Strengthening nonparametric Bayesian methods with structured kernels

Abstract

This thesis covers an assortment of topics at the intersection of Bayesian nonparametrics and kernel machines: that is, to propose more efficient, kernel-based solutions to nonparametric Bayesian machine learning tasks. In chronological order, we provide summaries for the 4 publications on 3 interconnected topics: (i) expressive and nonstationary covariance kernels for Gaussian processes (GPs); (ii) scalable approximate inference of GP models via pseudo-inputs; (iii) Bayesian sampling of un-normalized target distributions via the simulation of interacting particle systems (IPSs). GPs are flexible priors on functions, which inform the hypothesis spaces of infinitely wide neural networks. However, to fully exploit their tractable uncertainty measures, careful selection of flexible covariance kernels is required for pattern discovery. Highly parametrized, stationary kernels have been proposed for handling extrapolations in GPs, but the translation invariance implied by stationarity caps their expressiveness. We propose nonstationary generalizations of such expressiveness kernels, both in parametric and nonparametric forms, and explore the implications of those kernels with respect to their spectral properties. Another restrictive aspect of GP models lies upon the cumbersome cubic scaling in their inference. We can draw upon a smaller set of pseudo-inputs, or inducing points, to obtain a sparse and more scalable approximate posteriors. Myriad studies of sparse GPs have established a separation of model parameters, which can either be optimized or inferred, and variational parameters which only requires optimization and no priors. The inducing point locations, however, exist somewhat outside this dichotomy, but the common practice is to simply find point estimates via optimization. We demonstrate that a fully Bayesian treatment of inducing inputs is equally valid in sparse GPs, and leads to a more flexible inferential framework with measurable practical benefits. Lastly, we turn to the sampling of un-normalized densities, a ubiquitous task in Bayesian inference. Apart from Markov Chain Monte Carlo (MCMC) sampling, we can also draw samples by deterministically transporting a set of interacting particles, i.e., the simulation of IPSs. Despite their ostensible differences in mechanism, a duality exists between the subtypes of the two sampling regimes, namely Langevin diffusion (LD) and Stein variational gradient descent (SVGD), where SVGD can be seen as a "kernelized" counterpart of LD. We demonstrate that kernelized, deterministic approximations exist for all diffusion-based MCMCs, which we denote as MCMC dynamics. Drawing upon this extended duality, we obtain deterministic samplers that emulate the behavior of other MCMC diffusion processes.