Gradient-adjusted underdamped Langevin dynamics for sampling

TL;DR

This paper introduces gradient-adjusted underdamped Langevin dynamics (GAUL), a novel stochastic process inspired by optimization techniques, which improves sampling efficiency and convergence speed for complex target distributions.

Contribution

The paper proposes GAUL, a new SDE that incorporates optimization-inspired damping, and demonstrates its theoretical correctness and practical advantages over existing Langevin methods.

Findings

GAUL converges faster than traditional Langevin dynamics.

The mixing time depends on the square root of the condition number.

Numerical experiments show improved sampling in Bayesian models.

Abstract

Sampling from a target distribution is a fundamental problem. Traditional Markov chain Monte Carlo (MCMC) algorithms, such as the unadjusted Langevin algorithm (ULA), derived from the overdamped Langevin dynamics, have been extensively studied. From an optimization perspective, the Kolmogorov forward equation of the overdamped Langevin dynamics can be treated as the gradient flow of the relative entropy in the space of probability densities embedded with Wassrstein-2 metrics. Several efforts have also been devoted to including momentum-based methods, such as underdamped Langevin dynamics for faster convergence of sampling algorithms. Recent advances in optimizations have demonstrated the effectiveness of primal-dual damping and Hessian-driven damping dynamics for achieving faster convergence in solving optimization problems. Motivated by these developments, we introduce a class of stochastic differential equations (SDEs) called gradient-adjusted underdamped Langevin dynamics (GAUL), which add stochastic perturbations in primal-dual damping dynamics and Hessian-driven damping dynamics from optimization. We prove that GAUL admits the correct stationary distribution, whose marginal is the target distribution. The proposed method outperforms overdamped and underdamped Langevin dynamics regarding convergence speed in the total variation distance for Gaussian target distributions. Moreover, using the Euler-Maruyama discretization, we show that the mixing time towards a biased target distribution only depends on the square root of the condition number of the target covariance matrix. Numerical experiments for non-Gaussian target distributions, such as Bayesian regression problems and Bayesian neural networks, further illustrate the advantages of our approach.