Non-ergodic Convergence Analysis of Heavy-Ball Algorithms

TL;DR

This paper provides new non-ergodic convergence rate results for the Heavy-ball optimization algorithm, including linear convergence under relaxed conditions and extensions to multi-block and decentralized settings.

Contribution

It introduces the first non-ergodic O(1/k) convergence rate for the Heavy-ball method with constant step size in convex optimization.

Findings

First non-ergodic O(1/k) rate for Heavy-ball with constant step size

Linear convergence under relaxed strong convexity assumptions

Extensions to multi-block and decentralized optimization

Abstract

In this paper, we revisit the convergence of the Heavy-ball method, and present improved convergence complexity results in the convex setting. We provide the first non-ergodic O(1/k) rate result of the Heavy-ball algorithm with constant step size for coercive objective functions. For objective functions satisfying a relaxed strongly convex condition, the linear convergence is established under weaker assumptions on the step size and inertial parameter than made in the existing literature. We extend our results to multi-block version of the algorithm with both the cyclic and stochastic update rules. In addition, our results can also be extended to decentralized optimization, where the ergodic analysis is not applicable.