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Non-asymptotic analysis of Langevin-type Monte Carlo algorithms

Shogo Nakakita

TL;DR

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\'s inequalities is applied to show that the Langevin Monte Carlo algorithm can approximate Gibbs distributions with arbitrary accuracy.

Abstract

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 Poincaré 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.

Non-asymptotic analysis of Langevin-type Monte Carlo algorithms

TL;DR

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\'s inequalities is applied to show that the Langevin Monte Carlo algorithm can approximate Gibbs distributions with arbitrary accuracy.

Abstract

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 Poincaré 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.
Paper Structure (29 sections, 33 theorems, 154 equations)

This paper contains 29 sections, 33 theorems, 154 equations.

Table of Contents

  1. Introduction
  2. Related works
  3. Contributions
  4. Outline
  5. Notations
  6. Main results
  7. Estimate of the errors of general SG-LMC
  8. Concise representation of Theorem \ref{['thm:sglmc']}
  9. Sampling complexities of Langevin-type algorithms
  10. Analysis of the LMC algorithm for $U$ of class $\mathcal{C}^{1}$ with the uniformly continuous gradient
  11. The sampling complexity of the LMC algorithm
  12. The spherically smoothed Langevin Monte Carlo (SS-LMC) algorithm
  13. The sampling complexity of SS-LMC
  14. Examples of distributions with the regularity conditions
  15. The spherically smoothed stochastic gradient Langevin Monte Carlo (SS-SG-LMC) algorithm
  16. ...and 14 more sections

Key Result

Theorem 2.1.1

Assume (A1)--(A6) and $\eta\in(0,1\wedge(\Tilde{m}/2((\omega_{\widetilde{G}}(1))^{2}+\delta_{\mathbf{v},2}))]$. It holds that for any $r\in(0,1]$ and $k\in\mathbf{N}$, where $f$ is the function defined as $C_{0},C_{1},C_{2},\kappa_{\infty}>0$ are the positive constants defined as $\Bar{b}:=b+\omega_{\nabla U}(1)$, $U_{0}:=\left\|U\right\|_{L^{\infty}(B_{1}(\mathbf{0}))}$, $\|\nabla U\|_{\mathbb

Theorems & Definitions (60)

  • Remark 1
  • Theorem 2.1.1: error estimate of general SG-LMC
  • Remark 2
  • Corollary 3.1.1: error estimate of LMC
  • Proposition 3.1.2
  • proof
  • Corollary 3.2.1: error estimate of SS-LMC
  • Corollary 3.2.2
  • Proposition 3.2.3
  • Corollary 3.3.1: error estimate of SS-SG-LMC
  • ...and 50 more