Rates of Convergence for Sparse Variational Gaussian Process Regression


Excellent variational approximations to Gaussian process posteriors have been developed which avoid the $mathcalOłeft(N^3i̊ght)$ scaling with dataset size $N$. They reduce the computational cost to $mathcalOłeft(NM^2g̊ht)$, with $Młl N$ the number of inducing variables, which summarise the process. While the computational cost seems to be linear in $N$, the true complexity of the algorithm depends on how $M$ must increase to ensure a certain quality of approximation. We show that with high probability the KL divergence can be made arbitrarily small by growing $M$ more slowly than $N$. A particular case is that for regression with normally distributed inputs in D-dimensions with the Squared Exponential kernel, $M=mathcalO(łog^D N)$ suffices. Our results show that as datasets grow, Gaussian process posteriors can be approximated cheaply, and provide a concrete rule for how to increase $M$ in continual learning scenarios.

Proceedings of the 36th International Conference on Machine Learning (ICML)