Agnostic Learning by Refuting
This work provides a foundational equivalence between refutation complexity and efficient agnostic learning, impacting computational learning theory by bridging gaps in lower bounds and connecting to prior frameworks.
The paper tackles the problem of characterizing the sample complexity of efficient agnostic learning for Boolean-valued function classes by introducing refutation complexity, which exactly matches this sample complexity, addressing a gap where Rademacher complexity only characterizes learning without efficiency constraints.
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.