Sub-linear convergence of a stochastic proximal iteration method in Hilbert space

TL;DR

This paper proves that a stochastic proximal point algorithm converges at a sub-linear rate in infinite-dimensional Hilbert spaces under certain regularity conditions, extending finite-dimensional results to more general settings.

Contribution

It establishes the first convergence rate results for a stochastic proximal method in infinite-dimensional spaces with weak assumptions on the problem structure.

Findings

Proves sub-linear convergence rate in infinite-dimensional Hilbert spaces.

Shows convergence under weak regularity assumptions on the objective.

Demonstrates applicability to infinite-dimensional classification problems.

Abstract

We consider a stochastic version of the proximal point algorithm for optimization problems posed on a Hilbert space. A typical application of this is supervised learning. While the method is not new, it has not been extensively analyzed in this form. Indeed, most related results are confined to the finite-dimensional setting, where error bounds could depend on the dimension of the space. On the other hand, the few existing results in the infinite-dimensional setting only prove very weak types of convergence, owing to weak assumptions on the problem. In particular, there are no results that show convergence with a rate. In this article, we bridge these two worlds by assuming more regularity of the optimization problem, which allows us to prove convergence with an (optimal) sub-linear rate also in an infinite-dimensional setting. In particular, we assume that the objective function is the expected value of a family of convex differentiable functions. While we require that the full objective function is strongly convex, we do not assume that its constituent parts are so. Further, we require that the gradient satisfies a weak local Lipschitz continuity property, where the Lipschitz constant may grow polynomially given certain guarantees on the variance and higher moments near the minimum. We illustrate these results by discretizing a concrete infinite-dimensional classification problem with varying degrees of accuracy.