Random features and polynomial rules
This work provides theoretical insights into neural network behavior near infinite-width limits, but it is incremental as it builds on existing statistical mechanics tools and extends prior scaling results.
The authors tackled the generalization performance of random features models in supervised learning with Gaussian data, mapping them to polynomial models and deriving average generalization curves as functions of the number of features and training set size, with results extending beyond proportional scaling and showing quantitative agreement with numerical experiments across many orders of magnitude.
Random features models play a distinguished role in the theory of deep learning, describing the behavior of neural networks close to their infinite-width limit. In this work, we present a thorough analysis of the generalization performance of random features models for generic supervised learning problems with Gaussian data. Our approach, built with tools from the statistical mechanics of disordered systems, maps the random features model to an equivalent polynomial model, and allows us to plot average generalization curves as functions of the two main control parameters of the problem: the number of random features $N$ and the size $P$ of the training set, both assumed to scale as powers in the input dimension $D$. Our results extend the case of proportional scaling between $N$, $P$ and $D$. They are in accordance with rigorous bounds known for certain particular learning tasks and are in quantitative agreement with numerical experiments performed over many order of magnitudes of $N$ and $P$. We find good agreement also far from the asymptotic limits where $D\to \infty$ and at least one between $P/D^K$, $N/D^L$ remains finite.