Surprises in Proper Positive-Only Learning

The fundamental theorem of statistical learning states that, under suitable measurability assumptions, finite Vapnik--Chervonenkis (VC) dimension guarantees that every proper consistent learning rule is probably approximately correct (PAC). Blumer, Ehrenfeucht, Haussler, and Warmuth showed, assuming the Continuum Hypothesis, that the "well-behavedness" condition of the concept class cannot be omitted: they constructed a concept class of Borel sets of VC dimension one admitting a consistent learning rule that is not PAC. We show that the Continuum Hypothesis is unnecessary. Working in Zermelo--Fraenkel set theory with the Axiom of Choice (ZFC) alone, we construct a concept class of Borel sets on $[0,1]$ of VC dimension one and a proper consistent learning rule that is not PAC. More precisely, for a suitable Borel probability measure and target concept, the rule has true risk one at every sample size on a set of samples of outer probability one. Consequently, finite VC dimension and Borel measurability of the individual concepts do not suffice to guarantee that every proper consistent learning rule is PAC. The result shows, with no need of extra set-theoretical assumptions, that the additional regularity assumption in the fundamental theorem cannot in general be omitted.

Valiant's 1984 paper is widely credited with introducing the PAC learning model, but it, in fact, introduced a different model: unlike PAC learning, the learner receives only positives, may issue membership queries, and must output a hypothesis with no false positives. Prior work characterized variants, including the case without queries. We revisit Valiant's original model and ask: Which classes are learnable in it? For every finite domain, including Valiant's Boolean-hypercube setting, we show that a class is learnable if and only if every realizable positive sample can be certified by a poly-size adaptive query-compression scheme. This is a new variant of sample compression where the learner certifies samples via a short interaction with the membership oracle. Our characterization shows that learnability in Valiant's model is strictly sandwiched between learnability in the PAC model and the variant of Valiant's model without membership queries. This is one of the rare cases where introducing membership queries changes the set of learnable classes, and not just the sample or computational complexity. Next, we study the natural extension of the model to arbitrary domains. While we do not obtain an exact characterization, our techniques readily generalize and show that the same strict sandwiching persists. Finally, we show that $d$-dimensional halfspaces, which are not learnable without queries, are learnable with queries: we give a $\mathrm{poly}(d) \tilde{O}(1/ε)$ sample and $\mathrm{poly}(d) \mathrm{polylog}(1/ε)$ query algorithm, and prove that at least $Ω(d)$ samples or queries are necessary. To our knowledge, this is the first algorithm for halfspaces in Valiant's model. Together, these results uncover a surprisingly rich theory behind Valiant's original notion of learnability and introduce ideas that may be of independent interest in learning theory.

We study the problem of \emph{adversarially robust} PAC learning. In this framework, the learner observes independent samples from an unknown distribution over $\mathcal{X} \times \{0,1\}$, as in classical PAC learning. However, given a perturbation map $\mathcal{U} : \mathcal{X} \to 2^{\mathcal{X}}$ known to the learner, the goal is to output, with high probability, a predictor that correctly classifies \emph{every} perturbation $z \in \mathcal{U}(x)$ of most future examples $(x,y)$ drawn from the same underlying distribution. We determine the \emph{optimal} $\mathcal{U}$-independent sample complexity of this problem in both the realizable and agnostic settings. More specifically, for every concept class $\mathcal{H}$ of $\operatorname{VC}$ dimension $d$, we prove upper bounds of $\mathcal{O} \big( d/ε+ \log(1/δ)/ε\big)$ in the realizable setting and $\mathcal{O} \big( d/ε^2 + \log(1/δ)/ε^2 \big)$ in the agnostic setting, together with an optimal first-order refinement of the latter. These bounds match the corresponding lower bounds for classical PAC learning. Consequently, and perhaps surprisingly, adversarial robustness incurs \emph{no additional} distribution-free statistical cost, uniformly over all perturbation maps. Our bounds improve exponentially on those of [Montasser, Hanneke, and Srebro; COLT '19]. On the technical side, we present short and elementary proofs based on a new algorithmic principle that we call \emph{binomial-bagging}. We believe that binomial-bagging and its analysis may be of independent interest.