Staggered Time Average Algorithm for Stochastic Non-smooth Optimization with O(1/T) Convergence

TL;DR

This paper introduces a staggered time average algorithm for stochastic non-smooth convex optimization, achieving an optimal O(1/T) convergence rate under certain structural assumptions, with broader results for general cases.

Contribution

It proposes a novel algorithm with proven convergence rates for non-smooth stochastic optimization, extending analysis beyond locally polyhedral functions.

Findings

Achieves O(1/T) convergence rate for locally polyhedral functions.

Provides convergence bounds depending on curvature for general functions.

Shows improved convergence beyond O(1/T) in deterministic special cases.

Abstract

Stochastic non-smooth convex optimization constitutes a class of problems in machine learning and operations research. This paper considers minimization of a non-smooth function based on stochastic subgradients. When the function has a locally polyhedral structure, a staggered time average algorithm is proven to have O(1/T) convergence rate. A more general convergence result is proven when the locally polyhedral assumption is removed. In that case, the convergence bound depends on the curvature of the function near the minimum. Finally, the locally polyhedral assumption is shown to improve convergence beyond O(1/T) for a special case of deterministic problems.