Modelling the influence of data structure on learning in neural networks: the hidden manifold model

Understanding the reasons for the success of deep neural networks trained using stochastic gradient-based methods is a key open problem for the nascent theory of deep learning. The types of data where these networks are most successful, such as images or sequences of speech, are characterised by intricate correlations. Yet, most theoretical work on neural networks does not explicitly model training data, or assumes that elements of each data sample are drawn independently from some factorised probability distribution. These approaches are thus by construction blind to the correlation structure of real-world data sets and their impact on learning in neural networks. Here, we introduce a generative model for structured data sets that we call the hidden manifold model (HMM). The idea is to construct high-dimensional inputs that lie on a lower-dimensional manifold, with labels that depend only on their position within this manifold, akin to a single layer decoder or generator in a generative adversarial network. We demonstrate that learning of the hidden manifold model is amenable to an analytical treatment by proving a "Gaussian Equivalence Property" (GEP), and we use the GEP to show how the dynamics of two-layer neural networks trained using one-pass stochastic gradient descent is captured by a set of integro-differential equations that track the performance of the network at all times. This permits us to analyse in detail how a neural network learns functions of increasing complexity during training, how its performance depends on its size and how it is impacted by parameters such as the learning rate or the dimension of the hidden manifold.