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Non-asymptotic Analysis of Poisson randomized midpoint Langevin Monte Carlo

Tian Shen, Zhonggen Su

TL;DR

This work provides a non-asymptotic analysis of Poisson randomized midpoint Langevin Monte Carlo (PRLMC) as a discretized, stochastic approximation to Langevin diffusion for sampling from $π(dx) ∝ e^{-U(x)}dx$. It establishes the existence and uniqueness of the stationary distribution $π_η$ for PRLMC with a constant step, proves convergence in total variation to $π_η$, and derives a Wasserstein distance bound between $π_η$ and the target $π$, with tighter rates when higher derivatives of $U$ are bounded. The paper also develops a decreasing-step variant of PRLMC and shows near-optimal convergence to $π$ in $2$-Wasserstein distance and a $ ext{d}_{ ext{G}}$–type functional distance. The results rely on drift/minorization arguments, Lindeberg-type replacement, and coupling techniques to obtain explicit, non-asymptotic error bounds applicable in high dimensions. These findings provide actionable error controls for PRLMC algorithms in Bayesian inference and machine learning settings where precise sampling guarantees are essential.

Abstract

The task of sampling from a high-dimensional distribution $π$ on $\R^d$ is a fundamental algorithmic problem with applications throughout statistics, engineering, and the sciences. Consider the Langevin diffusion on $\R^d$ \begin{align*} \dif X_t=-\nabla U(X_t)dt+\sqrt{2}dB_t, \end{align*} under mild conditions, it admits $π(\dif x)\propto \exp(-U(x))\dif x$ as its unique stationary distribution. Recently, Kandasamy and Nagaraj (2024) introduced a stochastic algorithm called Poisson Randomized Midpoint Langevin Monte Carlo (PRLMC) to enhance the rate of convergence towards the target distribution $π$. In this paper, we first show that under mild conditions, the PRLMC, as a Markov chain, admits a unique stationary distribution $π_η$ ($η$ is the step size) and obtain the convergence rate of PRLMC to $π_η$ in total variation distance. Then we establish a sharp error bound between $π_η$ and $π$ under the 2-Wasserstein distance. Finally, we propose a decreasing-step size version of PRLMC and provide its convergence rate to $π$ which is nearly optimal.

Non-asymptotic Analysis of Poisson randomized midpoint Langevin Monte Carlo

TL;DR

This work provides a non-asymptotic analysis of Poisson randomized midpoint Langevin Monte Carlo (PRLMC) as a discretized, stochastic approximation to Langevin diffusion for sampling from . It establishes the existence and uniqueness of the stationary distribution for PRLMC with a constant step, proves convergence in total variation to , and derives a Wasserstein distance bound between and the target , with tighter rates when higher derivatives of are bounded. The paper also develops a decreasing-step variant of PRLMC and shows near-optimal convergence to in -Wasserstein distance and a –type functional distance. The results rely on drift/minorization arguments, Lindeberg-type replacement, and coupling techniques to obtain explicit, non-asymptotic error bounds applicable in high dimensions. These findings provide actionable error controls for PRLMC algorithms in Bayesian inference and machine learning settings where precise sampling guarantees are essential.

Abstract

The task of sampling from a high-dimensional distribution on is a fundamental algorithmic problem with applications throughout statistics, engineering, and the sciences. Consider the Langevin diffusion on \begin{align*} \dif X_t=-\nabla U(X_t)dt+\sqrt{2}dB_t, \end{align*} under mild conditions, it admits as its unique stationary distribution. Recently, Kandasamy and Nagaraj (2024) introduced a stochastic algorithm called Poisson Randomized Midpoint Langevin Monte Carlo (PRLMC) to enhance the rate of convergence towards the target distribution . In this paper, we first show that under mild conditions, the PRLMC, as a Markov chain, admits a unique stationary distribution ( is the step size) and obtain the convergence rate of PRLMC to in total variation distance. Then we establish a sharp error bound between and under the 2-Wasserstein distance. Finally, we propose a decreasing-step size version of PRLMC and provide its convergence rate to which is nearly optimal.

Paper Structure

This paper contains 14 sections, 17 theorems, 177 equations.

Table of Contents

  1. Introduction
  2. Assumptions and Main Results
  3. Proofs of Theorem \ref{['ms']} and Proposition \ref{['ms2']}
  4. Preliminaries
  5. Auxiliary results
  6. Proof of Main results
  7. Convergence rate in $d_{\mathcal{G}}$ distance for PRLMC
  8. Auxiliary results
  9. Proof of Theorem \ref{['ms3']}
  10. Non-asymptotic bounds in 2-Wasserstein distance for PRLMC
  11. Proofs of Theorem \ref{['ms4']}, Corollary \ref{['ms5']} and \ref{['ms6']}
  12. Proof of Lemma \ref{['s4']}
  13. Proof of the Lyapunov condition
  14. Proofs of the auxiliary results in Section \ref{['2127']} and \ref{['sec5']}

Key Result

Theorem 2.3

Under Assumption ass, there exists a constant $\eta_0\in(0,1)$ depending only on $m$ and $L$, such that for any $\eta\in(0,\eta_0)$, the Markov chain $(\tilde{X}_n)_{n\geqslant0}$ defined in prlmc has a unique stationary distribution $\pi_{\eta}$ which satisfies Furthermore, for any initial state $\tilde{X}_0=x\in\mathbb{R}^d$, we have where $\delta>1$ is a constant depending on step size $\eta$

Theorems & Definitions (39)

  • Remark 2.2
  • Theorem 2.3
  • Proposition 2.4
  • Remark 2.5
  • Theorem 2.8
  • Remark 2.9
  • Theorem 2.10
  • Corollary 2.11
  • Corollary 2.12
  • Definition 3.1: Accessible set, small set
  • ...and 29 more