Complexity of sampling truncated log-concave measures, and the role of stochastic localization

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• Yuansi Chen (ETH Zurich)
• Friday 06 June 2025, 14:00-15:00
• MR12, Centre for Mathematical Sciences.

If you have a question about this talk, please contact Qingyuan Zhao.

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

This talk is part of the Statistics series.

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