Abstract

Motivated by computational challenges in Bayesian models with indicator variables, such as probit/tobit regression, we study the computational complexity of drawing samples from a truncated log-concave measure. We discuss two problems. In the first part, using stochastic localization as a way to reduce the sampling problem to truncated Gaussians, we analyze the hit-and-run algorithm for sampling uniformly from an isotropic convex body in n dimensions and establish $n^{2$ mixing time. In the second part, building on interior point methods, we analyze the mixing time of regularized Dikin walks for sampling log-concave measures truncated on a polytope. For a logconcave and log-smooth distribution with condition number $\kappa$, truncated on a polytope in $R}n$ defined with $m$ linear constraints, we prove that the soft-threshold Dikin walk mixes in $O((m+\kappa)n)$ 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. Here, stochastic localization allows us to extend the analysis to weakly log-concave measures. https://arxiv.org/abs/2212.00297 https://arxiv.org/abs/2412.11303