Deep Variational Implicit Processes

Implicit processes (IPs) are a generalization of Gaussian processes (GPs). IPs may lack a closed-form expression but are easy to sample from. Examples include, among others, Bayesian neural networks or neural samplers. IPs can be used as priors over functions, resulting in flexible models with well-calibrated prediction uncertainty estimates. Methods based on IPs usually carry out function-space approximate inference, which overcomes some of the difficulties of parameterspace approximate inference. Nevertheless, the approximations employed often limit the expressiveness of the final model, resulting, e.g., in a Gaussian predictive distribution, which can be restrictive. We propose here a multi-layer generalization of IPs called the Deep Variational Implicit process (DVIP). This generalization is similar to that of deep GPs over GPs, but it is more flexible due to the use of IPs as the prior distribution over the latent functions. We describe a scalable variational inference algorithm for training DVIP and show that it outperforms previous IPbased methods and also deep GPs. We support these claims via extensive regression and classification experiments. We also evaluate DVIP on large datasets with up to several million data instances to illustrate its good scalability and performance. The Bayesian approach has become popular for capturing the uncertainty associated to the predictions made by models that otherwise provide point-wise estimates, such as neural networks (NNs) (Gelman et al., 2013; Gal, 2016; Murphy, 2012). However, when carrying out Bayesian inference, obtaining the posterior distribution in the space of parameters can become a limiting factor since it is often intractable. Symmetries and strong dependencies between parameters make the approximate inference problem much more complex. This is precisely the case in large deep NNs. Nevertheless, all these issues can be alleviated by carrying out approximate inference in the space of functions, which presents certain advantages due to the simplified problem.