Shifted Composition IV: Toward Ballistic Acceleration for Log-Concave Sampling

TL;DR

This paper introduces a new analysis framework for underdamped Langevin dynamics, achieving the first ballistic acceleration for log-concave sampling and providing improved iteration complexity guarantees.

Contribution

It develops a coupling-based local error framework for degenerate diffusions, enabling the first discrete-time ballistic acceleration results in log-concave sampling.

Findings

First ballistic acceleration result for log-concave sampling

Established $d^{1/3}$ iteration complexity for total variation error

Extended analysis framework to degenerate diffusions

Abstract

Acceleration is a celebrated cornerstone of convex optimization, enabling gradient-based algorithms to converge sublinearly in the condition number. A major open question is whether an analogous acceleration phenomenon is possible for log-concave sampling. Underdamped Langevin dynamics (ULD) has long been conjectured to be the natural candidate for acceleration, but a central challenge is that its degeneracy necessitates the development of new analysis approaches, e.g., the theory of hypocoercivity. Although recent breakthroughs established ballistic acceleration for the (continuous-time) ULD diffusion via space-time Poincare inequalities, (discrete-time) algorithmic results remain entirely open: the discretization error of existing analysis techniques dominates any continuous-time acceleration. In this paper, we give a new coupling-based local error framework for analyzing ULD and its numerical discretizations in KL divergence. This extends the framework in Shifted Composition III from uniformly elliptic diffusions to degenerate diffusions, and shares its virtues: the framework is user-friendly, applies to sophisticated discretization schemes, and does not require contractivity. Applying this framework to the randomized midpoint discretization of ULD establishes the first ballistic acceleration result for log-concave sampling (i.e., sublinear dependence on the condition number). Along the way, we also obtain the first $d^{1/3}$ iteration complexity guarantee for sampling to constant total variation error in dimension $d$.