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We consider the optimal approximate posterior over the top-layer weights in aBayesian neural network for regression, and show that it exhibits strongdependencies on the lower-layer weights. We adapt this result to develop acorrelated approximate posterior over the weights at all layers in a Bayesianneural network. We extend this approach to deep Gaussian processes, unifyinginference in the two model classes. Our approximate posterior uses learned"global" inducing points, which are defined only at the input layer andpropagated through the network to obtain inducing inputs at subsequent layers.By contrast, standard, "local", inducing point methods from the deep Gaussianprocess literature optimise a separate set of inducing inputs at every layer,and thus do not model correlations across layers. Our method givesstate-of-the-art performance for a variational Bayesian method, without dataaugmentation or tempering, on CIFAR-10 of 86.7%, which is comparable to SGMCMCwithout tempering but with data augmentation (88% in Wenzel et al. 2020).

Global inducing point variational posteriors for Bayesian neural networks and deep Gaussian processes