The Continuous Stochastic Gradient Method: Part I -- Convergence Theory

TL;DR

This paper provides a comprehensive convergence analysis of the continuous stochastic gradient (CSG) method, including for constant step sizes and line search, addressing challenges in stochastic optimization involving integration and high-dimensional problems.

Contribution

It offers a complete convergence theory for CSG with constant step sizes and introduces new techniques for calculating integration weights, broadening its applicability.

Findings

Convergence established for constant step sizes and line search.

New methods for integration weight calculation are proposed.

Extended applicability to high-dimensional integrals and distributed data.

Abstract

In this contribution, we present a full overview of the continuous stochastic gradient (CSG) method, including convergence results, step size rules and algorithmic insights. We consider optimization problems in which the objective function requires some form of integration, e.g., expected values. Since approximating the integration by a fixed quadrature rule can introduce artificial local solutions into the problem while simultaneously raising the computational effort, stochastic optimization schemes have become increasingly popular in such contexts. However, known stochastic gradient type methods are typically limited to expected risk functions and inherently require many iterations. The latter is particularly problematic, if the evaluation of the cost function involves solving multiple state equations, given, e.g., in form of partial differential equations. To overcome these drawbacks, a recent article introduced the CSG method, which reuses old gradient sample information via the calculation of design dependent integration weights to obtain a better approximation to the full gradient. While in the original CSG paper convergence of a subsequence was established for a diminishing step size, here, we provide a complete convergence analysis of CSG for constant step sizes and an Armijo-type line search. Moreover, new methods to obtain the integration weights are presented, extending the application range of CSG to problems involving higher dimensional integrals and distributed data.