Stochastic Gradient-Push for Strongly Convex Functions on Time-Varying Directed Graphs

TL;DR

This paper analyzes the convergence rate of the subgradient-push method for distributed optimization over time-varying directed graphs, showing it achieves an asymptotically faster rate for strongly convex functions with stochastic gradients.

Contribution

It establishes a faster convergence rate of $O(( ext{ln} t)/t)$ for strongly convex functions with stochastic gradients, improving upon previous results for convex functions.

Findings

Convergence rate of $O(( ext{ln} t)/t)$ for strongly convex functions.

Method works without knowledge of total agents or graph sequence.

Applicable with only stochastic gradient samples.

Abstract

We investigate the convergence rate of the recently proposed subgradient-push method for distributed optimization over time-varying directed graphs. The subgradient-push method can be implemented in a distributed way without requiring knowledge of either the number of agents or the graph sequence; each node is only required to know its out-degree at each time. Our main result is a convergence rate of $O \left((\ln t)/t \right)$ for strongly convex functions with Lipschitz gradients even if only stochastic gradient samples are available; this is asymptotically faster than the $O \left((\ln t)/\sqrt{t} \right)$ rate previously known for (general) convex functions.