超越多面体上狄金游走的\(d^{2.5}\)混合界
Beyond the $d^{2.5}$-mixing bound for Dikin walks on polytopes
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中文总结 AI 辅助
研究多面体上狄金游走的混合时间，通过改进\(d^{2.5}\)混合界取得进展，证明带缩放Lee - Sidford度量的狄金游走从热启动在\(d^{2.25}\)次迭代中混合，改进冷启动复杂度，还发展了高阶分析方法。
中文摘要 AI 辅助
受结构化凸优化内点法（IPM）启发，2009年Kannan和Narayanan引入狄金游走用于从多面体均匀采样。狄金游走仿射不变，收敛受局部提议的障碍几何控制。2017年Chen等人用Lewis权重障碍将混合时间改进到\(d^{2.5}\)，并猜想应为\(d^{2}\)。本文通过改进\(d^{2.5}\)混合界取得进展。对多面体上指数采样，证明带缩放Lee - Sidford度量的狄金游走从热启动在\(d^{2.25}\)次迭代中混合，还通过退火框架改进冷启动复杂度。主要技术要素是改进Lee - Sidford度量平均自和谐性，使Metropolis滤波器沿随机狄金提议有高接受概率。此前分析限于二阶控制，本文发展了有原则的高阶分析，证明结合递归瓶颈项的选择性高阶展开、Lewis权重高阶导数的移动正交框架演算和通过多重随机积分的维纳混沌分解来控制高斯多项式。
英文摘要
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.
发表机构
- Georgia Tech（佐治亚理工学院）
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