A Law of Robustness beyond Isoperimetry

Yihan Wu, Heng Huang, Hongyang Zhang
Proceedings of the 40th International Conference on Machine Learning, PMLR 202:37439-37455, 2023.

Abstract

We study the robust interpolation problem of arbitrary data distributions supported on a bounded space and propose a two-fold law of robustness. Robust interpolation refers to the problem of interpolating $n$ noisy training data points in $R^d$ by a Lipschitz function. Although this problem has been well understood when the samples are drawn from an isoperimetry distribution, much remains unknown concerning its performance under generic or even the worst-case distributions. We prove a Lipschitzness lower bound $\Omega(\sqrt{n/p})$ of the interpolating neural network with $p$ parameters on arbitrary data distributions. With this result, we validate the law of robustness conjecture in prior work by Bubeck, Li and Nagaraj on two-layer neural networks with polynomial weights. We then extend our result to arbitrary interpolating approximators and prove a Lipschitzness lower bound $\Omega(n^{1/d})$ for robust interpolation. Our results demonstrate a two-fold law of robustness: a) we show the potential benefit of overparametrization for smooth data interpolation when $n=poly(d)$, and b) we disprove the potential existence of an $O(1)$-Lipschitz robust interpolating function when $n=\exp(\omega(d))$.

Cite this Paper

BibTeX
@InProceedings{pmlr-v202-wu23g, title = {A Law of Robustness beyond Isoperimetry}, author = {Wu, Yihan and Huang, Heng and Zhang, Hongyang}, booktitle = {Proceedings of the 40th International Conference on Machine Learning}, pages = {37439--37455}, year = {2023}, editor = {Krause, Andreas and Brunskill, Emma and Cho, Kyunghyun and Engelhardt, Barbara and Sabato, Sivan and Scarlett, Jonathan}, volume = {202}, series = {Proceedings of Machine Learning Research}, month = {23--29 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v202/wu23g/wu23g.pdf}, url = {https://proceedings.mlr.press/v202/wu23g.html}, abstract = {We study the robust interpolation problem of arbitrary data distributions supported on a bounded space and propose a two-fold law of robustness. Robust interpolation refers to the problem of interpolating $n$ noisy training data points in $R^d$ by a Lipschitz function. Although this problem has been well understood when the samples are drawn from an isoperimetry distribution, much remains unknown concerning its performance under generic or even the worst-case distributions. We prove a Lipschitzness lower bound $\Omega(\sqrt{n/p})$ of the interpolating neural network with $p$ parameters on arbitrary data distributions. With this result, we validate the law of robustness conjecture in prior work by Bubeck, Li and Nagaraj on two-layer neural networks with polynomial weights. We then extend our result to arbitrary interpolating approximators and prove a Lipschitzness lower bound $\Omega(n^{1/d})$ for robust interpolation. Our results demonstrate a two-fold law of robustness: a) we show the potential benefit of overparametrization for smooth data interpolation when $n=poly(d)$, and b) we disprove the potential existence of an $O(1)$-Lipschitz robust interpolating function when $n=\exp(\omega(d))$.} }
Endnote
%0 Conference Paper %T A Law of Robustness beyond Isoperimetry %A Yihan Wu %A Heng Huang %A Hongyang Zhang %B Proceedings of the 40th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2023 %E Andreas Krause %E Emma Brunskill %E Kyunghyun Cho %E Barbara Engelhardt %E Sivan Sabato %E Jonathan Scarlett %F pmlr-v202-wu23g %I PMLR %P 37439--37455 %U https://proceedings.mlr.press/v202/wu23g.html %V 202 %X We study the robust interpolation problem of arbitrary data distributions supported on a bounded space and propose a two-fold law of robustness. Robust interpolation refers to the problem of interpolating $n$ noisy training data points in $R^d$ by a Lipschitz function. Although this problem has been well understood when the samples are drawn from an isoperimetry distribution, much remains unknown concerning its performance under generic or even the worst-case distributions. We prove a Lipschitzness lower bound $\Omega(\sqrt{n/p})$ of the interpolating neural network with $p$ parameters on arbitrary data distributions. With this result, we validate the law of robustness conjecture in prior work by Bubeck, Li and Nagaraj on two-layer neural networks with polynomial weights. We then extend our result to arbitrary interpolating approximators and prove a Lipschitzness lower bound $\Omega(n^{1/d})$ for robust interpolation. Our results demonstrate a two-fold law of robustness: a) we show the potential benefit of overparametrization for smooth data interpolation when $n=poly(d)$, and b) we disprove the potential existence of an $O(1)$-Lipschitz robust interpolating function when $n=\exp(\omega(d))$.
APA
Wu, Y., Huang, H. & Zhang, H.. (2023). A Law of Robustness beyond Isoperimetry. Proceedings of the 40th International Conference on Machine Learning, in Proceedings of Machine Learning Research 202:37439-37455 Available from https://proceedings.mlr.press/v202/wu23g.html.

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