Asymptotics of Learning with Deep Structured (Random) Features
This work provides theoretical insights for understanding learning in deep neural networks with structured randomness, which is incremental to existing asymptotic analyses.
The authors tackled the problem of characterizing the test error for learning with deep structured random features in high-dimensional settings, deriving a tight asymptotic expression in terms of feature covariance and applying it to Gaussian rainbow neural networks to provide closed-form formulas.
For a large class of feature maps we provide a tight asymptotic characterisation of the test error associated with learning the readout layer, in the high-dimensional limit where the input dimension, hidden layer widths, and number of training samples are proportionally large. This characterization is formulated in terms of the population covariance of the features. Our work is partially motivated by the problem of learning with Gaussian rainbow neural networks, namely deep non-linear fully-connected networks with random but structured weights, whose row-wise covariances are further allowed to depend on the weights of previous layers. For such networks we also derive a closed-form formula for the feature covariance in terms of the weight matrices. We further find that in some cases our results can capture feature maps learned by deep, finite-width neural networks trained under gradient descent.