On the Power of Localized Perceptron for Label-Optimal Learning of Halfspaces with Adversarial Noise

Jie Shen
Proceedings of the 38th International Conference on Machine Learning, PMLR 139:9503-9514, 2021.

Abstract

We study {\em online} active learning of homogeneous halfspaces in $\mathbb{R}^d$ with adversarial noise where the overall probability of a noisy label is constrained to be at most $\nu$. Our main contribution is a Perceptron-like online active learning algorithm that runs in polynomial time, and under the conditions that the marginal distribution is isotropic log-concave and $\nu = \Omega(\epsilon)$, where $\epsilon \in (0, 1)$ is the target error rate, our algorithm PAC learns the underlying halfspace with near-optimal label complexity of $\tilde{O}\big(d \cdot \polylog(\frac{1}{\epsilon})\big)$ and sample complexity of $\tilde{O}\big(\frac{d}{\epsilon} \big)$. Prior to this work, existing online algorithms designed for tolerating the adversarial noise are subject to either label complexity polynomial in $\frac{1}{\epsilon}$, or suboptimal noise tolerance, or restrictive marginal distributions. With the additional prior knowledge that the underlying halfspace is $s$-sparse, we obtain attribute-efficient label complexity of $\tilde{O}\big( s \cdot \polylog(d, \frac{1}{\epsilon}) \big)$ and sample complexity of $\tilde{O}\big(\frac{s}{\epsilon} \cdot \polylog(d) \big)$. As an immediate corollary, we show that under the agnostic model where no assumption is made on the noise rate $\nu$, our active learner achieves an error rate of $O(OPT) + \epsilon$ with the same running time and label and sample complexity, where $OPT$ is the best possible error rate achievable by any homogeneous halfspace.

Cite this Paper

BibTeX
@InProceedings{pmlr-v139-shen21a, title = {On the Power of Localized Perceptron for Label-Optimal Learning of Halfspaces with Adversarial Noise}, author = {Shen, Jie}, booktitle = {Proceedings of the 38th International Conference on Machine Learning}, pages = {9503--9514}, year = {2021}, editor = {Meila, Marina and Zhang, Tong}, volume = {139}, series = {Proceedings of Machine Learning Research}, month = {18--24 Jul}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v139/shen21a/shen21a.pdf}, url = {https://proceedings.mlr.press/v139/shen21a.html}, abstract = {We study {\em online} active learning of homogeneous halfspaces in $\mathbb{R}^d$ with adversarial noise where the overall probability of a noisy label is constrained to be at most $\nu$. Our main contribution is a Perceptron-like online active learning algorithm that runs in polynomial time, and under the conditions that the marginal distribution is isotropic log-concave and $\nu = \Omega(\epsilon)$, where $\epsilon \in (0, 1)$ is the target error rate, our algorithm PAC learns the underlying halfspace with near-optimal label complexity of $\tilde{O}\big(d \cdot \polylog(\frac{1}{\epsilon})\big)$ and sample complexity of $\tilde{O}\big(\frac{d}{\epsilon} \big)$. Prior to this work, existing online algorithms designed for tolerating the adversarial noise are subject to either label complexity polynomial in $\frac{1}{\epsilon}$, or suboptimal noise tolerance, or restrictive marginal distributions. With the additional prior knowledge that the underlying halfspace is $s$-sparse, we obtain attribute-efficient label complexity of $\tilde{O}\big( s \cdot \polylog(d, \frac{1}{\epsilon}) \big)$ and sample complexity of $\tilde{O}\big(\frac{s}{\epsilon} \cdot \polylog(d) \big)$. As an immediate corollary, we show that under the agnostic model where no assumption is made on the noise rate $\nu$, our active learner achieves an error rate of $O(OPT) + \epsilon$ with the same running time and label and sample complexity, where $OPT$ is the best possible error rate achievable by any homogeneous halfspace.} }
Endnote
%0 Conference Paper %T On the Power of Localized Perceptron for Label-Optimal Learning of Halfspaces with Adversarial Noise %A Jie Shen %B Proceedings of the 38th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2021 %E Marina Meila %E Tong Zhang %F pmlr-v139-shen21a %I PMLR %P 9503--9514 %U https://proceedings.mlr.press/v139/shen21a.html %V 139 %X We study {\em online} active learning of homogeneous halfspaces in $\mathbb{R}^d$ with adversarial noise where the overall probability of a noisy label is constrained to be at most $\nu$. Our main contribution is a Perceptron-like online active learning algorithm that runs in polynomial time, and under the conditions that the marginal distribution is isotropic log-concave and $\nu = \Omega(\epsilon)$, where $\epsilon \in (0, 1)$ is the target error rate, our algorithm PAC learns the underlying halfspace with near-optimal label complexity of $\tilde{O}\big(d \cdot \polylog(\frac{1}{\epsilon})\big)$ and sample complexity of $\tilde{O}\big(\frac{d}{\epsilon} \big)$. Prior to this work, existing online algorithms designed for tolerating the adversarial noise are subject to either label complexity polynomial in $\frac{1}{\epsilon}$, or suboptimal noise tolerance, or restrictive marginal distributions. With the additional prior knowledge that the underlying halfspace is $s$-sparse, we obtain attribute-efficient label complexity of $\tilde{O}\big( s \cdot \polylog(d, \frac{1}{\epsilon}) \big)$ and sample complexity of $\tilde{O}\big(\frac{s}{\epsilon} \cdot \polylog(d) \big)$. As an immediate corollary, we show that under the agnostic model where no assumption is made on the noise rate $\nu$, our active learner achieves an error rate of $O(OPT) + \epsilon$ with the same running time and label and sample complexity, where $OPT$ is the best possible error rate achievable by any homogeneous halfspace.
APA
Shen, J.. (2021). On the Power of Localized Perceptron for Label-Optimal Learning of Halfspaces with Adversarial Noise. Proceedings of the 38th International Conference on Machine Learning, in Proceedings of Machine Learning Research 139:9503-9514 Available from https://proceedings.mlr.press/v139/shen21a.html.

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