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.

The work to be presented in this talk proposes a novel framework seamlessly providing key properties of both neural nets (learning) and symbolic logic (knowledge and reasoning). Every neuron has a meaning as a component of a formula in a weighted real-valued logic, yielding a highly interpretable disentangled representation. Inference is omnidirectional rather than focused on predefined target variables, and corresponds to logical reasoning, including classical first-order logic theorem proving as a special case. The model is end-to-end differentiable, and learning minimizes a novel loss function capturing logical contradiction, yielding resilience to inconsistent knowledge. It also enables the open-world assumption by maintaining bounds on truth values which can have probabilistic semantics, yielding resilience to incomplete knowledge.