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We study the theoretical properties of a variational Bayes method in theGaussian Process regression model. We consider the inducing variables methodintroduced by Titsias (2009a) and derive sufficient conditions for obtainingcontraction rates for the corresponding variational Bayes (VB) posterior. Asexamples we show that for three particular covariance kernels (Mat\'ern,squared exponential, random series prior) the VB approach can achieve optimal,minimax contraction rates for a sufficiently large number of appropriatelychosen inducing variables. The theoretical findings are demonstrated bynumerical experiments.

Contraction rates for sparse variational approximations in Gaussian process regression