Hit-and-Run Mixes as Fast as the Ball Walk

2026-08-13

AI Summary

This study addresses the mixing time of the hit-and-run random walk for sampling uniformly from an isotropic convex body, starting from an arbitrary $M$-warm initial distribution. By integrating tools from probability theory, convex geometry, and Markov chain mixing analysis—particularly leveraging recent advances related to the KLS conjecture—the authors establish that the walk achieves $\varepsilon$-accuracy in total variation distance within $O(n^2 \psi_n^{-2} \log^3(M/\varepsilon))$ steps. This result improves the dependence on the warmness parameter $M$ and the target accuracy $\varepsilon$ from polynomial to polylogarithmic factors, thereby fully resolving an open problem posed by Chen and Eldan. The derived upper bound matches, up to logarithmic factors, the best-known guarantee for the ball walk under optimal warm starts, significantly advancing the theoretical understanding of the hit-and-run algorithm’s efficiency.

Abstract

Let $K\subset\mathbb{R}^n$ be an isotropic convex body. We prove that the hit-and-run walk, started from any $M$-warm distribution, reaches total-variation distance $\varepsilon$ from the uniform distribution on $K$ in $O\!\left(n^2ψ_n^{-2}\log^3(M/\varepsilon)\right)$ steps, where $ψ_n^{-1}$ is the Kannan-Lovász-Simonovits (KLS) constant. Up to logarithmic factors, this matches the best-known warm-start mixing time for the ball walk. Chen and Eldan [Discrete Comput. Geom. 2026] obtained the same $n^2ψ_n^{-2}$ dependence for hit-and-run, but with polynomial dependence on $M/\varepsilon$. Our result improves that polynomial dependence to a polylogarithmic one, fully resolving their open question about warm-start mixing of hit-and-run in isotropic convex bodies.

Problem

• hit-and-run
• mixing time
• isotropic convex body
• warm start
• KLS constant

Innovation

• hit-and-run
• mixing time
• isotropic convex body
• KLS conjecture
• Markov chain Monte Carlo