Understanding Variational Inference in Function-Space
This work addresses fundamental theoretical limitations in applying KL divergence for function-space variational inference, which is crucial for researchers developing new inference methods for Bayesian models, especially Bayesian neural networks.
This paper investigates the use of Kullback-Leibler divergence for directly approximating function-space posterior distributions in Bayesian models, particularly for Bayesian neural networks. It demonstrates that minimizing KL divergence between a wide class of parametric distributions and a Gaussian process prior leads to an ill-defined objective function. The authors propose featurized Bayesian linear regression as a benchmark to measure approximation quality.
Recent work has attempted to directly approximate the `function-space' or predictive posterior distribution of Bayesian models, without approximating the posterior distribution over the parameters. This is appealing in e.g. Bayesian neural networks, where we only need the former, and the latter is hard to represent. In this work, we highlight some advantages and limitations of employing the Kullback-Leibler divergence in this setting. For example, we show that minimizing the KL divergence between a wide class of parametric distributions and the posterior induced by a (non-degenerate) Gaussian process prior leads to an ill-defined objective function. Then, we propose (featurized) Bayesian linear regression as a benchmark for `function-space' inference methods that directly measures approximation quality. We apply this methodology to assess aspects of the objective function and inference scheme considered in Sun, Zhang, Shi, and Grosse (2018), emphasizing the quality of approximation to Bayesian inference as opposed to predictive performance.