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.

翻译：瓦利安特1984年的论文被广泛认为提出了PAC学习模型，但实际上它引入了一个不同的模型：与PAC学习不同，学习者仅接收正例，可以发起成员查询，并且必须输出一个无假正例的假设。先前的研究刻画了该模型的各种变体，包括无查询的情况。我们重新审视瓦利安特原始模型并提出问题：*哪些概念类在该模型中是可学习的？*对于每个有限域，包括瓦利安特的布尔超立方体设定，我们证明一个概念类是可学习的当且仅当每个可实现的正样本都能通过一个多项式大小的自适应查询压缩方案进行认证。这是一种新的样本压缩变体，其中学习者通过与成员资格谕示的简短交互来认证样本。我们的刻画表明，瓦利安特模型中的可学习性严格介于PAC模型的可学习性与无成员查询的瓦利安特模型变体之间。这是引入成员查询会改变可学习概念类集合（而不仅仅是样本或计算复杂度）的罕见案例之一。接下来，我们研究该模型在任意域上的自然推广。尽管未能获得精确刻画，但我们的技术可轻松推广并表明相同的严格夹逼关系依然成立。最后，我们证明$d$维半空间（无查询时不可学习）在有查询时是可学习的：我们给出一个$\mathrm{poly}(d) \tilde{O}(1/ε)$样本和$\mathrm{poly}(d) \mathrm{polylog}(1/ε)$查询的算法，并证明至少需要$Ω(d)$个样本或查询。据我们所知，这是瓦利安特模型中首个针对半空间的算法。这些结果共同揭示了瓦利安特原始可学习概念背后令人惊讶的丰富理论，并引入了可能在机器学习理论中具有独立价值的想法。