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Convergence of Sparse Variational Inference in Gaussian Processes Regression

David R. Burt, Carl Edward Rasmussen, Mark van der Wilk; 21(131):1−63, 2020.


Gaussian processes are distributions over functions that are versatile and mathematically convenient priors in Bayesian modelling. However, their use is often impeded for data with large numbers of observations, $N$, due to the cubic (in $N$) cost of matrix operations used in exact inference. Many solutions have been proposed that rely on $M \ll N$ inducing variables to form an approximation at a cost of $\mathcal{O}\left(NM^2\right)$. While the computational cost appears linear in $N$, the true complexity depends on how $M$ must scale with $N$ to ensure a certain quality of the approximation. In this work, we investigate upper and lower bounds on how $M$ needs to grow with $N$ to ensure high quality approximations. We show that we can make the KL-divergence between the approximate model and the exact posterior arbitrarily small for a Gaussian-noise regression model with $M \ll N$. Specifically, for the popular squared exponential kernel and $D$-dimensional Gaussian distributed covariates, $M = \mathcal{O}((\log N)^D)$ suffice and a method with an overall computational cost of $\mathcal{O}\left(N(\log N)^{2D}(\log \log N)^2\right)$ can be used to perform inference.

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