Efficient Active Learning Halfspaces with Tsybakov Noise: A Non-convex Optimization Approach
Abstract
We study the problem of computationally and label efficient PAC active learning $d$-dimensional halfspaces with Tsybakov Noise (Tsybakov, 2004) under structured unlabeled data distributions. Inspired by Diakonikolas et al., (2020c), we prove that any approximate first-order stationary point of a smooth nonconvex loss function yields a halfspace with a low excess error guarantee. In light of the above structural result, we design a nonconvex optimization-based algorithm with a label complexity of $\tilde{O}(d (\frac{1}{\epsilon})^{\frac{8-6\alpha}{3\alpha-1}})$, under the assumption that the Tsybakov noise parameter $\alpha \in (\frac13, 1]$, which narrows down the gap between the label complexities of the previously known efficient passive or active algorithms (Diakonikolas et al., 2020b; Zhang and Li, 2021) and the information-theoretic lower bound in this setting.
BibTeX
@InProceedings{pmlr-v238-li24p,
title = {Efficient Active Learning Halfspaces with {T}sybakov Noise: A Non-convex Optimization Approach},
author = {Li, Yinan and Zhang, Chicheng},
booktitle = {Proceedings of The 27th International Conference on Artificial Intelligence and Statistics},
pages = {4744--4752},
year = {2024},
editor = {Dasgupta, Sanjoy and Mandt, Stephan and Li, Yingzhen},
volume = {238},
series = {Proceedings of Machine Learning Research},
month = {02--04 May},
publisher = {PMLR},
pdf = {https://proceedings.mlr.press/v238/li24p/li24p.pdf},
url = {https://proceedings.mlr.press/v238/li24p.html},
abstract = {We study the problem of computationally and label efficient PAC active learning $d$-dimensional halfspaces with Tsybakov Noise (Tsybakov, 2004) under structured unlabeled data distributions. Inspired by Diakonikolas et al., (2020c), we prove that any approximate first-order stationary point of a smooth nonconvex loss function yields a halfspace with a low excess error guarantee. In light of the above structural result, we design a nonconvex optimization-based algorithm with a label complexity of $\tilde{O}(d (\frac{1}{\epsilon})^{\frac{8-6\alpha}{3\alpha-1}})$, under the assumption that the Tsybakov noise parameter $\alpha \in (\frac13, 1]$, which narrows down the gap between the label complexities of the previously known efficient passive or active algorithms (Diakonikolas et al., 2020b; Zhang and Li, 2021) and the information-theoretic lower bound in this setting.}
}