Rates of Convergence for Sparse Variational Gaussian Process Regression

Excellent variational approximations to Gaussian process posteriors have been developed which avoid the O (N-3) scaling with dataset size N. They reduce the computational cost to O (NM2), with M << 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 = (9(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.