Beyond the $d^{2.5}$-mixing bound for Dikin walks on polytopes

2026-07-15

AI Summary

This work addresses the challenge of accelerating the mixing time of the Dikin walk on polytopes, overcoming the longstanding bottleneck of $d^{2.5}$ iteration complexity. By introducing the Lee–Sidford barrier metric and, for the first time, incorporating higher-order derivative control into the analysis of the Dikin walk, the study transcends the limitations of conventional second-order methods. The approach integrates selective higher-order expansions, moving orthonormal frame calculus, Wiener chaos decomposition, and multiple stochastic integrals to improve the warm-start mixing bound to $d^{2.25}$. Furthermore, an annealing framework is employed to effectively mitigate cold-start difficulties. This advancement substantially enhances the convergence rate of exponential sampling over high-dimensional polytopes.

Abstract

Inspired by interior-point methods (IPM) for structured convex optimization, Kannan and Narayanan introduced the Dikin walk for sampling uniformly from polytopes in 2009. As in IPMs, the Dikin walk is affine-invariant, and its convergence is governed by the barrier geometry used to define its local proposal. They showed that the Dikin walk with the logarithmic barrier for a polytope in $\mathbb{R}^{d}$ with $m$ linear inequalities mixes in $md$ iterations. In 2017, Chen, Dwivedi, Wainwright, and Yu improved this to $d^{2.5}$ using a Lewis-weight barrier, and conjectured that the correct mixing time should be $d^{2}$. We make progress toward this conjecture by improving the previous $d^{2.5}$-mixing bound. For exponential sampling over a polytope, we prove that the Dikin walk with a scaled Lee--Sidford metric mixes from a warm start in $d^{2.25}$ iterations. This also yields an improved cold-start complexity via a known annealing framework. The main technical ingredient is improved average self-concordance of the Lee--Sidford metric, which gives high acceptance probability for the Metropolis filter along a random Dikin proposal. While previous analyses were effectively limited to second-order control due to technical difficulties, we develop a principled higher-order analysis. The proof combines a selective higher-order expansion of recursive bottleneck terms, a moving orthonormal-frame calculus for higher derivatives of the Lewis weights, and Wiener-chaos decompositions via multiple stochastic integrals to control the resulting Gaussian polynomials.

Problem

• Dikin walk
• mixing time
• polytopes
• self-concordance
• Lewis weights

Innovation

• Dikin walk
• Lee–Sidford metric
• higher-order analysis
• self-concordance
• mixing time