AF: Small: New Techniques for Optimal Bounds on MCMC Algorithms

University Of California-Santa Barbara, Santa Barbara CA

Investigators

Abstract

Markov Chain Monte Carlo (MCMC) algorithms are a widely used tool in a variety of scientific fields for sampling problems. Understanding the convergence rate of Markov chains to their equilibrium distribution is crucial for the efficiency and accuracy of scientific studies utilizing MCMC algorithms. This project focuses on the design and analysis of fast algorithms for randomly sampling from distributions defined on exponentially large combinatorial sets. This is a fundamental task that arises in a variety of scientific fields; some common examples include: Bayesian inference which is a key tool in machine learning, computer vision, and evolutionary biology; the study of the equilibrium state of idealized physical systems in statistical physics; and the design of algorithms for counting and sampling problems in theoretical computer science. The education plan of this project includes an interdisciplinary summer school at the University of California Santa Barbara to train graduate students on recent developments in the research area. This project will introduce new techniques for proving optimal convergence rates of Markov chains. The focus is an exciting new technique known as spectral independence, which measures the pairwise influences in graphical models or spin systems, and implies optimal mixing time bounds for a variety of Markov chains. This project will enhance the technique by strengthening the implications of spectral independence and extend its applicability by presenting new tools for establishing spectral independence. These improved techniques will yield new connections between various algorithmic approaches for approximate sampling and counting problems. In addition, this project will formalize connections between the computational complexity of approximate counting problems on general graphs with statistical physics phase transitions on trees. 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.