Contractive kinetic Langevin samplers beyond global Lipschitz continuity

TL;DR

The paper addresses sampling from π ∝ exp(-u) when ∇u can grow super-linearly, a regime where standard globally Lipschitz analyses fail. It introduces a monotonicity-preserving taming drift h_λ and develops two discretizations of the kinetic Langevin SDE: a stochastic exponential-type scheme and an OBABO splitting scheme, both shown to be contractive and to satisfy log-Sobolev inequalities. The authors prove explicit contraction rates in a weighted norm, derive uniform LSIs for iterates, and obtain non-asymptotic 2-Wasserstein bounds with an iteration complexity of O(log(1/ε)/ε^{2.5}), improving upon previous taming approaches. These results extend provable, efficient kinetic-Langevin sampling to ill-conditioned problems with super-linear gradients and have potential implications for differential privacy and high-dimensional Bayesian inference.

Abstract

In this paper, we examine the problem of sampling from log-concave distributions with (possibly) superlinear gradient growth under kinetic (underdamped) Langevin algorithms. Using a carefully tailored taming scheme, we propose two novel discretizations of the kinetic Langevin SDE, and we show that they are both contractive and satisfy a log-Sobolev inequality. Building on this, we establish a series of non-asymptotic bounds in $2$-Wasserstein distance between the law reached by each algorithm and the underlying target measure.

Theorems & Definitions (64)

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