Generalization in Deep Learning

This paper provides theoretical insights into why and how deep learning can generalize well, despite its large capacity, complexity, possible algorithmic instability, nonrobustness, and sharp minima, responding to an open question in the literature. We also discuss approaches to provide non-vacuous generalization guarantees for deep learning. Based on theoretical observations, we propose new open problems and discuss the limitations of our results.Comment: To appear in Mathematics of Deep Learning, Cambridge University Press. All previous results remain unchange

Bayesian Dropout

Dropout has recently emerged as a powerful and simple method for training neural networks preventing co-adaptation by stochastically omitting neurons. Dropout is currently not grounded in explicit modelling assumptions which so far has precluded its adoption in Bayesian modelling. Using Bayesian entropic reasoning we show that dropout can be interpreted as optimal inference under constraints. We demonstrate this on an analytically tractable regression model providing a Bayesian interpretation of its mechanism for regularizing and preventing co-adaptation as well as its connection to other Bayesian techniques. We also discuss two general approximate techniques for applying Bayesian dropout for general models, one based on an analytical approximation and the other on stochastic variational techniques. These techniques are then applied to a Baysian logistic regression problem and are shown to improve performance as the model become more misspecified. Our framework roots dropout as a theoretically justified and practical tool for statistical modelling allowing Bayesians to tap into the benefits of dropout training.Comment: 21 pages, 3 figures. Manuscript prepared 2014 and awaiting submissio