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We analyze the properties of adversarial training for learning adversarially robust halfspaces in the presence of agnostic label noise. Denoting $\mathsf{OPT}_{p,r}$ as the best robust classification error achieved by a halfspace that is…

Machine Learning · Computer Science 2021-04-20 Difan Zou , Spencer Frei , Quanquan Gu

We give a $2^{\tilde{O}(\sqrt{n}/\epsilon)}$-time algorithm for properly learning monotone Boolean functions under the uniform distribution over $\{0,1\}^n$. Our algorithm is robust to adversarial label noise and has a running time nearly…

Data Structures and Algorithms · Computer Science 2023-03-29 Jane Lange , Ronitt Rubinfeld , Arsen Vasilyan

In the classic point location problem, one is given an arbitrary dataset $X \subset \mathbb{R}^d$ of $n$ points with query access to an unknown halfspace $f : \mathbb{R}^d \to \{0,1\}$, and the goal is to learn the label of every point in…

Data Structures and Algorithms · Computer Science 2025-09-26 Hadley Black , Kasper Green Larsen , Arya Mazumdar , Barna Saha , Geelon So

The goal of active learning is to achieve the same accuracy achievable by passive learning, while using much fewer labels. Exponential savings in terms of label complexity have been proved in very special cases, but fundamental lower bounds…

Machine Learning · Statistics 2026-01-01 Yinglun Zhu , Robert Nowak

We study the task of agnostically learning halfspaces under the Gaussian distribution. Specifically, given labeled examples $(\mathbf{x},y)$ from an unknown distribution on $\mathbb{R}^n \times \{ \pm 1\}$, whose marginal distribution on…

Machine Learning · Computer Science 2023-02-14 Ilias Diakonikolas , Daniel M. Kane , Lisheng Ren

We study the problem of {\em properly} learning large margin halfspaces in the agnostic PAC model. In more detail, we study the complexity of properly learning $d$-dimensional halfspaces on the unit ball within misclassification error…

Machine Learning · Computer Science 2019-08-30 Ilias Diakonikolas , Daniel M. Kane , Pasin Manurangsi

We pose a fundamental question in computational learning theory: can we efficiently test whether a training set satisfies the assumptions of a given noise model? This question has remained unaddressed despite decades of research on learning…

Machine Learning · Computer Science 2026-05-11 Surbhi Goel , Adam R. Klivans , Konstantinos Stavropoulos , Arsen Vasilyan

With the explosion of massive, widely available unlabeled data in the past years, finding label and time efficient, robust learning algorithms has become ever more important in theory and in practice. We study the paradigm of active…

Machine Learning · Computer Science 2020-01-17 Max Hopkins , Daniel Kane , Shachar Lovett , Gaurav Mahajan

We study the learnability of linear separators in $\Re^d$ in the presence of bounded (a.k.a Massart) noise. This is a realistic generalization of the random classification noise model, where the adversary can flip each example $x$ with…

Machine Learning · Computer Science 2015-03-13 Pranjal Awasthi , Maria-Florina Balcan , Nika Haghtalab , Ruth Urner

This paper presents a novel online learning method that aims at finding a separator hyperplane between data points labelled as either positive or negative. Since weights and biases of artificial neurons can directly be related to…

Machine Learning · Computer Science 2023-09-13 Ákos Hajnal

Active learning is a subfield of machine learning, in which the learning algorithm is allowed to choose the data from which it learns. In some cases, it has been shown that active learning can yield an exponential gain in the number of…

Machine Learning · Computer Science 2020-12-22 Ori Kelner

We study the problem of agnostically learning halfspaces which is defined by a fixed but unknown distribution $\mathcal{D}$ on $\mathbb{Q}^n\times \{\pm 1\}$. We define $\mathrm{Err}_{\mathrm{HALF}}(\mathcal{D})$ as the least error of a…

Computational Complexity · Computer Science 2016-03-15 Amit Daniely

We consider the problem of training a model under the presence of label noise. Current approaches identify samples with potentially incorrect labels and reduce their influence on the learning process by either assigning lower weights to…

Machine Learning · Computer Science 2019-06-04 Duc Tam Nguyen , Thi-Phuong-Nhung Ngo , Zhongyu Lou , Michael Klar , Laura Beggel , Thomas Brox

We study the problem of PAC learning $\gamma$-margin halfspaces with Massart noise. We propose a simple proper learning algorithm, the Perspectron, that has sample complexity $\widetilde{O}((\epsilon\gamma)^{-2})$ and achieves…

Machine Learning · Computer Science 2025-01-20 Gautam Chandrasekaran , Vasilis Kontonis , Konstantinos Stavropoulos , Kevin Tian

Understanding noise tolerance of machine learning algorithms is a central quest in learning theory. In this work, we study the problem of computationally efficient PAC learning of halfspaces in the presence of malicious noise, where an…

Machine Learning · Computer Science 2025-02-18 Jie Shen

In this work, we give the first algorithms for tolerant testing of nontrivial classes in the active model: estimating the distance of a target function to a hypothesis class C with respect to some arbitrary distribution D, using only a…

Machine Learning · Statistics 2017-11-02 Avrim Blum , Lunjia Hu

Given a finite set $X \subset \mathbb{R}^d$ and a binary linear classifier $c: \mathbb{R}^d \to \{0,1\}$, how many queries of the form $c(x)$ are required to learn the label of every point in $X$? Known as \textit{point location}, this…

Computational Geometry · Computer Science 2020-04-27 Max Hopkins , Daniel M. Kane , Shachar Lovett , Gaurav Mahajan

We study the problem of learning a single neuron with respect to the $L_2^2$-loss in the presence of adversarial label noise. We give an efficient algorithm that, for a broad family of activations including ReLUs, approximates the optimal…

Machine Learning · Computer Science 2023-06-14 Puqian Wang , Nikos Zarifis , Ilias Diakonikolas , Jelena Diakonikolas

We study the complexity of PAC learning halfspaces in the presence of Massart noise. In this problem, we are given i.i.d. labeled examples $(\mathbf{x}, y) \in \mathbb{R}^N \times \{ \pm 1\}$, where the distribution of $\mathbf{x}$ is…

Machine Learning · Computer Science 2022-07-29 Ilias Diakonikolas , Daniel M. Kane , Pasin Manurangsi , Lisheng Ren

We study active learning where the labeler can not only return incorrect labels but also abstain from labeling. We consider different noise and abstention conditions of the labeler. We propose an algorithm which utilizes abstention…

Machine Learning · Computer Science 2016-11-01 Songbai Yan , Kamalika Chaudhuri , Tara Javidi