Robust Regression of General ReLUs with Queries

Abstract

We study the task of agnostically learning general (as opposed to homogeneous) ReLUs under the Gaussian distribution with respect to the squared loss. In the passive learning setting, recent work gave a computationally efficient algorithm that uses $poly(d,1/ε)$ labeled examples and outputs a hypothesis with error $O(opt)+ε$, where $opt$ is the squared loss of the best fit ReLU. Here we focus on the interactive setting, where the learner has some form of query access to the labels of unlabeled examples. Our main result is the first computationally efficient learner that uses $d polylog(1/ε)+\tilde{O}(\min\{1/p, 1/ε\})$ black-box label queries, where $p$ is the bias of the target function, and achieves error $O(opt)+ε$. We complement our algorithmic result by showing that its query complexity bound is qualitatively near-optimal, even ignoring computational constraints. Finally, we establish that query access is essentially necessary to improve on the label complexity of passive learning. Specifically, for pool-based active learning, any active learner requires $\tildeΩ(d/ε)$ labels, unless it draws a super-polynomial number of unlabeled examples.