Dimension-free convergence rates for gradient Langevin dynamics in RKHS

• URL: http://arxiv.org/abs/2003.00306v2
• Date: Thu, 26 Mar 2020 10:05:35 GMT
• Title: Dimension-free convergence rates for gradient Langevin dynamics in RKHS
• Authors: Boris Muzellec, Kanji Sato, Mathurin Massias, Taiji Suzuki
• Abstract summary: Gradient Langevin dynamics (GLD) and SGLD have attracted considerable attention lately. We provide a convergence analysis GLD and SGLD when the space is an infinite dimensional space.
• Score: 47.198067414691174
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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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