CAREER: Towards Tight Guarantees of Markov Chain Sampling Algorithms in High Dimensional Statistical Inference

Duke University, Durham NC

Investigators

Abstract

Drawing samples from a distribution is a core computational challenge in fields such as Bayesian statistics, machine learning, statistical physics, and many other areas involving stochastic models. Among all methods, Markov Chain Monte Carlo (MCMC) algorithms stand out as the most widely used class of sampling algorithms with a broad range of applications, notably in high dimensional Bayesian inference. While MCMC algorithms have been proposed, studied, and implemented since the foundational work of Metropolis et al. in 1953, many convergence properties of algorithms used in practice are not well understood. Practitioners in Bayesian statistics are often faced with a series of key challenges to be addressed rigorously: the choice of algorithm hyper-parameters, the estimated computational cost and the choice of the best algorithm, etc. This project focuses on developing theoretical guarantees of MCMC sampling algorithms that arise in large-scale Bayesian statistical inference problems. The project will also offer numerous interdisciplinary research training, outreach and mentoring opportunities for the next generation of statisticians and data scientists at all levels, from undergraduate to doctoral students. This project will address three specific research problems centered around MCMC algorithms in high dimensional inference. First, the project intends to rigorously rank the efficiency of MCMC algorithms for sampling log-concave distributions and to provide succinct non-asymptotic mini-max analysis of mixing time. Log-concave distributions in sampling are as important as convex functions in optimization, and one cannot expect to build a foundational theory basis without determining the fundamental limits of sampling algorithms on log-concave distributions. Widely-used algorithms such as Hamiltonian Monte Carlo, Gibbs sampling and hit-and-run will be studied rigorously. Second, as concentration inequalities constitute an essential component in understanding the efficiency of MCMC sampling algorithms, the project will develop a fine-grained understanding of concentration of high dimensional log-concave distributions via new technical tools such as stochastic localization. Finally, the project will unify the existing theoretical tools for studying discrete-state and continuous-state sampling algorithms through localization schemes. The proposed research aims to advance the field with a comprehensive understanding of MCMC sampling algorithms and their optimal settings in both discrete and continuous cases. The project will provide a wide range of interdisciplinary initiatives to enhance professional development of undergraduate and graduate students in statistical sciences. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.