The query complexity of sampling from strongly log-concave distributions in one dimension

TL;DR

This paper proves a tight lower bound on the number of queries needed to sample from strongly log-concave distributions in one dimension, and introduces a rejection sampling algorithm that outperforms existing methods.

Contribution

It establishes the first tight lower bound on query complexity and presents a novel rejection sampling algorithm that improves over MCMC-based methods.

Findings

Lower bound of Ω(log log κ) on query complexity.

A new rejection sampling algorithm that matches the lower bound.

Improved efficiency over polynomially scaling MCMC algorithms.

Abstract

We establish the first tight lower bound of $\Omega(\log\log\kappa)$ on the query complexity of sampling from the class of strongly log-concave and log-smooth distributions with condition number $\kappa$ in one dimension. Whereas existing guarantees for MCMC-based algorithms scale polynomially in $\kappa$, we introduce a novel algorithm based on rejection sampling that closes this doubly exponential gap.