Active Learning Polynomial Threshold Functions

TL;DR

This paper explores active learning of polynomial threshold functions, showing that access to derivatives enables efficient learning in univariate cases but not in multivariate cases, with new algorithms and bounds.

Contribution

It introduces derivative-based active learning algorithms for univariate PTFs and demonstrates their limitations in multivariate settings.

Findings

Efficient active learning algorithms for univariate PTFs using derivatives.

Lower bounds showing derivatives are insufficient for multivariate PTF active learning.

Near-optimal algorithms for average case settings of PTFs.

Abstract

We initiate the study of active learning polynomial threshold functions (PTFs). While traditional lower bounds imply that even univariate quadratics cannot be non-trivially actively learned, we show that allowing the learner basic access to the derivatives of the underlying classifier circumvents this issue and leads to a computationally efficient algorithm for active learning degree-$d$ univariate PTFs in $\tilde{O}(d^3\log(1/\varepsilon\delta))$ queries. We also provide near-optimal algorithms and analyses for active learning PTFs in several average case settings. Finally, we prove that access to derivatives is insufficient for active learning multivariate PTFs, even those of just two variables.