Tighter Bounds on the Log Marginal Likelihood of Gaussian Process Regression Using Conjugate Gradients

TL;DR

This paper introduces a new lower bound for the log marginal likelihood in Gaussian process regression that avoids full matrix factorization, enabling more efficient and less biased parameter learning.

Contribution

It presents a novel lower bound leveraging conjugate gradients, unifying variational and iterative methods for Gaussian process training.

Findings

Improved predictive performance over comparable conjugate gradient methods.

Reduced bias in parameter estimation.

Efficient computation without full kernel matrix factorization.

Abstract

We propose a lower bound on the log marginal likelihood of Gaussian process regression models that can be computed without matrix factorisation of the full kernel matrix. We show that approximate maximum likelihood learning of model parameters by maximising our lower bound retains many of the sparse variational approach benefits while reducing the bias introduced into parameter learning. The basis of our bound is a more careful analysis of the log-determinant term appearing in the log marginal likelihood, as well as using the method of conjugate gradients to derive tight lower bounds on the term involving a quadratic form. Our approach is a step forward in unifying methods relying on lower bound maximisation (e.g. variational methods) and iterative approaches based on conjugate gradients for training Gaussian processes. In experiments, we show improved predictive performance with our model for a comparable amount of training time compared to other conjugate gradient based approaches.