Efficient active learning of sparse halfspaces with arbitrary bounded noise

We study active learning of homogeneous s-sparse halfspaces in R^d under the setting where the unlabeled data distribution is isotropic log-concave and each label is flipped with probability at most ? for a parameter ? 2 [0, ¹₂), known as the bounded noise. Even in the presence of mild label noise, i.e. ? is a small constant, this is a challenging problem and only recently have label complexity bounds of the form Õ (s · polylog (d, ¹_e⁾⁾ 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 Õ((^{s ln}_e^d)^2poly(1/(1-2?⁾⁾⁾, which is label-efficient only when the noise rate ? is a fixed constant. In this work, we substantially improve on it by designing a polynomial time algorithm for active learning of ssparse halfspaces, with a label complexity of Õ((1-^s_2?)4 polylog (d, ¹_e⁾⁾. This is the first efficient algorithm with label complexity polynomial in _1-¹_2? in this setting, which is label-efficient even for ? 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.