ON SCALABLE AND FAST LANGEVIN-DYNAMICS-BASED SAMPLING ALGORITHMS
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Langevin dynamics-based sampling algorithms are arguably among the most widelyused Markov Chain Monte Carlo (MCMC) methods. Two main directions of the modern study of MCMC methods are (i) How to scale MCMC methods to big data applications, and (ii) Tight convergence analysis of MCMC algorithms, with explicit dependence on various characteristics of target distribution, in a non-asymptotic manner. This thesis continues the previous efforts in this two lines and consists of three parts. In the ﬁrst part, we study stochastic gradient MCMC methods for large scale application. We propose a non-uniform subsampling of gradients scheme to approximately match the transition kernel of a base MCMC base with full gradient, aiming for better sample quality. The demonstration is based on underdamped Langevin dynamics. In the second part, we consider an analog of Nesterov’s accelerated algorithm in optimization for sampling. We derive a dynamics termed Hessian-Free-High-Resolution (HFHR) dynamics, from a high-resolution ordinary differential equation description of the Nesterov’s accelerated algorithm. We then quantify the acceleration of HFHR over underdamped Langevin dynamics at both continuous dynamics level and discrete algorithm level. In the third part, we study a broad family of bounded, contractive-SDE-based sampling algorithms via mean-square analysis. We show how to extend the applicability of classical mean-square analysis from ﬁnite time to inﬁnite time. Iteration complexity in 2-Wasserstein distance is also characterized and when applied to Langevin Monte Carlo algorithm, we obtain an improved iteration complexity bound.