[论文] On two proofs of d² mixing of weighted Dikin walks

论文概要

研究领域: ML 作者: Yuansi Chen, Yunbum Kook 发布时间: 2026-08-28 arXiv: 2608.28566

中文摘要

我们研究了加权Dikin游走在多面体和截断半正定（PSD）锥上从指数分布采样的混合时间。我们的第一个结果在强自协调、ν̄对称和局部度量的混合迹正则性下给出了总变差混合界。关键思想是在高概率区域上控制Metropolis-Hastings接受概率，而非在每个点上。将此框架应用于Lee-Sidford、Lewis权重和John度量，对多面体采样产生Õ(d²)的混合界，而应用于混合障碍函数对截断PSD锥采样产生Õ(d⁴)的混合界。我们的第二个结果使用新的四阶bootstrap条件建立了更强的χ²散度保证和逐点接受控制。对于适当缩放的Lee-Sidford度量，这在χ²散度中产生Õ(d²)的混合界，改进了之前的Õ(d^{9/4})界。

原文摘要

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 accep...

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