Efficient Quantum Agnostic Improper Learning of Decision Trees

The agnostic setting is the hardest generalization of the PAC model since it is akin to learning with adversarial noise. We study an open question on the existence of efficient quantum boosting algorithms in this setting. We answer this question in the affirmative by providing a quantum version of the Kalai-Kanade potential boosting algorithm. This algorithm shows the standard quadratic speedup in the VC dimension of the weak learner compared to the classical case. Using our boosting algorithm as a subroutine, we give a quantum algorithm for agnostically learning decision trees in polynomial running time without using membership queries. To the best of our knowledge, this is the first algorithm (quantum or classical) to do so. Learning decision trees without membership queries is hard (and an open problem) in the standard classical realizable setting. In general, even coming up with weak learners in the agnostic setting is a challenging task. We show how to construct a quantum agnostic weak learner using standard quantum algorithms, which is of independent interest for designing ensemble learning setups.