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arXiv · 2003.00306

Dimension-free convergence rates for gradient Langevin dynamics in RKHS

Boris Muzellec·Kanji Sato·Mathurin Massias·Taiji Suzuki
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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.

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BibTeXRIS

Boris Muzellec, Kanji Sato, Mathurin Massias, Taiji Suzuki. 2020-02-29. Dimension-free convergence rates for gradient Langevin dynamics in RKHS. https://arxiv.org/abs/2003.00306

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