Non-convex learning via Stochastic Gradient Langevin Dynamics: a\n nonasymptotic analysis

2017/02/13 by Maxim Raginsky, Alexander Rakhlin, Raginsky, Maxim +3 · 27 citations
Computer Science · Engineering · Mathematics · Medicine · #Advanced Neuroimaging Techniques and Applications #FOS: Computer and information sciences #FOS: Mathematics #Machine Learning (cs.LG) #Machine Learning (stat.ML) #Markov Chains and Monte Carlo Methods #Optimization and Control (math.OC) #Probability (math.PR) #Sparse and Compressive Sensing Techniques #Stochastic Gradient Optimization Techniques

paper · pdf · doi:10.48550/arxiv.1702.03849

openalex publication_date 2017/02/13 · openalex created_date 2022/09/02 · openalex updated_date 2026/07/28

Abstract

Stochastic Gradient Langevin Dynamics (SGLD) is a popular variant of\nStochastic Gradient Descent, where properly scaled isotropic Gaussian noise is\nadded to an unbiased estimate of the gradient at each iteration. This modest\nchange allows SGLD to escape local minima and suffices to guarantee asymptotic\nconvergence to global minimizers for sufficiently regular non-convex objectives\n(Gelfand and Mitter, 1991). The present work provides a nonasymptotic analysis\nin the context of non-convex learning problems, giving finite-time guarantees\nfor SGLD to find approximate minimizers of both empirical and population risks.\nAs in the asymptotic setting, our analysis relates the discrete-time SGLD\nMarkov chain to a continuous-time diffusion process. A new tool that drives the\nresults is the use of weighted transportation cost inequalities to quantify the\nrate of convergence of SGLD to a stationary distribution in the Euclidean\n2-Wasserstein distance.\n

Citations

• Escaping From Saddle Points --- Online Stochastic Gradient for Tensor Decomposition

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