Sampling and Integration of Logconcave Functions by Algorithmic Diffusion

TL;DR

This paper introduces a novel algorithmic approach to sampling, rounding, and integrating logconcave functions, achieving significant complexity improvements and stronger guarantees compared to prior methods.

Contribution

It presents the first major complexity improvements in two decades for these problems on general logconcave functions, matching best-known results for convex bodies.

Findings

Significant complexity reductions in sampling, rounding, and integrating logconcave functions.

Stronger output guarantees for sampling, enhancing statistical estimation.

Simplified analysis of dependent random samples in statistical applications.

Abstract

We study the complexity of sampling, rounding, and integrating arbitrary logconcave functions. Our new approach provides the first complexity improvements in nearly two decades for general logconcave functions for all three problems, and matches the best-known complexities for the special case of uniform distributions on convex bodies. For the sampling problem, our output guarantees are significantly stronger than previously known, and lead to a streamlined analysis of statistical estimation based on dependent random samples.