Underdamped Langevin MCMC: A non-asymptotic analysis (1707.03663v7)

Abstract: We study the underdamped Langevin diffusion when the log of the target distribution is smooth and strongly concave. We present a MCMC algorithm based on its discretization and show that it achieves $\varepsilon$ error (in 2-Wasserstein distance) in $\mathcal{O}(\sqrt{d}/\varepsilon)$ steps. This is a significant improvement over the best known rate for overdamped Langevin MCMC, which is $\mathcal{O}(d/\varepsilon^2)$ steps under the same smoothness/concavity assumptions. The underdamped Langevin MCMC scheme can be viewed as a version of Hamiltonian Monte Carlo (HMC) which has been observed to outperform overdamped Langevin MCMC methods in a number of application areas. We provide quantitative rates that support this empirical wisdom.

• The paper shows that underdamped Langevin MCMC achieves an ε-error in 2-Wasserstein distance after O(√d/ε) steps, outperforming overdamped methods.
• It analyzes the continuous SDE process and discretization error, ensuring exponential contraction to the invariant distribution.
• The study demonstrates robustness under noisy gradient conditions, laying a theoretical foundation for advanced Bayesian inference and machine learning.

Analysis and Performance of Underdamped Langevin MCMC

This paper presents a critical investigation into the field of sampling algorithms, focusing specifically on the underdamped Langevin Markov chain Monte Carlo (MCMC) method. The paper advances the theoretical understanding of sampling in high-dimensional spaces by offering a non-asymptotic convergence analysis of the underdamped Langevin diffusion, under conditions of log-smooth and strongly log-concave distributions. This analysis is crucial for establishing run-time guarantees and efficiency in computational tasks involving Bayesian inference and machine learning.

At its core, the paper introduces a new MCMC algorithm based on the discretization of the underdamped Langevin process. The authors rigorously prove that this algorithm achieves an $\varepsilon$ error rate in the 2-Wasserstein distance after $\mathcal{O}(\sqrt{d}/\varepsilon)$ steps. This result is significant, representing a substantial improvement over the overdamped Langevin MCMC method, which requires $\mathcal{O}(d/\varepsilon^2)$ steps under the same distributional assumptions. The underdamped Langevin MCMC thereby exhibits enhanced convergence characteristics, especially in scenarios where high-dimensional sampling is prevalent.

The analysis leverages the stochastic differential equation (SDE) framework of the underdamped Langevin diffusion. The authors begin by studying the continuous-time process and then delineate its convergence properties, establishing that it contracts exponentially fast to its invariant distribution. This is a pivotal property, providing theoretical justification for the empirical fast mixing capabilities observed in Hamiltonian Monte Carlo (HMC), an algorithm widely recognized for its efficiency.

Further contributions include examining the discretization error introduced when transforming the continuous diffusion into a practical algorithm. Theoretical bounds on the discrepancy due to this discretization are provided, ensuring that the overall convergence rate remains favorable. Additionally, the paper explores the application of the algorithm in settings where only noisy gradient information is available, demonstrating robustness in the presence of stochastic perturbations in the gradient estimates.

Indications for future research are manifold. Understanding the underpinnings of the acceleration mechanisms in stochastic optimization, as well as extending this framework to non-convex distributions, presents a promising avenue. Additionally, exploring adaptive methods to further optimize step sizes dynamically could yield further efficiencies. The condition number dependence posed by the algorithm, though improved, still leaves room for further reduction, potentially leading to faster MCMC methods.

In conclusion, this paper provides a substantive advancement in understanding and improving the performance of sampling methods in AI and machine learning. The theoretical contributions offered not only improve sampling efficiency but also align with the empirical successes observed in using HMC, offering a formal foundation for these widely practiced computational techniques. Both the implications for high-dimensional Bayesian inference tasks and the broader applicability to machine learning underscore the importance of this research in shaping future developments in probabilistic computation.