Agnostic Learning by Refuting

TL;DR

This paper introduces refutation complexity as a computational measure that exactly characterizes the sample complexity of efficient agnostic learning for Boolean functions, linking it to the difficulty of distinguishing structured data from noise.

Contribution

It defines refutation complexity and proves its equivalence to the sample complexity of efficient agnostic learning, connecting learning theory with refutation of random constraint satisfaction problems.

Findings

Refutation complexity characterizes efficient agnostic learning.

The relationship between refutation and learning is made explicit.

Connections to previous work on PAC learning and CSP refutation.

Abstract

The sample complexity of learning a Boolean-valued function class is precisely characterized by its Rademacher complexity. This has little bearing, however, on the sample complexity of \emph{efficient} agnostic learning. We introduce \emph{refutation complexity}, a natural computational analog of Rademacher complexity of a Boolean concept class and show that it exactly characterizes the sample complexity of \emph{efficient} agnostic learning. Informally, refutation complexity of a class $\mathcal{C}$ is the minimum number of example-label pairs required to efficiently distinguish between the case that the labels correlate with the evaluation of some member of $\mathcal{C}$ (\emph{structure}) and the case where the labels are i.i.d. Rademacher random variables (\emph{noise}). The easy direction of this relationship was implicitly used in the recent framework for improper PAC learning lower bounds of Daniely and co-authors via connections to the hardness of refuting random constraint satisfaction problems. Our work can be seen as making the relationship between agnostic learning and refutation implicit in their work into an explicit equivalence. In a recent, independent work, Salil Vadhan discovered a similar relationship between refutation and PAC-learning in the realizable (i.e. noiseless) case.