Classification Under Misspecification: Halfspaces, Generalized Linear Models, and Evolvability

Abstract

In this paper, we revisit the problem of distribution-independently learning halfspaces under Massart noise with rate . Recent work resolved a long-standing problem in this model of efficiently learning to error + for any > 0, by giving an improper learner that partitions space into poly(d,1/) regions. Here we give a much simpler algorithm and settle a number of outstanding open questions: (1) We give the first proper learner for Massart halfspaces that achieves + . (2) Based on (1), we develop a blackbox knowledge distillation procedure to convert an arbitrarily complex classifier to an equally good proper classifier. (3) By leveraging a simple but overlooked connection to evolvability, we show any SQ algorithm requires super-polynomially many queries to achieve OPT + . We then zoom out to study generalized linear models and give an efficient algorithm for learning under a challenging new corruption model generalizing Massart noise. Finally we study our algorithm for learning halfspaces under Massart noise empirically and find that it exhibits some appealing fairness properties as a byproduct of its strong provable robustness guarantees.

Tasks

Reproductions