Yunbum Kook
Yunbum Kook — researcher or builder tracked in the Angestrom contributor network.
Papers · 2
Beyond the $d^{2.5}$-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 $\mathbb{R}^{d}$ 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
Spectral Gaps of Hit-and-Run and Coordinate Hit-and-Run
For any convex body $\mathcal{K}\subset\mathbb{R}^{n}$ containing a unit ball, the spectral gap of Hit-and-Run is $Ω(1/(n^2 C_{\mathsf{PI}}))$, where $C_{\mathsf{PI}}$ is the Poincaré constant of the uniform distribution $π$ over $\mathcal{K}$. This implies that Hit-and-Run converges to a distribution within $χ^2$-divergence $\varepsilon$ of the uniform distribution $π$ in $O(n^2 C_{\mathsf{PI}}\log(M/\varepsilon))$ steps from any starting distribution $π_0$ with $M=χ^2(π_{0}\,\|\,π)$, thus refining the known bound of $O(n^2 R^2 \log(M/\varepsilon))$ by Lovász and Vempala (2004) in terms of th