关于加权Dikin游走的$d^2$混合的两种证明
On two proofs of $d^2$ mixing of weighted Dikin walks
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中文总结 AI 辅助
该研究针对加权Dikin游走的混合时间问题，提出两种证明方法：一是通过高概率区域接受概率控制得到多面体等场景的混合界，二是借助四阶自举条件将Lee--Sidford度量下的$\u03c7^2$散度混合界提升至$\tilde O(d^2)$。
中文摘要 AI 辅助
我们研究用于从多面体上的指数分布和截断半正定（PSD）锥采样的加权Dikin游走的混合时间。第一项结果给出了在强自和谐、$\barν$-对称性和局部度量的混合迹正则性条件下的通用总变差混合界。核心思路是在高概率区域而非每个点上控制Metropolis--Hastings接受概率。将该框架应用于Lee--Sidford度量、Lewis权重度量和John度量，得到多面体采样的$\tilde O(d^2)$混合界；将其应用于混合障碍函数，得到截断PSD锥采样的$\tilde O(d^4)$混合界。第二项结果利用新的四阶自举条件，建立了更强的$\u03c7^2$散度保证和逐点接受控制。对于适当缩放的Lee--Sidford度量，这得到了$\u03c7^2$散度下的$\tilde O(d^2)$混合界，改进了此前的$\tilde O(d^{9/4})$界。
英文摘要
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 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.
