Do PAC-Learners Learn the Marginal Distribution?

We study a foundational variant of Valiant and Vapnik and Chervonenkis' Probably Approximately Correct (PAC)-Learning in which the adversary is restricted to a known family of marginal distributions $\mathscr{P}$. In particular, we study how the PAC-learnability of a triple $(\mathscr{P},X,H)$ relates to the learners ability to infer \emph{distributional} information about the adversary's choice of $D \in \mathscr{P}$. To this end, we introduce the `unsupervised' notion of \emph{TV-Learning}, which, given a class $(\mathscr{P},X,H)$, asks the learner to approximate $D$ from unlabeled samples with respect to a natural class-conditional total variation metric. In the classical distribution-free setting, we show that TV-learning is \emph{equivalent} to PAC-Learning: in other words, any learner must infer near-maximal information about $D$. On the other hand, we show this characterization breaks down for general $\mathscr{P}$, where PAC-Learning is strictly sandwiched between two approximate variants we call `Strong' and `Weak' TV-learning, roughly corresponding to unsupervised learners that estimate most relevant distances in $D$ with respect to $H$, but differ in whether the learner \emph{knows} the set of well-estimated events. Finally, we observe that TV-learning is in fact equivalent to the classical notion of \emph{uniform estimation}, and thereby give a strong refutation of the uniform convergence paradigm in supervised learning.