Dissipativity Theory for Accelerating Stochastic Variance Reduction: A Unified Analysis of SVRG and Katyusha Using Semidefinite Programs
This work offers a new theoretical perspective for understanding accelerated stochastic optimization methods, which is incremental as it builds on existing algorithms without introducing new ones.
The paper tackles the analysis of variance-reduction algorithms like SVRG and Katyusha for convex finite-sum problems by applying dissipativity theory from control, providing a unified framework that recovers existing convergence results and generalizes to alternative parameters.
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.