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Parallelized Midpoint Randomization for Langevin Monte Carlo

Lu Yu, Arnak Dalalyan

TL;DR

The paper addresses accelerating Langevin-based Monte Carlo sampling for smooth, strongly log-concave targets by introducing parallelized randomized midpoint discretizations for both vanilla and kinetic Langevin diffusions. The pRLMC and pRKLMC algorithms partition parallel gradient evaluations to achieve nonasymptotic Wasserstein-2 guarantees with explicit dependence on step size $h$, parallel width $R$, and sequential iterations $Q$, enabling quantified time-speedups. The main contributions include strengthened ${ m W}_2$ bounds with small constants, reduced initialization requirements for kinetic Langevin, and a detailed time-versus-gradient-query trade-off analysis, including limits as $R\to\infty$ and comparisons to concurrent work. The results offer practical guidance on choosing $R$, $Q$, and $h$ to balance convergence speed and computational resources, and they establish improved rates over prior randomized midpoint analyses while highlighting memory considerations for large $p$ and high condition numbers.

Abstract

We study the problem of sampling from a target probability density function in frameworks where parallel evaluations of the log-density gradient are feasible. Focusing on smooth and strongly log-concave densities, we revisit the parallelized randomized midpoint method and investigate its properties using recently developed techniques for analyzing its sequential version. Through these techniques, we derive upper bounds on the Wasserstein distance between sampling and target densities. These bounds quantify the substantial runtime improvements achieved through parallel processing.

Parallelized Midpoint Randomization for Langevin Monte Carlo

TL;DR

The paper addresses accelerating Langevin-based Monte Carlo sampling for smooth, strongly log-concave targets by introducing parallelized randomized midpoint discretizations for both vanilla and kinetic Langevin diffusions. The pRLMC and pRKLMC algorithms partition parallel gradient evaluations to achieve nonasymptotic Wasserstein-2 guarantees with explicit dependence on step size , parallel width , and sequential iterations , enabling quantified time-speedups. The main contributions include strengthened bounds with small constants, reduced initialization requirements for kinetic Langevin, and a detailed time-versus-gradient-query trade-off analysis, including limits as and comparisons to concurrent work. The results offer practical guidance on choosing , , and to balance convergence speed and computational resources, and they establish improved rates over prior randomized midpoint analyses while highlighting memory considerations for large and high condition numbers.

Abstract

We study the problem of sampling from a target probability density function in frameworks where parallel evaluations of the log-density gradient are feasible. Focusing on smooth and strongly log-concave densities, we revisit the parallelized randomized midpoint method and investigate its properties using recently developed techniques for analyzing its sequential version. Through these techniques, we derive upper bounds on the Wasserstein distance between sampling and target densities. These bounds quantify the substantial runtime improvements achieved through parallel processing.
Paper Structure (16 sections, 16 theorems, 167 equations, 1 figure, 2 algorithms)

This paper contains 16 sections, 16 theorems, 167 equations, 1 figure, 2 algorithms.

Table of Contents

  1. Introduction
  2. Parallelized randomized midpoint discretization: the vanilla Langevin diffusion
  3. Parallelized Randomized midpoint method for the kinetic Langevin diffusion
  4. Discussion
  5. Consequences in the setting of an unlimited number of parallel units
  6. Consequences in the setting of a limited number of parallel units
  7. Time complexity versus computational complexity
  8. Acknowledgements
  9. Proof of the upper bound on the error of parallelized RLMC
  10. Proof of technical lemmas
  11. Proof of the upper bound on the error of parallelized RKLMC
  12. Auxiliary lemmas
  13. Proof of Theorem \ref{['thm:rklmc']}
  14. Proofs of the technical lemmas
  15. Proof of \ref{['lem:2']} (Discretization error)
  16. ...and 1 more sections

Key Result

Theorem 1

If the function $f:\mathbb R^p\to \mathbb R$ satisfies Assumption asm:A-scgl and the parameters $(h,Q,R)$ are such that $\bar{h}^{Q}+\bar{h}/R+(\bar{h}^{Q-1}+\bar{h}/R^{3/2})\sqrt{\kappa \bar{h}}\leqslant 0.1$ with $\bar{h}=Mh,$ then, for every $n\geqslant 1$, the the distribution $\nu_n^{\textup{\s

Figures (1)

  • Figure 1: Visualization of Parallelized RLMC.

Theorems & Definitions (25)

  • Theorem 1
  • Theorem 2
  • Corollary 1: Mixing time for the vanilla Langevin with parallelization and randomized midpoint discretization
  • Corollary 2: Mixing time for the kinetic Langevin with parallelization and randomized midpoint discretization
  • Corollary 3
  • proof : Proof of Theorem \ref{['thm:rlmc']}
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • ...and 15 more