Modelling the influence of data structure on learning in neural networks: the hidden manifold model
Résumé
The lack of crisp mathematical models that capture the structure of real-world data sets is a major obstacle to the detailed theoretical understanding of deep neural networks. Here, we introduce a generative model for data sets that we call the hidden manifold model (HMM). The idea is to have high-dimensional inputs 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 first demonstrate the effect of structured data sets by experimentally comparing the dynamics and the performance of two-layer neural networks trained on three different data sets: (i) an unstructured synthetic data set containing random i.i.d. inputs, (ii) a structured data set drawn from the HMM and (iii) a simple canonical data set containing MNIST images. We pinpoint two phenomena related to the dynamics of the networks and their ability to generalise that only appear when training on structured data sets, and we experimentally demonstrate that training networks on data sets drawn from the HMM reproduces both the phenomena seen during training on real dataset. Our main theoretical result is that we show that the learning dynamics in the hidden manifold model is amenable to an analytical treatment by proving a "Gaussian Equivalence Theorem", opening the way to further detailed theoretical studies. In particular, we show how the dynamics of stochastic gradient descent for a two-layer network is captured by a set of ordinary differential equations that track the generalisation error at all times.
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