# On two proofs of d2 mixing of weighted Dikin walks

arXiv: https://arxiv.org/abs/2608.28566 · HTML: https://mlsanity.com/paper/2608.28566/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 12 of its 34 references that are in the map.
- First posted on arXiv 28 Aug 2026.

## Builds on (most cited first, 12 of 12)

- [In-and-Out: Algorithmic Diffusion for Sampling Convex Bodies](https://mlsanity.com/paper/2405.01425/) (2024)
- [Fast MCMC sampling algorithms on polytopes](https://mlsanity.com/paper/1710.08165/) (2017)
- [When does Metropolized Hamiltonian Monte Carlo provably outperform Metropolis-adjusted Langevin algorithm?](https://mlsanity.com/paper/2304.04724/) (2023)
- [Sampling and Integration of Logconcave Functions by Algorithmic Diffusion](https://mlsanity.com/paper/2411.13462/) (2024)
- [Strong Self-Concordance and Sampling](https://mlsanity.com/paper/1911.05656/) (2019)
- [Reducing Isotropy and Volume to KLS: Faster Rounding and Volume Algorithms](https://mlsanity.com/paper/2008.02146/) (2020)
- [The Mixing Time of the Dikin Walk in a Polytope - A Simple Proof](https://mlsanity.com/paper/1508.01977/) (2015)
- [Zeroth-order Logconcave Sampling](https://mlsanity.com/paper/2507.18021/) (2025)
- [A unified complexity bound for logconcave sampling](https://mlsanity.com/paper/2606.12694/) (2026)
- [Regularized Dikin Walks for Sampling Truncated Logconcave Measures, Mixed Isoperimetry and Beyond Worst-Case Analysis](https://mlsanity.com/paper/2412.11303/) (2024)
- [Beyond the d2.5-mixing bound for Dikin walks on polytopes](https://mlsanity.com/paper/2607.13943/) (2026)
- [Spectral Gaps of Hit-and-Run and Coordinate Hit-and-Run](https://mlsanity.com/paper/2608.16878/) (2026)

## Abstract (arXiv)

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, barnu-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(d2) mixing bound for sampling from polytopes, while applying it to a hybrid barrier yields an widetilde O(d4) mixing bound for sampling from truncated PSD cones. Our second result establishes stronger chi2-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(d2) mixing bound in chi2-divergence, improving on the previous widetilde O(d9/4) bound.