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Non-asymptotic analysis of Langevin-type Monte Carlo algorithms

arXiv 2303.12407 ↗First posted 22 Mar 2023math.ST math.PR stat.MLcited by 0

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Abstract (arXiv)

We study Langevin-type algorithms for sampling from Gibbs distributions such that the potentials are dissipative and their weak gradients have finite moduli of continuity not necessarily convergent to zero. Our main result is a non-asymptotic upper bound of the 2-Wasserstein distance between a Gibbs distribution and the law of general Langevin-type algorithms based on the Liptser--Shiryaev theory and Poincar'e inequalities. We apply this bound to show that the Langevin Monte Carlo algorithm can approximate Gibbs distributions with arbitrary accuracy if the potentials are dissipative and their gradients are uniformly continuous. We also propose Langevin-type algorithms with spherical smoothing for distributions whose potentials are not convex or continuously differentiable.

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References · 14

  1. 2017Non-convex learning via Stochastic Gradient Langevin Dynamics: a nonasymptotic analysiscited by 242
  2. 2014Theoretical guarantees for approximate sampling from smooth and log-concave densitiescited by 206
  3. 2015Non-asymptotic convergence analysis for the Unadjusted Langevin Algorithmcited by 195
  4. 2016High-dimensional Bayesian inference via the Unadjusted Langevin Algorithmcited by 144
  5. 2018Analysis of Langevin Monte Carlo via convex optimizationcited by 126
  6. 2017Global Convergence of Langevin Dynamics Based Algorithms for Nonconvex Optimizationcited by 96
  7. 2020On the Convergence of Langevin Monte Carlo: The Interplay between Tail Growth and Smoothnesscited by 55
  8. 2013Proximal Markov chain Monte Carlo algorithmscited by 46
  9. 2018Global Non-convex Optimization with Discretized Diffusionscited by 44
  10. 2019Nonasymptotic estimates for Stochastic Gradient Langevin Dynamics under local conditions in nonconvex optimizationcited by 38
  11. 2017The Tamed Unadjusted Langevin Algorithmcited by 31
  12. 2019Langevin Monte Carlo without smoothnesscited by 27
2 more
  1. 2021The Langevin Monte Carlo algorithm in the non-smooth log-concave casecited by 26
  2. 2021Unadjusted Langevin algorithm for sampling a mixture of weakly smooth potentialscited by 3

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