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When Langevin Monte Carlo Meets Randomization: Non-asymptotic Error Bounds beyond Log-Concavity and Gradient Lipschitzness

Xiaojie Wang, Bin Yang

TL;DR

The paper tackles non-asymptotic sampling for high-dimensional, potentially non-convex targets by analyzing randomized Langevin Monte Carlo (RLMC) and a projected variant (pRLMC). It establishes a uniform-in-time $W_2$ error bound of $O(\sqrt{d}\,h)$ for RLMC under gradient Lipschitzness and a log-Sobolev inequality, matching the best rates in the log-concave regime without extra smoothness assumptions. For non-globally Lipschitz gradients with superlinear growth, it introduces a modified RLMC (pRLMC) with an explicit non-asymptotic bound that reveals the dimension dependence $d^{(11\gamma+2)/4}$ and provides a mixing-time guarantee. The analysis hinges on combining finite-time mean-square error bounds with exponential ergodicity and uniform moment bounds to achieve time-uniform convergence in $\text{W}_2$. Overall, the work extends non-asymptotic Langevin sampling guarantees beyond log-concavity, enabling robust, theory-backed sampling for broader non-convex targets.

Abstract

Efficient sampling from complex and high dimensional target distributions turns out to be a fundamental task in diverse disciplines such as scientific computing, statistics and machine learning. In this paper, we revisit the randomized Langevin Monte Carlo (RLMC) for sampling from high dimensional distributions without log-concavity. Under the gradient Lipschitz condition and the log-Sobolev inequality, we prove a uniform-in-time error bound in $\mathcal{W}_2$-distance of order $O(\sqrt{d}h)$ for the RLMC sampling algorithm, which matches the best one in the literature under the log-concavity condition. Moreover, when the gradient of the potential $U$ is non-globally Lipschitz with superlinear growth, modified RLMC algorithms are proposed and analyzed, with non-asymptotic error bounds established. To the best of our knowledge, the modified RLMC algorithms and their non-asymptotic error bounds are new in the non-globally Lipschitz setting.

When Langevin Monte Carlo Meets Randomization: Non-asymptotic Error Bounds beyond Log-Concavity and Gradient Lipschitzness

TL;DR

The paper tackles non-asymptotic sampling for high-dimensional, potentially non-convex targets by analyzing randomized Langevin Monte Carlo (RLMC) and a projected variant (pRLMC). It establishes a uniform-in-time error bound of for RLMC under gradient Lipschitzness and a log-Sobolev inequality, matching the best rates in the log-concave regime without extra smoothness assumptions. For non-globally Lipschitz gradients with superlinear growth, it introduces a modified RLMC (pRLMC) with an explicit non-asymptotic bound that reveals the dimension dependence and provides a mixing-time guarantee. The analysis hinges on combining finite-time mean-square error bounds with exponential ergodicity and uniform moment bounds to achieve time-uniform convergence in . Overall, the work extends non-asymptotic Langevin sampling guarantees beyond log-concavity, enabling robust, theory-backed sampling for broader non-convex targets.

Abstract

Efficient sampling from complex and high dimensional target distributions turns out to be a fundamental task in diverse disciplines such as scientific computing, statistics and machine learning. In this paper, we revisit the randomized Langevin Monte Carlo (RLMC) for sampling from high dimensional distributions without log-concavity. Under the gradient Lipschitz condition and the log-Sobolev inequality, we prove a uniform-in-time error bound in -distance of order for the RLMC sampling algorithm, which matches the best one in the literature under the log-concavity condition. Moreover, when the gradient of the potential is non-globally Lipschitz with superlinear growth, modified RLMC algorithms are proposed and analyzed, with non-asymptotic error bounds established. To the best of our knowledge, the modified RLMC algorithms and their non-asymptotic error bounds are new in the non-globally Lipschitz setting.

Paper Structure

This paper contains 18 sections, 15 theorems, 136 equations, 1 table.

Table of Contents

  1. Introduction
  2. Related works
  3. Contributions of this work
  4. Exponential ergodicity of the Langevin dynamics without log-concavity
  5. Notation
  6. Exponential ergodicity under a log-Sobolev inequality
  7. Main results
  8. Main results for randomized Langevin Monte Carlo
  9. Main results for a modified randomized Langevin Monte Carlo
  10. Technical Overview
  11. Conclusion and future work
  12. Proofs of results in Subsection \ref{['Randomization-subsec:main_result_for_RLMC']}
  13. Proof of Lemma \ref{['lem:uniform_in_time_bounded_moments_to_RLMC']}
  14. Proof of Lemma \ref{['lem:finite_time_error_analysis_RLMC']}
  15. Proof of Theorem \ref{['thm:main_thm_for_RLMC']}
  16. ...and 3 more sections

Key Result

Lemma 2.4

Suppose that Assumption Randomization-ass:diss_cond_and_grad_Lip_cond holds. Let $\{X_t\}_{t\geq 0}$ be the solution of the Langevin SDE Randomization-eq:langevin_SDE. Then for any $p \in [1,\infty)$ it holds where $c\in(0,2 \mu)$ and $\mathcal{M}_1(p):=\frac{2(2p-1+\mu')^p}{cp} ( \frac{2p-2}{(2\mu-c)p} )^{p-1}$ are independent of $d$ and $t$.

Theorems & Definitions (19)

  • Lemma 2.4: Uniformly bounded moments
  • Proposition 2.5: Exponential ergodicity in $\mathcal{W}_2$-distance
  • Lemma 3.2: Uniformly bounded moments of RLMC
  • Lemma 3.3: Finite-time convergence of RLMC
  • Theorem 3.4: Main result for RLMC
  • Proposition 3.5
  • Lemma 3.7: Uniformly bounded moments of pRLMC
  • Lemma 3.8: Finite-time error analysis of pRLMC
  • Theorem 3.9: Main result for pRLMC
  • Proposition 3.10
  • ...and 9 more