Efficient Sampling Using Markov Chain Monte Carlo Methods

Abstract

This thesis explores the challenge of efficient sampling from target distributions, a problem at the heart of statistics, machine learning, and theoretical computer science with applications in Bayesian estimation, volume computation, and bandit optimization. The focus is on designing optimal samplers using the Markov Chain Monte Carlo (MCMC) method, leveraging a gradient or value oracle for a given smooth function, aiming to match the output distribution closely to the target distribution without direct access to the density function or its normalization constant. The research addresses the inefficiencies of current algorithms, especially in high-dimensional, ill-conditioned, structured, or constrained distributions, and introduces novel sampling algorithms that optimize query complexity. The thesis is structured into four main parts: The first part presents an improved discretization method for simulating stochastic differential equations like Langevin Diffusion, significantly enhancing sampling efficiency. The second part examines Metropolized Sampling Algorithms, offering new insights into their query complexity and establishing upper and lower bounds for widely used algorithms like Metropolized Hamiltonian Monte Carlo and Metropolis-adjusted Langevin Dynamics. The third part introduces a proximal sampler, improving condition number dependence and presenting efficient algorithms for various structured log-concave families. Finally, the fourth part tackles the challenging task of sampling from constrained sets, overcoming the difficulties posed by maintaining the random walk within the constraints and slow convergence rates in ill-conditioned sets. This work demonstrates theoretical advancements and practical efficiency in sampling algorithms. It contributes to understanding the fundamental aspects of sampling, the intricacies of discretization errors, and the impact of condition numbers on sampling complexity. This work improves over existing sampling methods, particularly in sampling from high-dimensional, constrained, ill-conditioned, and structured distributions.