Dissipativity Theory for Accelerating Stochastic Variance Reduction: A Unified Analysis of SVRG and Katyusha Using Semidefinite Programs

TL;DR

This paper introduces a dissipativity-based framework for analyzing and accelerating stochastic variance reduction algorithms like SVRG and Katyusha, providing a unified, intuitive, and automatable convergence analysis.

Contribution

It applies control theory, specifically dissipativity, to unify and automate the convergence analysis of SVRG and Katyusha algorithms, extending their theoretical understanding.

Findings

Recovers existing convergence results for SVRG and Katyusha.

Generalizes the theory to alternative parameter choices.

Provides a semidefinite programming approach for analysis.

Abstract

Techniques for reducing the variance of gradient estimates used in stochastic programming algorithms for convex finite-sum problems have received a great deal of attention in recent years. By leveraging dissipativity theory from control, we provide a new perspective on two important variance-reduction algorithms: SVRG and its direct accelerated variant Katyusha. Our perspective provides a physically intuitive understanding of the behavior of SVRG-like methods via a principle of energy conservation. The tools discussed here allow us to automate the convergence analysis of SVRG-like methods by capturing their essential properties in small semidefinite programs amenable to standard analysis and computational techniques. Our approach recovers existing convergence results for SVRG and Katyusha and generalizes the theory to alternative parameter choices. We also discuss how our approach complements the linear coupling technique. Our combination of perspectives leads to a better understanding of accelerated variance-reduced stochastic methods for finite-sum problems.