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Langevin Monte Carlo Beyond Lipschitz Gradient Continuity

Matej Benko, Iwona Chlebicka, Jørgen Endal, Błażej Miasojedow

TL;DR

This work introduces the Inexact Proximal Langevin Algorithm (IPLA), a proximal-splitting Langevin method designed to sample from $\mu^*(x) \propto \exp(-V(x))$ for convex potentials with polynomial or super-quadratic growth, where the gradient is not globally Lipschitz. By modeling the optimization of the associated functional $\mathcal{F}[\mu]=\mathcal{F}_V[\mu]+\mathcal{F}_{\mathcal{E}}[\mu]$ as a Wasserstein gradient flow and employing an inexact proximal step, IPLA achieves controlled complexity and convergence guarantees: $\text{KL}(\nu_n^N\|\mu^*)$ scales with $d^{(q_V+1)/2}$ and $\varepsilon^{-2}$, and in the strongly convex regime, $W_2$-convergence guarantees lead to similar complexity up to logarithmic factors. The authors prove moment bounds for the Markov chain, provide precise KL and Wasserstein convergence bounds, and establish complexity results that match or improve existing LMC bounds under weaker smoothness assumptions. Through three high-dimensional experiments (light tails, Ginzburg--Landau, and Bayesian image deconvolution) IPLA demonstrates robust performance and feasible computation, with practical proximal-operator approximations enabling scalability. Overall, IPLA broadens the applicability of Langevin Monte Carlo to a wider class of convex and super-quadratic potentials while maintaining rigorous convergence guarantees.

Abstract

We present a significant advancement in the field of Langevin Monte Carlo (LMC) methods by introducing the Inexact Proximal Langevin Algorithm (IPLA). This novel algorithm broadens the scope of problems that LMC can effectively address while maintaining controlled computational costs. IPLA extends LMC's applicability to potentials that are convex, strongly convex in the tails, and exhibit polynomial growth, beyond the conventional $L$-smoothness assumption. Moreover, we extend LMC's applicability to super-quadratic potentials and offer improved convergence rates over existing algorithms. Additionally, we provide bounds on all moments of the Markov chain generated by IPLA, enhancing its analytical robustness.

Langevin Monte Carlo Beyond Lipschitz Gradient Continuity

TL;DR

This work introduces the Inexact Proximal Langevin Algorithm (IPLA), a proximal-splitting Langevin method designed to sample from for convex potentials with polynomial or super-quadratic growth, where the gradient is not globally Lipschitz. By modeling the optimization of the associated functional as a Wasserstein gradient flow and employing an inexact proximal step, IPLA achieves controlled complexity and convergence guarantees: scales with and , and in the strongly convex regime, -convergence guarantees lead to similar complexity up to logarithmic factors. The authors prove moment bounds for the Markov chain, provide precise KL and Wasserstein convergence bounds, and establish complexity results that match or improve existing LMC bounds under weaker smoothness assumptions. Through three high-dimensional experiments (light tails, Ginzburg--Landau, and Bayesian image deconvolution) IPLA demonstrates robust performance and feasible computation, with practical proximal-operator approximations enabling scalability. Overall, IPLA broadens the applicability of Langevin Monte Carlo to a wider class of convex and super-quadratic potentials while maintaining rigorous convergence guarantees.

Abstract

We present a significant advancement in the field of Langevin Monte Carlo (LMC) methods by introducing the Inexact Proximal Langevin Algorithm (IPLA). This novel algorithm broadens the scope of problems that LMC can effectively address while maintaining controlled computational costs. IPLA extends LMC's applicability to potentials that are convex, strongly convex in the tails, and exhibit polynomial growth, beyond the conventional -smoothness assumption. Moreover, we extend LMC's applicability to super-quadratic potentials and offer improved convergence rates over existing algorithms. Additionally, we provide bounds on all moments of the Markov chain generated by IPLA, enhancing its analytical robustness.

Paper Structure

This paper contains 16 sections, 11 theorems, 80 equations, 7 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Our Contribution
  3. Description of our setting
  4. Inexact Proximal Langevin Algorithm
  5. Theoretical results
  6. Experiments
  7. Example 1: Distribution with Light Tails
  8. Example 2: Ginzburg--Landau Model
  9. Example 3: Bayesian Image Deconvolution
  10. Effective implementation of IPLA
  11. Conclusion
  12. Auxiliary results
  13. Proofs of main results of the paper
  14. Explicit formulas for constants
  15. Discussion
  16. ...and 1 more sections

Key Result

Theorem 5.1

Let $m\geq 0$. Suppose that $V$ satisfies (V) and $\varrho_0$ satisfies ($\boldsymbol{\varrho_0}$), and, for $k=1,\dots,n$, let $X_k$ be as in Algorithm alg:upl. Then there exists a constant ${\mathcal{C}}_{m}>0$ such that for all $0<\tau<1/\lambda_V$, we have If the error and the stepsize satisfy $\delta= \kappa\tau^{1+\alpha}$ for some $\kappa>0$ and $\alpha\geq 0$, then ${\mathcal{C}}_{m}={\

Figures (7)

  • Figure 1: Trajectory of the $1^{\mathrm{st}}$ coordinate from Example 1 starting in a tail. Both plots are based on the same data.
  • Figure 2: Dependence RE of IPLA and TULA (for Example 1) on a stepsize $\tau$ starting in a tail.
  • Figure 3: Result of the Bayesian Image Denoising from Example 3. The original photo by Zbyszko Siemaszko 1955-56.
  • Figure 4: Example 1, starting in tail. Trajectory of the 1$^{\mathrm{st}}$ coordinate using IPLA with burn-in time $10\,000$ samples.
  • Figure 5: Example 1, starting in tail. Trajectory of the 1$^{\mathrm{st}}$ coordinate using TULA (a) and reference Metropolis--Hastings Algorithm (MH) (b) with burn-in time $10\,000$ samples.
  • ...and 2 more figures

Theorems & Definitions (28)

  • Theorem 5.1: Moment bound
  • Remark 5.2
  • Remark 5.3
  • Proposition 5.4
  • Theorem 5.5: $\operatorname{KL}$-error bound
  • proof
  • proof
  • Theorem 5.7: Wasserstein error bound
  • proof
  • Remark 5.9
  • ...and 18 more