Deep kernel learning (DKL) leverages the connection between Gaussian process (GP) and neural networks (NN) to build an end-to-end, hybrid model. It combines the capability of NN to learn rich representations under massive data and the non-parametric property of GP to achieve automatic regularization that incorporates a trade-off between model fit and model complexity. However, the deterministic encoder may weaken the model regularization of the following GP part, especially on small datasets, due to the free latent representation. We therefore present a complete deep latent-variable kernel learning (DLVKL) model wherein the latent variables perform stochastic encoding for regularized representation. We further enhance the DLVKL from two aspects: (i) the expressive variational posterior through neural stochastic differential equation (NSDE) to improve the approximation quality, and (ii) the hybrid prior taking knowledge from both the SDE prior and the posterior to arrive at a flexible trade-off. Intensive experiments imply that the DLVKL-NSDE performs similarly to the well calibrated GP on small datasets, and outperforms existing deep GPs on large datasets.
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