Towards a Unified Analysis of Random Fourier Features

Zhu Li, Jean-Francois Ton, Dino Oglic, Dino Sejdinovic
Proceedings of the 36th International Conference on Machine Learning, PMLR 97:3905-3914, 2019.

Abstract

Random Fourier features is a widely used, simple, and effective technique for scaling up kernel methods. The existing theoretical analysis of the approach, however, remains focused on specific learning tasks and typically gives pessimistic bounds which are at odds with the empirical results. We tackle these problems and provide the first unified risk analysis of learning with random Fourier features using the squared error and Lipschitz continuous loss functions. In our bounds, the trade-off between the computational cost and the expected risk convergence rate is problem specific and expressed in terms of the regularization parameter and the number of effective degrees of freedom. We study both the standard random Fourier features method for which we improve the existing bounds on the number of features required to guarantee the corresponding minimax risk convergence rate of kernel ridge regression, as well as a data-dependent modification which samples features proportional to ridge leverage scores and further reduces the required number of features. As ridge leverage scores are expensive to compute, we devise a simple approximation scheme which provably reduces the computational cost without loss of statistical efficiency.

Cite this Paper

BibTeX
@InProceedings{pmlr-v97-li19k, title = {Towards a Unified Analysis of Random {F}ourier Features}, author = {Li, Zhu and Ton, Jean-Francois and Oglic, Dino and Sejdinovic, Dino}, booktitle = {Proceedings of the 36th International Conference on Machine Learning}, pages = {3905--3914}, year = {2019}, editor = {Chaudhuri, Kamalika and Salakhutdinov, Ruslan}, volume = {97}, series = {Proceedings of Machine Learning Research}, month = {09--15 Jun}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v97/li19k/li19k.pdf}, url = {https://proceedings.mlr.press/v97/li19k.html}, abstract = {Random Fourier features is a widely used, simple, and effective technique for scaling up kernel methods. The existing theoretical analysis of the approach, however, remains focused on specific learning tasks and typically gives pessimistic bounds which are at odds with the empirical results. We tackle these problems and provide the first unified risk analysis of learning with random Fourier features using the squared error and Lipschitz continuous loss functions. In our bounds, the trade-off between the computational cost and the expected risk convergence rate is problem specific and expressed in terms of the regularization parameter and the number of effective degrees of freedom. We study both the standard random Fourier features method for which we improve the existing bounds on the number of features required to guarantee the corresponding minimax risk convergence rate of kernel ridge regression, as well as a data-dependent modification which samples features proportional to ridge leverage scores and further reduces the required number of features. As ridge leverage scores are expensive to compute, we devise a simple approximation scheme which provably reduces the computational cost without loss of statistical efficiency.} }
Endnote
%0 Conference Paper %T Towards a Unified Analysis of Random Fourier Features %A Zhu Li %A Jean-Francois Ton %A Dino Oglic %A Dino Sejdinovic %B Proceedings of the 36th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2019 %E Kamalika Chaudhuri %E Ruslan Salakhutdinov %F pmlr-v97-li19k %I PMLR %P 3905--3914 %U https://proceedings.mlr.press/v97/li19k.html %V 97 %X Random Fourier features is a widely used, simple, and effective technique for scaling up kernel methods. The existing theoretical analysis of the approach, however, remains focused on specific learning tasks and typically gives pessimistic bounds which are at odds with the empirical results. We tackle these problems and provide the first unified risk analysis of learning with random Fourier features using the squared error and Lipschitz continuous loss functions. In our bounds, the trade-off between the computational cost and the expected risk convergence rate is problem specific and expressed in terms of the regularization parameter and the number of effective degrees of freedom. We study both the standard random Fourier features method for which we improve the existing bounds on the number of features required to guarantee the corresponding minimax risk convergence rate of kernel ridge regression, as well as a data-dependent modification which samples features proportional to ridge leverage scores and further reduces the required number of features. As ridge leverage scores are expensive to compute, we devise a simple approximation scheme which provably reduces the computational cost without loss of statistical efficiency.
APA
Li, Z., Ton, J., Oglic, D. & Sejdinovic, D.. (2019). Towards a Unified Analysis of Random Fourier Features. Proceedings of the 36th International Conference on Machine Learning, in Proceedings of Machine Learning Research 97:3905-3914 Available from https://proceedings.mlr.press/v97/li19k.html.

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