Tighter Bounds on the Log Marginal Likelihood of Gaussian Process Regression Using Conjugate Gradients
This work addresses a computational bottleneck for researchers and practitioners using Gaussian process models, offering an incremental improvement by unifying variational and iterative methods.
The paper tackles the computational challenge of estimating the log marginal likelihood in Gaussian process regression by proposing a lower bound that avoids full kernel matrix factorisation, showing improved predictive performance with comparable training time to other conjugate gradient methods.
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.