New Bounds for Sparse Variational Gaussian Processes

TL;DR

This paper introduces a new, tighter variational bound for sparse Gaussian processes that improves hyperparameter learning and predictive accuracy, with minimal changes to existing implementations.

Contribution

It relaxes the traditional variational assumption in sparse GPs by optimizing a more general distribution, resulting in a tighter bound and better performance.

Findings

Tighter evidence lower bound improves hyperparameter estimation.

Enhanced predictive performance demonstrated on multiple datasets.

Method extends to non-Gaussian likelihoods.

Abstract

Sparse variational Gaussian processes (GPs) construct tractable posterior approximations to GP models. At the core of these methods is the assumption that the true posterior distribution over training function values ${\bf f}$ and inducing variables ${\bf u}$ is approximated by a variational distribution that incorporates the conditional GP prior $p({\bf f} | {\bf u})$ in its factorization. While this assumption is considered as fundamental, we show that for model training we can relax it through the use of a more general variational distribution $q({\bf f} | {\bf u})$ that depends on $N$ extra parameters, where $N$ is the number of training examples. In GP regression, we can analytically optimize the evidence lower bound over the extra parameters and express a tractable collapsed bound that is tighter than the previous bound. The new bound is also amenable to stochastic optimization and its implementation requires minor modifications to existing sparse GP code. Further, we also describe extensions to non-Gaussian likelihoods. On several datasets we demonstrate that our method can reduce bias when learning the hyperparameters and can lead to better predictive performance.