Global convergence of Langevin dynamics based algorithms for nonconvex optimization

Abstract

We present a unified framework to analyze the global convergence of Langevin dynamics based algorithms for nonconvex finite-sum optimization with n component functions. At the core of our analysis is a direct analysis of the ergodicity of the numerical approximations to Langevin dynamics, which leads to faster convergence rates. Specifically, we show that gradient Langevin dynamics (GLD) and stochastic gradient Langevin dynamics (SGLD) converge to the almost minimizer2 within Õe(nd/(λε)) and Õe(d7/(λ5ε5)) stochastic gradient evaluations respectively3, where d is the problem dimension, and λ is the spectral gap of the Markov chain generated by GLD. Both results improve upon the best known gradient complexity4 results [45]. Furthermore, for the first time we prove the global convergence guarantee for variance reduced stochastic gradient Langevin dynamics (SVRG-LD) to the almost minimizer within Õe(pnd5/(λ4ε5/2)) stochastic gradient evaluations, which outperforms the gradient complexities of GLD and SGLD in a wide regime. Our theoretical analyses shed some light on using Langevin dynamics based algorithms for nonconvex optimization with provable guarantees.