Modelling the influence of data structure on learning in neural networks
Authors: S. Goldt, M. Mézard, F. Krzakala, L. Zdeborová
Keywords: Neural Networks, Generative models, Synthetic data sets, Generalisation, Stochastic Gradient descent
TL;DR: We demonstrate how structure in data sets impacts neural networks and introduce a generative model for synthetic data sets that reproduces this impact.
Abstract: 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 first demonstrate the effect of structured data sets
by experimentally comparing the dynamics and the performance of two-layer
networks trained on two different data sets: (i) an unstructured synthetic data
set containing random i.i.d. inputs, and (ii) a simple canonical data set such
as MNIST images. Our analysis reveals two phenomena related to the dynamics of
the networks and their ability to generalise that only appear when training on
structured data sets. Second, we introduce a generative model for data sets,
where high-dimensional inputs lie on a lower-dimensional manifold and have
labels that depend only on their position within this manifold. We call it the
*hidden manifold model* and we experimentally demonstrate that training
networks on data sets drawn from this model reproduces both the phenomena seen
during training on MNIST.