Accelerating Langevin Monte Carlo Sampling: A Large Deviations Analysis

TL;DR

This work develops a unified large deviations framework for generalized Langevin dynamics used in sampling from $\mu(\theta) \propto e^{-U(\theta)}$ in high dimensions. It derives an explicit rate function $I_{\tau}(\nu)$ that decomposes into symmetric and antisymmetric components, enabling direct comparison of convergence speed across overdamped and several variants, including mirror Langevin, high-order Langevin, and Hessian-free high-resolution dynamics. Each variant is shown to admit a tailored rate function (e.g., $I_{M}$, $I_{H}$, $I_{R}$) under hypoellipticity, controllability, and Lyapunov conditions, with Poisson equations linking the antisymmetric parts. The theoretical results are complemented by Bayesian logistic regression experiments on synthetic and real data, demonstrating acceleration or comparable performance for many variants given appropriate hyperparameter choices, thereby informing practical algorithm selection and tuning. Overall, the paper provides a principled, quantitative framework to analyze and compare Langevin-based samplers beyond overdamped dynamics.

Abstract

Langevin algorithms are popular Markov chain Monte Carlo methods that are often used to solve high-dimensional large-scale sampling problems in machine learning. The most classical Langevin Monte Carlo algorithm is based on the overdamped Langevin dynamics. There are many variants of Langevin dynamics that often show superior performance in practice. In this paper, we provide a unified approach to study the acceleration of the variants of the overdamped Langevin dynamics through the lens of large deviations theory. Numerical experiments using both synthetic and real data are provided to illustrate the efficiency of these variants.

Figures (4)

• Figure 1: The plots show the accuracy over the synthetic data with dimension 569 $\times$ 31, in which all variants of the Langevin algorithms outperform overdamped Langevin algorithm in Figure \ref{['fig:OD_s']} with an appropriate choice of hyperparameters.
• Figure 2: With a slight change of hyperparameters, we can see from this figure that underdamped Langevin (Figure \ref{['fig:UD2_s']}), high-order Langevin (Figure \ref{['fig:HO2_s']}) and Hessian-free high-resolution (Figure \ref{['fig:HFHR2_s']}) can outperform overdamped Langevin (Figure \ref{['fig:OD2_s']}); however, mirror Langevin (Figure \ref{['fig:ML2_s']}) and non-reversible (Figure \ref{['fig:NR2_s']}) cannot, even though their performance is comparable with overdamped Langevin (Figure \ref{['fig:OD2_s']}).
• Figure 3: The plots show the accuracy over the real data with dimension 569 $\times$ 31, in which all variants of the Langevin algorithms outperform overdamped Langevin algorithm in Figure \ref{['fig:OD']} with an appropriate choice of hyperparameters.
• Figure 4: With a slight change of hyperparameters, we can see from this figure that underdamped Langevin (Figure \ref{['fig:UD2']}), high-order Langevin (Figure \ref{['fig:HO2']}) and Hessian-free high-resolution (Figure \ref{['fig:HFHR2_s']}) can outperform overdamped Langevin (Figure \ref{['fig:OD2']}); however, mirror Langevin (Figure \ref{['fig:ML2']}) and non-reversible (Figure \ref{['fig:NR2']}) cannot, even though their performance is comparable with overdamped Langevin (Figure \ref{['fig:OD2']}).

Theorems & Definitions (35)

• Proposition 2.1: Proposition 2.9 in LDP-GG
• Proposition 2.2: Proposition 2.10. in LDP-GG
• Lemma 2.1: Theorem 3.3 in LDP-GG
• Theorem 2.1
• Lemma 2.2: Proposition 4.1 in LDP-GG
• Lemma 2.3
• Lemma 2.4: Proposition 4.3 in LDP-GG
• Theorem 2.2
• Lemma 2.5
• Lemma 2.6