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Dimension-free convergence rates for gradient Langevin dynamics in RKHS

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 under the dissipative condition. 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, dimensionfree 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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Dates and versions

hal-03920387 , version 1 (03-01-2023)

Identifiers

  • HAL Id : hal-03920387 , version 1

Cite

Boris Muzellec, Kanji Sato, Mathurin Massias, Taiji Suzuki. Dimension-free convergence rates for gradient Langevin dynamics in RKHS. COLT 2022 - 35th Annual Conference on Learning Theory, Jul 2022, London, United Kingdom. ⟨hal-03920387⟩
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