I will present a novel comprehensive theory of large scale learning with Deep Neural Networks, based on the correspondence between Deep Learning and the Information Bottleneck framework. The new theory has the following components: (1) rethinking Learning theory; I will prove a new generalization bound, the input-compression bound, which shows that compression of the representation of input variable is far more important for good generalization than the dimension of the network hypothesis class, an ill defined notion for deep learning. (2) I will prove that for large scale Deep Neural Networks the mutual information on the input and the output variables, for the last hidden layer, provide a complete characterization of the sample complexity and accuracy of the network. This makes the information Bottleneck bound for the problem as the optimal trade-off between sample complexity and accuracy with ANY learning algorithm. (3) I will show how Stochastic Gradient Descent, as used in Deep Learning, achieves this optimal bound. In that sense, Deep Learning is a method for solving the Information Bottleneck problem for large scale supervised learning problems. The theory provide a new computational understating of the benefit of the hidden layers, and gives concrete predictions for the structure of the layers of Deep Neural Networks and their design principles. These turn out to depend solely on the joint distribution of the input and output and on the sample size.

Based partly on joint works with Ravid Shwartz-Ziv, Noga Zaslavsky, and Amichai Painsky.