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While neural networks are used for classification tasks across domains, a long-standing open problem in machine learning is determining whether neural networks trained using standard procedures are consistent for classification, i.e., whether such models minimize the probability of misclassification for arbitrary data distributions. In this work, we identify and construct an explicit set of neural network classifiers that are consistent. Since effective neural networks in practice are typically both wide and deep, we analyze infinitely wide networks that are also infinitely deep. In particular, using the recent connection between infinitely wide neural networks and neural tangent kernels, we provide explicit activation functions that can be used to construct networks that achieve consistency. Interestingly, these activation functions are simple and easy to implement, yet differ from commonly used activations such as ReLU or sigmoid. More generally, we create a taxonomy of infinitely wide and deep networks and show that these models implement one of three well-known classifiers depending on the activation function used: 1) 1-nearest neighbor (model predictions are given by the label of the nearest training example); 2) majority vote (model predictions are given by the label of the class with the greatest representation in the training set); or 3) singular kernel classifiers (a set of classifiers containing those that achieve consistency). Our results highlight the benefit of using deep networks for classification tasks, in contrast to regression tasks, where excessive depth is harmful. 

Matrix completion problems arise in many applications including recommendation systems, computer vision, and genomics. Increasingly larger neural networks have been successful in many of these applications but at considerable computational costs. Remarkably, taking the width of a neural network to infinity allows for improved computational performance. In this work, we develop an infinite width neural network framework for matrix completion that is simple, fast, and flexible. Simplicity and speed come from the connection between the infinite width limit of neural networks and kernels known as neural tangent kernels (NTK). In particular, we derive the NTK for fully connected and convolutional neural networks for matrix completion. The flexibility stems from a feature prior, which allows encoding relationships between coordinates of the target matrix, akin to semisupervised learning. The effectiveness of our framework is demonstrated through competitive results for virtual drug screening and image inpainting/reconstruction. We also provide an implementation in Python to make our framework accessible on standard hardware to a broad audience. 

In the past decade the mathematical theory of machine learning has lagged far behind the triumphs of deep neural networks on practical challenges. However, the gap between theory and practice is gradually starting to close. In this paper I will attempt to assemble some pieces of the remarkable and still incomplete mathematical mosaic emerging from the efforts to understand the foundations of deep learning. The two key themes will be interpolation and its sibling over-parametrization. Interpolation corresponds to fitting data, even noisy data, exactly. Over-parametrization enables interpolation and provides flexibility to select a suitable interpolating model. As we will see, just as a physical prism separates colours mixed within a ray of light, the figurative prism of interpolation helps to disentangle generalization and optimization properties within the complex picture of modern machine learning. This article is written in the belief and hope that clearer understanding of these issues will bring us a step closer towards a general theory of deep learning and machine learning.