Faster logconcave sampling from a cold start in high dimension

TL;DR

This paper introduces a faster algorithm for warm-starting logconcave sampling in high dimensions, achieving sub-cubic complexity and relaxing warmness requirements by leveraging new divergence and inequality analyses.

Contribution

It provides the first sub-cubic sampling algorithm for arbitrary logconcave densities in near-isotropic position, improving warm-start conditions and refining key inequalities.

Findings

Achieves sub-cubic sampling complexity in high dimensions.

Reduces warm-start requirements from $ ext{infinity}$-Rényi divergence to $q$-Rényi divergence.

Generalizes the log-Sobolev inequality to broader geometric conditions.

Abstract

We present a faster algorithm to generate a warm start for sampling an arbitrary logconcave density specified by an evaluation oracle, leading to the first sub-cubic sampling algorithms for inputs in (near-)isotropic position. A long line of prior work incurred a warm-start penalty of at least linear in the dimension, hitting a cubic barrier, even for the special case of uniform sampling from convex bodies. Our improvement relies on two key ingredients of independent interest. (1) We show how to sample given a warm start in weaker notions of distance, in particular $q$-R\'enyi divergence for $q=\widetilde{\mathcal{O}}(1)$, whereas previous analyses required stringent $\infty$-R\'enyi divergence (with the exception of Hit-and-Run, whose known mixing time is higher). This marks the first improvement in the required warmness since Lov\'asz and Simonovits (1991). (2) We refine and generalize the log-Sobolev inequality of Lee and Vempala (2018), originally established for isotropic logconcave distributions in terms of the diameter of the support, to logconcave distributions in terms of a geometric average of the support diameter and the largest eigenvalue of the covariance matrix.