arXivDaily arXiv每日学术速递 周一至周五更新
arXiv 2202.11592cs.LGstat.ML
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超越等周性的鲁棒性定律

A Law of Robustness beyond Isoperimetry

University of Maryland at College Park(马里兰大学帕克分校);University of Waterloo(滑铁卢大学)

Yihan Wu, Heng Huang, Hongyang Zhang

AI总结:

本文研究任意数据分布下的鲁棒插值问题,证明了参数化插值神经网络的Lipschitz性下界,验证并扩展了鲁棒性定律猜想,揭示了过度参数化的潜在益处并否定了指数级样本下常数Lipschitz鲁棒插值函数的存在性。

摘要:

我们研究支撑在有界空间上的任意数据分布的鲁棒插值问题,并提出一个双重鲁棒性定律。鲁棒插值是指通过一个Lipschitz函数在$\mathbb{R}^d$中插值$n$个含噪训练数据点的问题。尽管当样本来自等周分布时该问题已被充分理解,但在一般分布甚至最坏情况分布下其表现仍有很多未知之处。我们证明了在任意数据分布上,具有$p$个参数的插值神经网络的Lipschitz性下界为$Ω(\sqrt{n/p})$。利用这一结果,我们验证了Bubeck、Li和Nagaraj先前工作中关于具有多项式权重的两层神经网络的鲁棒性定律猜想。随后我们将结果扩展到任意插值逼近器,并证明了鲁棒插值的Lipschitz性下界为$Ω(n^{1/d})$。我们的结果展示了一个双重鲁棒性定律:i) 我们证明了当$n=\mathrm{poly}(d)$时,过度参数化对光滑数据插值具有潜在益处;ii) 我们否定了当$n=\exp(ω(d))$时存在$O(1)$-Lipschitz鲁棒插值函数的可能性。

英文摘要:

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 $\mathbb{R}^d$ 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 $Ω(\sqrt{n/p})$ 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 $Ω(n^{1/d})$ for robust interpolation. Our results demonstrate a two-fold law of robustness: i) we show the potential benefit of overparametrization for smooth data interpolation when $n=\mathrm{poly}(d)$, and ii) we disprove the potential existence of an $O(1)$-Lipschitz robust interpolating function when $n=\exp(ω(d))$.

机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。

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