Neuronal network computation and computation by avalanche supporting networks are of interest to the fields of physics, computer science (computation theory as well as statistical or machine learning) and neuroscience. Here we show that computation of complex Boolean functions arises spontaneously in threshold networks as a function of connectivity and antagonism (inhibition), computed by logic automata (motifs) in the form of computational cascades. We explain the emergent inverse relationship between the computational complexity of the motifs and their rank-ordering by function probabilities due to motifs, and its relationship to symmetry in function space. We also show that the optimal fraction of inhibition observed here supports results in computational neuroscience, relating to optimal information processing.

Tali's work emphasized the tradeoff between compression and information preservation. In this talk I will explore this theme in the context of deep learning. Artificial neural networks have recently revolutionized the field of machine learning. However, we still do not have sufficient theoretical understanding of how such models can be successfully learned. Two specific questions in this context are: how can neural nets be learned despite the non-convexity of the learning problem, and how can they generalize well despite often having more parameters than training data. I will describe our recent work showing that gradient-descent optimization indeed leads to 'simpler' models, where simplicity is captured by lower weight norm and in some cases clustering of weight vectors. We demonstrate this for several teacher and student architectures, including learning linear teachers with ReLU networks, learning boolean functions and learning convolutional pattern detection architectures.