Convergence of Sparse Variational Inference in Gaussian Processes Regression

• URL: http://arxiv.org/abs/2008.00323v1
• Date: Sat, 1 Aug 2020 19:23:34 GMT
• Title: Convergence of Sparse Variational Inference in Gaussian Processes Regression
• Authors: David R. Burt and Carl Edward Rasmussen and Mark van der Wilk
• Abstract summary: We show that a method with an overall computational cost of $mathcalO(log N)2D(loglog N)2)$ can be used to perform inference.
• Score: 29.636483122130027
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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 $\mathcal{O}(NM^2)$. 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 $\mathcal{O}(N(\log N)^{2D}(\log\log N)^2)$ can be used to perform inference.

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