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

arXiv:2608.28566v1 Announce Type: cross 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, ar{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 hi^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 hi^2-divergence, improving on the previous widetilde O(d^{9/4}) bound.

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Source: arXiv cs.LG | 2026-08-31