On the convergence of decentralized gradient descent with diminishing stepsize, revisited

TL;DR

This paper analyzes the convergence of decentralized gradient descent with a diminishing stepsize, showing it converges to the optimizer at a rate of O(t^{-p}) under certain conditions, even when local functions are not convex.

Contribution

It revisits and extends convergence analysis for decentralized gradient descent with diminishing stepsize, providing explicit convergence rates under strong convexity of the total cost.

Findings

Convergence rate of O(t^{-p}) established for the algorithm.

Suitable choices of parameters a and w ensure convergence.

The analysis applies even when local functions are not convex.

Abstract

Distributed optimization has received a lot of interest in recent years due to its wide applications in various fields. In this work, we revisit the convergence property of the decentralized gradient descent [A. Nedi{\'c}-A.Ozdaglar (2009)] on the whole space given by $$ x_i(t+1) = \sum^m_{j=1}w_{ij}x_j(t) - \alpha(t) \nabla f_i(x_i(t)), $$ where the stepsize is given as $\alpha (t) = \frac{a}{(t+w)^p}$ with $0< p\leq 1$. Under the strongly convexity assumption on the total cost function $f$ with local cost functions $f_i$ not necessarily being convex, we show that the sequence converges to the optimizer with rate $O(t^{-p})$ when the values of $a>0$ and $w>0$ are suitably chosen.