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

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, $\bar{\nu}$-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 $\chi^2$-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 $\chi^2$-divergence, improving on the previous $\widetilde O(d^{9/4})$ bound.

36 pages. AI disclosure included

Data Structures and Algorithms (cs.DS); Machine Learning (cs.LG); Optimization and Control (math.OC); Probability (math.PR); Computation (stat.CO)

arXiv:2608.28566 [cs.DS]

(or arXiv:2608.28566v1 [cs.DS] for this version)

https://doi.org/10.48550/arXiv.2608.28566 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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