Dimension-free convergence rates for gradient Langevin dynamics in RKHS

TL;DR

This paper establishes dimension-free convergence rates for gradient Langevin dynamics in infinite-dimensional RKHS, enabling non-convex optimization with guarantees unaffected by the space's dimensionality.

Contribution

It provides the first non-asymptotic, dimension-free convergence analysis of GLD/SGLD in infinite-dimensional Hilbert spaces for regularized non-convex optimization.

Findings

Derived non-asymptotic convergence rates independent of dimension

Analyzed the properties of stochastic differential equations and Markov chains

Established geometric ergodicity of the associated Markov processes

Abstract

Gradient Langevin dynamics (GLD) and stochastic GLD (SGLD) have attracted considerable attention lately, as a way to provide convergence guarantees in a non-convex setting. However, the known rates grow exponentially with the dimension of the space. In this work, we provide a convergence analysis of GLD and SGLD when the optimization space is an infinite dimensional Hilbert space. More precisely, we derive non-asymptotic, dimension-free convergence rates for GLD/SGLD when performing regularized non-convex optimization in a reproducing kernel Hilbert space. Amongst others, the convergence analysis relies on the properties of a stochastic differential equation, its discrete time Galerkin approximation and the geometric ergodicity of the associated Markov chains.