More data speeds up training time in learning halfspaces over sparse vectors
This addresses a foundational question in machine learning about using data as a computational resource, with implications for algorithm design in supervised learning, though it is incremental in extending computational-statistical gap methods.
The paper tackles the problem of whether more data can speed up training time for learning halfspaces over sparse vectors, showing that under hardness assumptions, efficient learning requires at least Ω̃(n²/ε²) examples, while O(n/ε²) examples are insufficient, establishing a tradeoff between sample and computational complexity.
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 $\{-1,1,0\}^n$. This class is inefficiently learnable using $O\left(n/ε^2\right)$ 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 $\mathrm{3CNF}$ formulas is hard, it is impossible to efficiently learn this class using only $O\left(n/ε^2\right)$ examples. We further show that under stronger hardness assumptions, even $O\left(n^{1.499}/ε^2\right)$ examples do not suffice. On the other hand, we show a new algorithm that learns this class efficiently using $\tildeΩ\left(n^2/ε^2\right)$ examples. This formally establishes the tradeoff between sample and computational complexity for a natural supervised learning problem.