Convergence in total variation for the kinetic Langevin algorithm (2407.09301v3)

Abstract: We prove non asymptotic total variation estimates for the kinetic Langevin algorithm in high dimension when the target measure satisfies a Poincar\'e inequality and has gradient Lipschitz potential. The main point is that the estimate improves significantly upon the corresponding bound for the non kinetic version of the algorithm, due to Dalalyan. In particular the dimension dependence drops from $O(n)$ to $O(\sqrt n)$.

• The paper establishes non-asymptotic total variation bounds for the kinetic Langevin algorithm, reducing the dimension dependence from O(n) to O(√n).
• It employs a hypocoercivity framework to manage the challenges of degenerate diffusion, ensuring robust convergence for potentials with gradient Lipschitz and log-concave properties.
• The paper provides practical guidance on parameter selection and efficient high-dimensional sampling, benefiting applications in Bayesian statistics and machine learning.

Convergence in Total Variation for the Kinetic Langevin Algorithm

Introduction

The paper "Convergence in total variation for the kinetic Langevin algorithm" by Joseph Lehec rigorously explores the efficiency of the kinetic Langevin algorithm for sampling from high-dimensional probability measures. Specifically, the paper targets measures satisfying a Poincaré inequality and with a gradient Lipschitz potential. The key contribution is the derivation of non-asymptotic total variation distance estimates, which are shown to significantly outperform bounds for the overdamped (non-kinetic) Langevin algorithm.

Problem Context and Motivation

The paper addresses the problem of sampling from a target probability measure $\mu$ on $\mathbb{R}^n$ expressed as $\mu(dx) = e^{-V(x)} \,dx$, where $V$ is a smooth potential function. Sampling from such distributions is pivotal in various fields, including Bayesian statistics, optimization, and machine learning. The kinetic Langevin algorithm, which includes an additional speed variable, is proposed as a more efficient alternative to the traditional Langevin algorithm.

Methodology

Langevin and Kinetic Langevin Algorithms

The Langevin algorithm is a Markov chain Monte Carlo (MCMC) method formed by discretizing the Langevin diffusion given by: $dX_t = \sqrt{2} \, dW_t - \nabla V (X_t) \, dt,$ where $(W_t)$ is a standard Brownian motion. This algorithm converges to the target measure $\mu$ but with an $O(n)$ dependence on the dimension.

The kinetic version, or underdamped Langevin diffusion, introduces an auxiliary velocity variable $Y_t$ and follows: $$\begin{cases} dX_t = Y_t \, dt \ dY_t = \sqrt{2\beta} \, dW_t - \beta Y_t \, dt - \nabla V(X_t) \, dt, \end{cases}$$ where $\beta$ is the friction parameter. This modification results in dimensions dropping from $O(n)$ to $O(\sqrt{n})$ in certain cases.

Main Results

Non-asymptotic Total Variation Bounds

The primary result (Theorem 1) establishes that the sampling error (in total variation) of the kinetic Langevin algorithm can be bounded and showcases a significant dimension dependence improvement from $O(n)$ to $O(\sqrt{n})$. The results hold under two contexts:

1. General potential satisfying gradient Lipschitz condition.
2. Log-concave potential.

The theorem specifies the number of steps $k$ required to achieve a total variation distance $\epsilon$: $$k \approx \epsilon^{-1} (L C_P)^{3/2} (\sqrt{n} + \sqrt{\log (1+\chi_2(x_0 \mid \mu))}) \log \left(\frac{\chi_2(x_0 \mid \mu)}{\epsilon}\right),$$ where $C_P$ is the Poincaré constant and $L$ is the Lipschitz constant of the gradient.

Hypocoercive Estimate

A significant part of the proof involves a hypocoercive estimate, crucial due to the degenerate nature of the diffusion process. Hypocoercivity pertains to the convergence rate of non-coercive diffusions and was addressed using methods inspired by Villani and later adapted by Cao, Lu, and Wang. Theorem 2 in the paper provides: $$\chi_2(\nu P_t \mid \pi) \leq 2 \cdot \exp\left(- c \cdot \frac{\beta t}{1 + (\beta^2 + \kappa)C_P}\right) \cdot \chi_2(\nu \mid \pi),$$ where $\kappa$ is the semi-convexity constant of the potential $V$.

Practical and Theoretical Implications

Practically, the findings advocate for the use of kinetic Langevin algorithms in high-dimensional sampling tasks, promising significantly improved efficiency. Theoretically, this work contributes to the understanding of non-coercive diffusion processes and offers a robust framework for analyzing MCMC algorithms.

Future Directions

Future research could extend these results to more complex potential functions and explore further optimizations in the choice of friction and time step parameters. The interactions between kinetic Langevin algorithms and other MCMC methods, such as Hamiltonian Monte Carlo, merit further investigation.

Conclusion

This paper solidifies the kinetic Langevin algorithm as a powerful tool for high-dimensional sampling, offering rigorous convergence guarantees that surpass traditional Langevin methods, particularly in terms of dimensional efficiency and robustness in total variation distance. The theoretical insights and practical guidelines provided are valuable contributions to algorithmic sampling and stochastic optimization.