We study the problem of drawing samples from a logconcave distribution truncated on a polytope, motivated by computational challenges in Bayesian statistical models with indicator variables, such as probit regression. Building on interior point methods and the Dikin walk for sampling from uniform distributions, we analyze the mixing time of regularized Dikin walks. Our contributions are threefold. First, for a logconcave and log-smooth distribution with condition number $\kappa$, truncated on a polytope in $\mathbb{R}^n$ defined with $m$ linear constraints, we prove that the soft-threshold Dikin walk mixes in $\widetilde{O}((m+\kappa)n)$ iterations from a warm initialization. It improves upon prior work which required the polytope to be bounded and involved a bound dependent on the radius of the bounded region. Moreover, we introduce the regularized Dikin walk using Lewis weights for approximating the John ellipsoid. We show that it mixes in $\widetilde{O}((n^{2.5}+\kappa n)$. Second, we extend the mixing time guarantees mentioned above to weakly log-concave distributions truncated on polytopes, provided that they have a finite covariance matrix. Third, going beyond worst-case mixing time analysis, we demonstrate that soft-threshold Dikin walk can mix significantly faster when only a limited number of constraints intersect the high-probability mass of the distribution, improving the $\widetilde{O}((m+\kappa)n)$ upper bound to $\widetilde{O}(m + \kappa n)$. Additionally, per-iteration complexity of regularized Dikin walk and ways to generate a warm initialization are discussed to facilitate practical implementation.

翻译：本文研究从多面体截断的对数凹分布中抽取样本的问题，该问题源于含指示变量贝叶斯统计模型（如概率回归）中的计算挑战。基于内点方法及用于均匀分布采样的Dikin游走，我们分析了正则化Dikin游走的混合时间。我们的贡献有三方面：首先，对于条件数为$\kappa$的对数凹且对数平滑的分布，在由$m$个线性约束定义的$\mathbb{R}^n$空间多面体上截断的情形，我们证明软阈值Dikin游走在热启动条件下以$\widetilde{O}((m+\kappa)n)$次迭代混合。该结果改进了先前要求多面体有界且依赖有界区域半径的工作。此外，我们引入使用Lewis权重逼近John椭球的正则化Dikin游走，证明其混合时间为$\widetilde{O}((n^{2.5}+\kappa n)$。其次，我们将上述混合时间保证扩展到多面体截断的弱对数凹分布，前提是其具有有限协方差矩阵。第三，超越最坏情况混合时间分析，我们证明当仅有有限数量的约束与分布的高概率质量区域相交时，软阈值Dikin游走可显著加速混合，将$\widetilde{O}((m+\kappa)n)$上界改进为$\widetilde{O}(m + \kappa n)$。此外，本文讨论了正则化Dikin游走的单步计算复杂度及热启动生成方法，以促进实际应用。