Rates of Convergence for Sparse Variational Gaussian Process Regression

Abstract

Excellent variational approximations to Gaussian process posteriors have been developed which avoid the O\left(N^3\right) scaling with dataset size N. They reduce the computational cost to O\left(NM^2\right), with M\ll 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=O(\log^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.