The shifted ODE method for underdamped Langevin MCMC

Abstract

In this paper, we consider the underdamped Langevin diffusion (ULD) and propose a numerical approximation using its associated ordinary differential equation (ODE).When used as a Markov Chain Monte Carlo (MCMC) algorithm, we show that the ODE approximation achieves a 2-Wasserstein error ofsteps under the standard smoothness and strong convexity assumptions on the target distribution.This matches the complexity of the randomized midpoint method proposed by Shen and Lee [NeurIPS 2019] which was shown to be order optimal by Cao, Lu and Wang.However, the main feature of the proposed numerical method is that it can utilize additional smoothness of the target log-density f . More concretely, we show that the ODE approximation achieves a 2-Wasserstein error of ε in O d 2 5 /ε 2 5 and O √ d/ε 1 3steps when Lipschitz continuity is assumed for the Hessian and third derivative of f .By discretizing this ODE using a third order Runge-Kutta method, we can obtain a practical MCMC method that uses just two additional gradient evaluations per step.In our experiment, where the target comes from a logistic regression, this method shows faster convergence compared to other unadjusted Langevin MCMC algorithms.

Wasserstein distance estimates for the distributions of numerical approximations to ergodic stochastic differential equations

“…It is well known that these two assumptions are equivalent to the Hessian of f , which we will denote by H : R d → R d×d , being positive definite and satisfying mI d×d H(x) LI d×d . In studies like the present one, Assumptions 2.1 and 2.2 are standard in the literature: see, among others, [15,14,16,17] for the overdamped Langevin dynamics and [20,21,22,24] for the underdamped case.…”

“…In the absence of noise, this integrator is the well-known Euler exponential integrator [37], based, via the variation of constants formula/Duhamel's principle, on the exact integration of the system dv/dt = −γv, dx/dt = v. In the stochastic scenario the algorithm is first order in both the weak and strong senses. The paper [24]…”

“…A bound similar to (6.5) is derived which is valid only for small values h ≤ O(κ −1 ) ∧ O(L 1/2 md −1/2 L −1 1 ). The reference [24] suggests a novel approach to obtaining high order discretizations of the underdamped Langevin dynamics (1.2). At each step, in addition to generating suitable random variables, one has to integrate a so-called "shifted ODE", whose solutions are smoother than the solutions of (1.2).…”

“…At each step, in addition to generating suitable random variables, one has to integrate a so-called "shifted ODE", whose solutions are smoother than the solutions of (1.2). The analysis in that reference examines the case where the integration is exact; in practice, the shifted ODE has of course to be discretized by a suitable numerical method and [24] provides numerical examples based on two different choices of such a method.…”

“…[14,15,16,17,18], while the paper [19] obtains similar bounds for implicit methods applied to (1.1). Similar nonasymptotic analyses for the case of the underdamped Langevin equation appear in [20,21,22,23,24]. One of the aims of all that literature is to study the number of steps n that the integrators require to achieve a target accuracy ǫ when applied to d-dimensional targets with condition number κ. Underdamped discretizations may lead to a better dependence of n on ǫ and d than their overdamped counterparts.…”