Physics-Aware Learnability: From Set-Theoretic Independence to Operational Constraints

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.

Whether quantum computation can reduce the amount of data sampled from an unknown probability distribution required to learn a prediction rule is a fundamental question in quantum machine learning. Quantum PAC learning studies this question using quantum data as a quantum state whose squared amplitudes encode the unknown distribution from which classical learning data are sampled. With only copies of such quantum data, the optimal worst-case sample complexity asymptotically matches that of classical PAC learning. In contrast, access to both a state-preparation unitary for this state and its inverse can improve the query-complexity dependence on the accuracy parameter in realizable learning. However, it has remained unclear whether forward-only access allows such an improvement. In this work, taking the worst case over compatible state-preparation unitaries and their finite ambient dimensions, we show that the optimal forward-only query complexities of realizable and agnostic learning are, respectively, $Θ((d+\log(1/δ))/\varepsilon)$ and $Θ((d+\log(1/δ))/\varepsilon^2)$, where $d$ is the VC dimension of the concept class, $\varepsilon$ the accuracy parameter, and $δ$ the failure probability. These bounds match the optimal sample complexities with classical data or quantum data copies. To prove them, we establish a reduction using $q$ copies of the prepared state to approximate the Haar-averaged output of any $q$-query forward-only algorithm. These results show that forward-only access cannot provide an asymptotic query-complexity advantage over learning from classical data or quantum data copies in this worst-case setting, and establish the essential role of inverse access in the known realizable-setting improvement. Our reduction also provides a new framework for analyzing limitations of forward state-preparation access via state-copy lower bounds.

Machine learning is usually evaluated after an evidence interface has been fixed. A dataset, sensor suite, query language, action set, or experimental protocol determines which observations can be obtained, and learning is judged by what it extracts from them. We study a complementary capability. A learner's state can determine which evidence-generating experiments it can reliably realise under bounded resources, even when primitive affordances remain fixed. We call this learner-relative experiment family its effective epistemic reach, and use Wide Learning for task-relevant learning-induced changes in that family.We formalise effective reach relative to learner state, deployment budget, reliability threshold, and evaluation distribution. In a controlled construction, two hidden worlds have exactly the same public observation law. An informative diagnostic exists in a fixed five-primitive substrate. Before calibration, one address attempt realises it with probability at most $2^{-10} = 1/1024$, below a pre-specified 0.95 threshold; after calibration, held-out realisation is 1. Public-channel total variation is 0, whereas the realised diagnostic has total variation 1, and sealed binary risk moves from approximately 1/2 to 0. The construction establishes that learning can change effective epistemic reach even when primitive affordances and deployment resources are held fixed. It opens a complementary evaluation question for learning systems: not only what they infer from available evidence, but what informative evidence experience teaches them to bring within reach.