic $ ext{ extcal{AB}}$ method with diminishing stepsize in distributed optimization, proving convergence to the global minimizer and establishing asymptotic normality under certain conditions.

Contribution

It provides the first analysis of the asymptotic properties of the stochastic $ ext{ extcal{AB}}$ method with diminishing stepsize, including convergence rates and normality results.

Findings

Iterates are bounded and converge to the global minimizer at rate $ ext{O}(1/\sqrt{k})$.

Asymptotic normality of Polyak-Ruppert averaged $ ext{ extcal{AB}}$ is established.

Numerical tests confirm the theoretical results.

Abstract

The popular $\mathcal{AB}$/push-pull method for distributed optimization problem may unify much of the existing decentralized first-order methods based on gradient tracking technique. More recently, the stochastic gradient variant of $\mathcal{AB}$/Push-Pull method ($\mathcal{S}$-$\mathcal{AB}$) has been proposed, which achieves the linear rate of converging to a neighborhood of the global minimizer when the step-size is constant. This paper is devoted to the asymptotic properties of $\mathcal{S}$-$\mathcal{AB}$ with diminishing stepsize. Specifically, under the condition that each local objective is smooth and the global objective is strongly-convex, we first present the boundedness of the iterates of $\mathcal{S}$-$\mathcal{AB}$ and then show that the iterates converge to the global minimizer with the rate $\mathcal{O}\left(1/\sqrt{k}\right)$. Furthermore, the asymptotic normality of Polyak-Ruppert averaged $\mathcal{S}$-$\mathcal{AB}$ is obtained and applications on statistical inference are discussed. Finally, numerical tests are conducted to demonstrate the theoretic results.