Beyond the d2.5-mixing bound for Dikin walks on polytopes

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 ℝd with m linear inequalities mixes in md iterations. In 2017, Chen, Dwivedi, Wainwright, and Yu improved this to d2.5 using a Lewis-weight barrier, and conjectured that the correct mixing time should be d2. We make progress toward this conjecture by improving the previous d2.5-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 d2.25 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.