Dimension-free convergence rates for gradient Langevin dynamics in RKHS

2020/02/29 by Boris Muzellec, Kanji Sato, Muzellec, Boris +5 · 5 citations
Computer Science · Mathematics · #Applied mathematics #Computer science #Convergence (economics) #Dimension (graph theory) #FOS: Computer and information sciences #FOS: Mathematics #Hilbert space #Langevin dynamics #Machine Learning (cs.LG) #Machine Learning (stat.ML) #Markov Chains and Monte Carlo Methods #Markov chain #Mathematical analysis #Mathematics #Probability (math.PR) #Pure mathematics #Rate of convergence #Reproducing kernel Hilbert space #Statistical Methods and Inference #Stochastic Gradient Optimization Techniques #Stochastic differential equation #cs.LG #math.PR #stat.ML

paper · pdf · doi:10.48550/arxiv.2003.00306

published in arXiv (Cornell University) (Cornell University)

openalex publication_date 2020/02/29 · arxiv created 2020/03/26 · arxiv updated 2020/03/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• ADADELTA: An Adaptive Learning Rate Method
• Adam: A Method for Stochastic Optimization
• Learning Tensors in Reproducing Kernel Hilbert Spaces with Multilinear Spectral Penalties
• A Stochastic Approximation Method
• A Coupling Approach¶to Randomly Forced Nonlinear PDE's. I
• Non-convex learning via Stochastic Gradient Langevin Dynamics: a nonasymptotic analysis
• A Hitting Time Analysis of Stochastic Gradient Langevin Dynamics
• Non-asymptotic convergence analysis for the Unadjusted Langevin Algorithm
• Global Convergence of Langevin Dynamics Based Algorithms for Nonconvex Optimization

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