Abstract

The increased availability of data in recent years has led several authors to ask whether it is possible to use data as a computational resource. That is, if more data is available, beyond the sample complexity limit, is it possible to use the extra examples to speed up the computation time required to perform the learning task? We give the first positive answer to this question for a natural supervised learning problem — we consider agnostic PAC learning of halfspaces over 3-sparse
vectors in This class is inefficiently learnable using O n/2 examples. Our main contribution is a novel, non-cryptographic, methodology for establishing computational-statistical gaps, which allows us to show that, under a widely believed assumption that refuting random 3CNF
formulas is hard, it is impossible to efficiently learn this class using only examples.
We further show that under stronger hardness assumptions, even examples do not suffice. On the other
hand, we show a new algorithm that learns this class efficiently using examples. This formally establishes the tradeoff between sample and computational complexity for a natural supervised learning problem.