On Sparse variational methods and the Kullback-Leibler divergence between stochastic processes
This work addresses theoretical gaps in variational inference for Gaussian processes, which is important for researchers in machine learning and statistics, though it is incremental in nature.
The paper generalizes the connection between sparse variational methods and Kullback-Leibler divergence for Gaussian processes, providing new proofs and conditions for consistency, and applies this to interdomain and Cox process approximations.
The variational framework for learning inducing variables (Titsias, 2009a) has had a large impact on the Gaussian process literature. The framework may be interpreted as minimizing a rigorously defined Kullback-Leibler divergence between the approximating and posterior processes. To our knowledge this connection has thus far gone unremarked in the literature. In this paper we give a substantial generalization of the literature on this topic. We give a new proof of the result for infinite index sets which allows inducing points that are not data points and likelihoods that depend on all function values. We then discuss augmented index sets and show that, contrary to previous works, marginal consistency of augmentation is not enough to guarantee consistency of variational inference with the original model. We then characterize an extra condition where such a guarantee is obtainable. Finally we show how our framework sheds light on interdomain sparse approximations and sparse approximations for Cox processes.