On two proofs of $d^2$ mixing of weighted Dikin walks

2026-08-28

AI Summary

本文研究了加权Dikin游走的混合时间，通过控制高概率区域的接受概率，对多面体和截断半正定锥上的指数分布采样提供了O(d^2)的混合界。

Abstract

We study the mixing time of weighted Dikin walks for sampling from exponential distributions on polytopes and truncated positive-semidefinite (PSD) cones. Our first result gives a general total-variation mixing bound under strong self-concordance, $\barν$-symmetry, and mixed-trace regularity on the local metric. The key idea is to control the Metropolis--Hastings acceptance probability on a high-probability region rather than at every point. Applying this framework to the Lee--Sidford, Lewis-weight, and John metrics yields an $\widetilde O(d^2)$ mixing bound for sampling from polytopes, while applying it to a hybrid barrier yields an $\widetilde O(d^4)$ mixing bound for sampling from truncated PSD cones. Our second result establishes stronger $χ^2$-divergence guarantees and pointwise acceptance control using a new fourth-order bootstrap condition. For a suitably scaled Lee--Sidford metric, this yields an $\widetilde O(d^2)$ mixing bound in $χ^2$-divergence, improving on the previous $\widetilde O(d^{9/4})$ bound.

Problem

• weighted Dikin walks
• mixing time
• exponential distributions
• polytopes
• truncated PSD cones

Innovation

• weighted Dikin walks
• Metropolis--Hastings acceptance probability
• high-probability region
• fourth-order bootstrap condition
• χ^2-divergence