On stochastic gradient Langevin dynamics with dependent data streams in the logconcave case

2018/12/06 by Mathias Barkhagen, Barkhagen, M., Nancy H. Chau +9 · 5 citations
Mathematics · #62L10 #62L20 #65C40 #FOS: Computer and information sciences #FOS: Mathematics #Machine Learning (stat.ML) #Markov Chains and Monte Carlo Methods #Point processes and geometric inequalities #Probability (math.PR) #Statistical Methods and Inference #Statistics Theory (math.ST)

paper · pdf · doi:10.48550/arxiv.1812.02709

openalex publication_date 2018/12/06 · openalex created_date 2020/12/07 · openalex updated_date 2026/07/28

Abstract

We study the problem of sampling from a probability distribution π on \rset^d which has a density \wrt the Lebesgue measure known up to a normalization factor x ↦ \rme^-U(x) / ∫_\rset^d \rme^-U(y) \rmd y. We analyze a sampling method based on the Euler discretization of the Langevin stochastic differential equations under the assumptions that the potential U is continuously differentiable, ∇ U is Lipschitz, and U is strongly concave. We focus on the case where the gradient of the log-density cannot be directly computed but unbiased estimates of the gradient from possibly dependent observations are available. This setting can be seen as a combination of a stochastic approximation (here stochastic gradient) type algorithms with discretized Langevin dynamics. We obtain an upper bound of the Wasserstein-2 distance between the law of the iterates of this algorithm and the target distribution π with constants depending explicitly on the Lipschitz and strong convexity constants of the potential and the dimension of the space. Finally, under weaker assumptions on U and its gradient but in the presence of independent observations, we obtain analogous results in Wasserstein-2 distance.

Citations

• Non-convex learning via Stochastic Gradient Langevin Dynamics: a\n nonasymptotic analysis
• On stochastic gradient Langevin dynamics with dependent data streams: the fully non-convex case
• Non-asymptotic convergence analysis for the Unadjusted Langevin Algorithm
• Global Convergence of Langevin Dynamics Based Algorithms for Nonconvex Optimization

Cited by

• Quantitative Convergence Analysis of Projected Stochastic Gradient Descent for Non-Convex Losses via the Goldstein Subdifferential
• Anchored Langevin Algorithms
• Mixing properties of some Markov chains models in random environments
• Contractive kinetic Langevin samplers beyond global Lipschitz continuity
• Regime-Switching Langevin Monte Carlo Algorithms

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• High-dimensional Bayesian inference via the Unadjusted Langevin Algorithm