Random Features Model with General Convex Regularization: A Fine Grained Analysis with Precise Asymptotic Learning Curves
This work addresses the challenge of theoretical analysis for random feature models with correlated data, offering a more tractable framework for researchers in machine learning theory.
The paper tackles the problem of analyzing learning curves for random feature models with convex regularization, providing precise asymptotic expressions that avoid intractable proximal operators and extending universality results to L1 regularization. The result is a computable 4-dimensional scalar optimization, validated numerically to be accurate even in non-asymptotic regimes.
We compute precise asymptotic expressions for the learning curves of least squares random feature (RF) models with either a separable strongly convex regularization or the $\ell_1$ regularization. We propose a novel multi-level application of the convex Gaussian min max theorem (CGMT) to overcome the traditional difficulty of finding computable expressions for random features models with correlated data. Our result takes the form of a computable 4-dimensional scalar optimization. In contrast to previous results, our approach does not require solving an often intractable proximal operator, which scales with the number of model parameters. Furthermore, we extend the universality results for the training and generalization errors for RF models to $\ell_1$ regularization. In particular, we demonstrate that under mild conditions, random feature models with elastic net or $\ell_1$ regularization are asymptotically equivalent to a surrogate Gaussian model with the same first and second moments. We numerically demonstrate the predictive capacity of our results, and show experimentally that the predicted test error is accurate even in the non-asymptotic regime.