A remarkable recent paper by Rubinfeld and Vasilyan (2022) initiated the
study of \emph{testable learning}, where the goal is to replace hard-to-verify
distributional assumptions (such as Gaussianity) with efficiently testable ones
and to require that the learner succeed whenever the unknown distribution
passes the corresponding test. In this model, they gave an efficient algorithm
for learning halfspaces under testable assumptions that are provably satisfied
by Gaussians.
In this paper we give a powerful new approach for developing algorithms for
testable learning using tools from moment matching and metric distances in
probability. We obtain efficient testable learners for any concept class that
admits low-degree \emph{sandwiching polynomials}, capturing most important
examples for which we have ordinary agnostic learners. We recover the results
of Rubinfeld and Vasilyan as a corollary of our techniques while achieving
improved, near-optimal sample complexity bounds for a broad range of concept
classes and distributions.
Surprisingly, we show that the information-theoretic sample complexity of
testable learning is tightly characterized by the Rademacher complexity of the
concept class, one of the most well-studied measures in statistical learning
theory. In particular, uniform convergence is necessary and sufficient for
testable learning. This leads to a fundamental separation from (ordinary)
distribution-specific agnostic learning, where uniform convergence is
sufficient but not necessary.Comment: 34 page