Part of Advances in Neural Information Processing Systems 33 (NeurIPS 2020)
Chicheng Zhang, Jie Shen, Pranjal Awasthi
We study active learning of homogeneous s-sparse halfspaces in Rd under the setting where the unlabeled data distribution is isotropic log-concave and each label is flipped with probability at most η for a parameter η∈[0,12), known as the bounded noise. Even in the presence of mild label noise, i.e. η is a small constant, this is a challenging problem and only recently have label complexity bounds of the form ˜O(s⋅polylog(d,1ϵ)) been established in [Zhang 2018] for computationally efficient algorithms. In contrast, under high levels of label noise, the label complexity bounds achieved by computationally efficient algorithms are much worse: the best known result [Awasthi et al. 2016] provides a computationally efficient algorithm with label complexity ˜O((slnd/ϵ)poly(1/(1−2η))), which is label-efficient only when the noise rate η is a fixed constant. In this work, we substantially improve on it by designing a polynomial time algorithm for active learning of s-sparse halfspaces, with a label complexity of ˜O(s(1−2η)4polylog(d,1ϵ)). This is the first efficient algorithm with label complexity polynomial in 11−2η in this setting, which is label-efficient even for η arbitrarily close to 12. Our active learning algorithm and its theoretical guarantees also immediately translate to new state-of-the-art label and sample complexity results for full-dimensional active and passive halfspace learning under arbitrary bounded noise.