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User-friendly guarantees for the Langevin Monte Carlo with inaccurate gradient

Arnak S. Dalalyan, Avetik G. Karagulyan

TL;DR

The paper addresses nonasymptotic sampling guarantees for Langevin Monte Carlo (LMC) targeting smooth, strongly log-concave densities π(θ) ∝ e^{-f(θ)} by analyzing Wasserstein-2 convergence. It develops user-friendly bounds for LMC with both constant and varying step sizes, extends guarantees to inaccurate gradient evaluations, and introduces second-order variants LMCO and LMCO' that leverage Hessian information to achieve improved rates in ill-conditioned regimes. The results cover mixtures of log-concave components (MLMC) and demonstrate that gradient noise can be tolerated with explicit bias and variance terms, while still yielding dimension-aware convergence rates. The work also clarifies connections between sampling and optimization, showing how diffusion-based sampling recovers gradient-descent and Newton-method behaviors in appropriate limits, and suggests practical guidelines for choosing step-sizes and iteration counts in high-dimensional settings. Overall, the contributions offer practical, nearly dimension-optimal guarantees for a broad family of Langevin-based samplers, including robust handling of approximate gradients and second-order discretizations.

Abstract

In this paper, we study the problem of sampling from a given probability density function that is known to be smooth and strongly log-concave. We analyze several methods of approximate sampling based on discretizations of the (highly overdamped) Langevin diffusion and establish guarantees on its error measured in the Wasserstein-2 distance. Our guarantees improve or extend the state-of-the-art results in three directions. First, we provide an upper bound on the error of the first-order Langevin Monte Carlo (LMC) algorithm with optimized varying step-size. This result has the advantage of being horizon free (we do not need to know in advance the target precision) and to improve by a logarithmic factor the corresponding result for the constant step-size. Second, we study the case where accurate evaluations of the gradient of the log-density are unavailable, but one can have access to approximations of the aforementioned gradient. In such a situation, we consider both deterministic and stochastic approximations of the gradient and provide an upper bound on the sampling error of the first-order LMC that quantifies the impact of the gradient evaluation inaccuracies. Third, we establish upper bounds for two versions of the second-order LMC, which leverage the Hessian of the log-density. We provide nonasymptotic guarantees on the sampling error of these second-order LMCs. These guarantees reveal that the second-order LMC algorithms improve on the first-order LMC in ill-conditioned settings.

User-friendly guarantees for the Langevin Monte Carlo with inaccurate gradient

TL;DR

The paper addresses nonasymptotic sampling guarantees for Langevin Monte Carlo (LMC) targeting smooth, strongly log-concave densities π(θ) ∝ e^{-f(θ)} by analyzing Wasserstein-2 convergence. It develops user-friendly bounds for LMC with both constant and varying step sizes, extends guarantees to inaccurate gradient evaluations, and introduces second-order variants LMCO and LMCO' that leverage Hessian information to achieve improved rates in ill-conditioned regimes. The results cover mixtures of log-concave components (MLMC) and demonstrate that gradient noise can be tolerated with explicit bias and variance terms, while still yielding dimension-aware convergence rates. The work also clarifies connections between sampling and optimization, showing how diffusion-based sampling recovers gradient-descent and Newton-method behaviors in appropriate limits, and suggests practical guidelines for choosing step-sizes and iteration counts in high-dimensional settings. Overall, the contributions offer practical, nearly dimension-optimal guarantees for a broad family of Langevin-based samplers, including robust handling of approximate gradients and second-order discretizations.

Abstract

In this paper, we study the problem of sampling from a given probability density function that is known to be smooth and strongly log-concave. We analyze several methods of approximate sampling based on discretizations of the (highly overdamped) Langevin diffusion and establish guarantees on its error measured in the Wasserstein-2 distance. Our guarantees improve or extend the state-of-the-art results in three directions. First, we provide an upper bound on the error of the first-order Langevin Monte Carlo (LMC) algorithm with optimized varying step-size. This result has the advantage of being horizon free (we do not need to know in advance the target precision) and to improve by a logarithmic factor the corresponding result for the constant step-size. Second, we study the case where accurate evaluations of the gradient of the log-density are unavailable, but one can have access to approximations of the aforementioned gradient. In such a situation, we consider both deterministic and stochastic approximations of the gradient and provide an upper bound on the sampling error of the first-order LMC that quantifies the impact of the gradient evaluation inaccuracies. Third, we establish upper bounds for two versions of the second-order LMC, which leverage the Hessian of the log-density. We provide nonasymptotic guarantees on the sampling error of these second-order LMCs. These guarantees reveal that the second-order LMC algorithms improve on the first-order LMC in ill-conditioned settings.

Paper Structure

This paper contains 17 sections, 17 theorems, 160 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Guarantees in the Wasserstein distance with accurate gradient
  3. Reminder on guarantees for the constant-step LMC
  4. Guarantees under strong convexity for the varying step LMC
  5. Extension to mixtures of strongly log-concave densities
  6. Guarantees for the inaccurate gradient version
  7. Guarantees under additional smoothness
  8. Relation with optimization
  9. Conclusion
  10. Proofs
  11. Proof of \ref{['thOne']}
  12. Proof of \ref{['thOneBis']}
  13. Proof of \ref{['th:three']}
  14. Proof of \ref{['thTwo']}
  15. Proof of \ref{['thFour']}
  16. ...and 2 more sections

Key Result

Theorem 1

Assume that $h\in(0,2/M)$ and $f$ satisfies condition 1. The following claims hold:

Figures (2)

  • Figure 1: Plots showing the logarithm of the number of iterations as function of dimension $p$ for several values of $\epsilon$. The plotted values are derived from \ref{['G1']}-\ref{['G3']} using the data $m=10$, $M=20$, $W_2^2(\nu_0,\pi)=p + (p/m)$.
  • Figure 2: Plots showing the logarithm of the number of iterations as function of dimension $p$ for several values of $\epsilon$. The plotted values are derived from \ref{['thFour']} and \ref{['DM2']} (referred to as DM bound) using the data $m=10$, $M=50$, $M_2=1$, $W_2^2(\nu_0,\pi)=p + (p/m)$, $\delta= \sigma=0$.

Theorems & Definitions (28)

  • Theorem 1
  • Remark 2.1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Lemma 1
  • Theorem 5
  • Theorem 6
  • Proposition 1
  • Proposition 2
  • ...and 18 more