Revisiting Agnostic PAC Learning

• Published in: Proceedings / annual Symposium on Foundations of Computer Science pp. 1968 - 1982
• Main Authors: Hanneke, Steve, Larsen, Kasper Green, Zhivotovskiy, Nikita
• Format: Conference Proceeding
• Language: English
• Published: IEEE 27.10.2024
• Subjects: agnostic; Classification algorithms; Computer science; learning theory; pac learning; Picture archiving and communication systems; Risk minimization; sample complexity; Supervised learning; Training; Training data; vc-dimension
• ISSN: 2575-8454
• DOI: 10.1109/FOCS61266.2024.00118

PAC learning, dating back to Valiant'84 and Vapnik and Chervonenkis'64,'74, is a classic model for studying supervised learning. In the agnostic setting, we have access to a hypothesis set \mathbf{H} and a training set of labeled samples drawn i,i \mathbf{d} . from an unknown data distribution D. The goal is to produce a classifier that is competitive with the hypothesis in \mathbf{H} having the least probability of mispredicting the label of a new sample from D. Empirical Risk Minimization (ERM) is a natural learning algorithm, where one simply outputs the hypothesis from \mathbf{H} making the fewest mistakes on the training data. This simple algorithm is known to have an optimal error in terms of the VC-dimension of \mathbf{H} and the number of samples. In this work, we revisit agnostic PAC learning and first show that ERM and any other proper learning algorithm is in fact sub-optimal if we treat the performance of the best hypothesis in \mathbf{H} , as a parameter. We then complement this lower bound with the first learning algorithm achieving an optimal error. Our algorithm introduces several new ideas that we hope may find further applications in learning theory.