# Adversarial Robustness of Nonparametric Regression

arXiv: https://arxiv.org/abs/2505.17356 · HTML: https://mlsanity.com/paper/2505.17356/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 30 of its 66 references that are in the map.
- First posted on arXiv 23 May 2025.

## Builds on (most cited first, 30 of 30)

- [Communication-Efficient Learning of Deep Networks from Decentralized Data](https://mlsanity.com/paper/1602.05629/) (2016)
- [Explaining and Harnessing Adversarial Examples](https://mlsanity.com/paper/1412.6572/) (2014)
- [Towards Deep Learning Models Resistant to Adversarial Attacks](https://mlsanity.com/paper/1706.06083/) (2017)
- [Intriguing properties of neural networks](https://mlsanity.com/paper/1312.6199/) (2013)
- [Towards Evaluating the Robustness of Neural Networks](https://mlsanity.com/paper/1608.04644/) (2016)
- [Megatron-LM: Training Multi-Billion Parameter Language Models Using Model Parallelism](https://mlsanity.com/paper/1909.08053/) (2019)
- [Certified Adversarial Robustness via Randomized Smoothing](https://mlsanity.com/paper/1902.02918/) (2019)
- [Byzantine-Robust Distributed Learning: Towards Optimal Statistical Rates](https://mlsanity.com/paper/1803.01498/) (2018)
- [KAN: Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2404.19756/) (2024)
- [Recent Advances in Adversarial Training for Adversarial Robustness](https://mlsanity.com/paper/2102.01356/) (2021)
- [Thin-Plate Spline Motion Model for Image Animation](https://mlsanity.com/paper/2203.14367/) (2022)
- [Robust Estimators in High Dimensions without the Computational Intractability](https://mlsanity.com/paper/1604.06443/) (2016)
- [Federated Optimization:Distributed Optimization Beyond the Datacenter](https://mlsanity.com/paper/1511.03575/) (2015)
- [Being Robust (in High Dimensions) Can Be Practical](https://mlsanity.com/paper/1703.00893/) (2017)
- [Training Convolutional Networks with Noisy Labels](https://mlsanity.com/paper/1406.2080/) (2014)
- [Provable tradeoffs in adversarially robust classification](https://mlsanity.com/paper/2006.05161/) (2020)
- [Analyzing the Robustness of Nearest Neighbors to Adversarial Examples](https://mlsanity.com/paper/1706.03922/) (2017)
- [Faster Algorithms for High-Dimensional Robust Covariance Estimation](https://mlsanity.com/paper/1906.04661/) (2019)
- [Adversarially Robust Estimate and Risk Analysis in Linear Regression](https://mlsanity.com/paper/2012.10278/) (2020)
- [Regularization properties of adversarially-trained linear regression](https://mlsanity.com/paper/2310.10807/) (2023)
- [Robustness for Non-Parametric Classification: A Generic Attack and Defense](https://mlsanity.com/paper/1906.03310/) (2019)
- [Robust Nonparametric Regression under Poisoning Attack](https://mlsanity.com/paper/2305.16771/) (2023)
- [Robust Kernel Density Estimation with Median-of-Means principle](https://mlsanity.com/paper/2006.16590/) (2020)
- [It Is All About Data: A Survey on the Effects of Data on Adversarial Robustness](https://mlsanity.com/paper/2303.09767/) (2023)
- [Coded Computing for Resilient Distributed Computing: A Learning-Theoretic Framework](https://mlsanity.com/paper/2406.00300/) (2024)
- [Game of Coding: Sybil Resistant Decentralized Machine Learning with Minimal Trust Assumption](https://mlsanity.com/paper/2410.05540/) (2024)
- [General Coded Computing: Adversarial Settings](https://mlsanity.com/paper/2502.08058/) (2025)
- [The Trimmed Mean in Non-parametric Regression Function Estimation](https://mlsanity.com/paper/1909.10734/) (2019)
- [Learning Efficient Surrogate Dynamic Models with Graph Spline Networks](https://mlsanity.com/paper/2310.16397/) (2023)
- [Adaptive Bayesian Multivariate Spline Knot Inference with Prior Specifications on Model Complexity](https://mlsanity.com/paper/2405.13353/) (2024)

## Built on by (most cited first, 1 of 1)

- [Adversarial Robustness of NTK Neural Networks](https://mlsanity.com/paper/2604.25965/) (2026)

## Abstract (arXiv)

In this paper, we investigate the adversarial robustness of nonparametric regression, a fundamental problem in machine learning, under the setting where an adversary can arbitrarily corrupt a subset of the input data. While the robustness of parametric regression has been extensively studied, its nonparametric counterpart remains largely unexplored. We characterize the adversarial robustness in nonparametric regression, assuming the regression function belongs to the second-order Sobolev space (i.e., it is square integrable up to its second derivative). The contribution of this paper is two-fold: (i) we establish a minimax lower bound on the estimation error, revealing a fundamental limit that no estimator can overcome, and (ii) we show that, perhaps surprisingly, the classical smoothing spline estimator, when properly regularized, exhibits robustness against adversarial corruption. These results imply that if o(n) out of n samples are corrupted, the estimation error of the smoothing spline vanishes as n → ∞. On the other hand, when a constant fraction of the data is corrupted, no estimator can guarantee vanishing estimation error, implying the optimality of the smoothing spline in terms of maximum tolerable number of corrupted samples.