Random Features Model with General Convex Regularization: A Fine Grained Analysis with Precise Asymptotic Learning Curves

David Bosch, Ashkan Panahi, Ayca Ozcelikkale, Devdatt Dubhashi
Proceedings of The 26th International Conference on Artificial Intelligence and Statistics, PMLR 206:11371-11414, 2023.

Abstract

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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v206-bosch23a, title = {Random Features Model with General Convex Regularization: A Fine Grained Analysis with Precise Asymptotic Learning Curves}, author = {Bosch, David and Panahi, Ashkan and Ozcelikkale, Ayca and Dubhashi, Devdatt}, booktitle = {Proceedings of The 26th International Conference on Artificial Intelligence and Statistics}, pages = {11371--11414}, year = {2023}, editor = {Ruiz, Francisco and Dy, Jennifer and van de Meent, Jan-Willem}, volume = {206}, series = {Proceedings of Machine Learning Research}, month = {25--27 Apr}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v206/bosch23a/bosch23a.pdf}, url = {https://proceedings.mlr.press/v206/bosch23a.html}, abstract = {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.} }
Endnote
%0 Conference Paper %T Random Features Model with General Convex Regularization: A Fine Grained Analysis with Precise Asymptotic Learning Curves %A David Bosch %A Ashkan Panahi %A Ayca Ozcelikkale %A Devdatt Dubhashi %B Proceedings of The 26th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2023 %E Francisco Ruiz %E Jennifer Dy %E Jan-Willem van de Meent %F pmlr-v206-bosch23a %I PMLR %P 11371--11414 %U https://proceedings.mlr.press/v206/bosch23a.html %V 206 %X 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.
APA
Bosch, D., Panahi, A., Ozcelikkale, A. & Dubhashi, D.. (2023). Random Features Model with General Convex Regularization: A Fine Grained Analysis with Precise Asymptotic Learning Curves. Proceedings of The 26th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 206:11371-11414 Available from https://proceedings.mlr.press/v206/bosch23a.html.

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