More data speeds up training time in learning halfspaces over sparse vectors

The increased availability of data in recent years has led several authors to ask whether it is possible to use data as a em 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 em natural supervised learning problem --- we consider agnostic PAC learning of halfspaces over 3-sparse vectors in -1,1,0n. This class is inefficiently learnable using Oleft(n/ε2right) 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 Oleft(n/ε2right) examples. We further show that under stronger hardness assumptions, even Oleft(n1.499/ε2right) examples do not suffice. On the other hand, we show a new algorithm that learns this class efficiently using tildeOmegaleft(n2/ε2right) examples. This formally establishes the tradeoff between sample and computational complexity for a natural supervised learning problem.