Large Deviation Principle

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Sharp dimensional analysis of midpoint methods for Langevin sampling

Fan Chen, Sinho Chewi, Jianfeng Lu, Matthew S. Zhang

Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology. · Department of Statistics and Data Science, Yale University. · Department of Mathematics, Duke University.+1

Langevin Dynamics Large Deviation Principle Optimal Sample Complexity

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We study deterministic and randomized midpoint discretizations of Langevin dynamics for a target

π\propto e^{-V}

, where

0 \prec αI\preceq\nabla^2V\preceqβI

and

κ=β/α

. To achieve

\sqrtα\,W_2\leqslant\varepsilon

, we show that deterministic Heun uses at most

\widetilde O(κ^{4/3}d^{1/3}\varepsilon^{-2/3})

gradient queries, and underdamped exponential midpoint uses

\widetilde O(κ^{5/4}d^{1/4}\varepsilon^{-1/2})

. The proofs exploit cancellation at stationarity and smoothing using techniques from Malliavin calculus, outperforming previous upper bounds based on standard couplings. At bounded condition number, a lower bound matches the

d

and

\varepsilon

powers of both deterministic methods. To contrast, for the randomized midpoint methods and Poisson midpoint with at least two grid points (both overdamped and underdamped variants), a simple Gaussian calculation yields a lower bound

d^{1/3}\varepsilon^{-1/3}

to get an

\varepsilon

-close sample despite starting at a benign initialization. This shows surprisingly that in high dimensions, deterministic discretizations can outperform their random counterparts.

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