Complexity of zigzag sampling algorithm for strongly log-concave distributions

TL;DR

This paper analyzes the computational complexity of the zigzag sampling algorithm for strongly log-concave distributions, highlighting its efficiency and dimension independence in convergence rate.

Contribution

It provides a complexity analysis showing that zigzag sampling achieves epsilon error with a specific gradient evaluation cost under certain conditions.

Findings

Achieves epsilon error with $O(\, ext{complexity})$ gradient evaluations.

Convergence rate is dimension independent.

Efficient for strongly log-concave distributions with favorable condition number.

Abstract

We study the computational complexity of zigzag sampling algorithm for strongly log-concave distributions. The zigzag process has the advantage of not requiring time discretization for implementation, and that each proposed bouncing event requires only one evaluation of partial derivative of the potential, while its convergence rate is dimension independent. Using these properties, we prove that the zigzag sampling algorithm achieves $\varepsilon$ error in chi-square divergence with a computational cost equivalent to $O\bigl(\kappa^2 d^\frac{1}{2}(\log\frac{1}{\varepsilon})^{\frac{3}{2}}\bigr)$ gradient evaluations in the regime $\kappa \ll \frac{d}{\log d}$ under a warm start assumption, where $\kappa$ is the condition number and $d$ is the dimension.