General Machine Learning: Theory for Learning Under Variable Regimes

Modern machine learning spans supervised, transfer, continual, meta-learning, and related regimes that often reuse the same hypothesis classes, architectures, and optimizers but differ in information access, objectives, memory, adaptation, and sample accounting. This makes it difficult to determine whether one paradigm is genuinely distinct, a special case of another, or part of a broader structural hierarchy. We introduce a protocol-objective-resource (POR) framework that separates representational capacity from these design choices. A paradigm is specified by an environment class, observation protocol, admissible learners, performance functional, and resource accounting rule. POR reductions combine environment embeddings, learner compilers, threshold maps, and calibrated resource overheads. Our main theorem shows that such reductions imply worst-case complexity domination on embedded comparison classes, transferring upper bounds forward and lower bounds backward; under labeled-example accounting, this yields sample complexity domination. We also show that calibrated nontrivial accuracy regimes are necessary to avoid vacuous comparisons, and that strengthening the objective can strictly increase minimax sample complexity even with unchanged protocols and learner classes. Instantiating the framework for supervised, transfer, continual, and meta-learning yields canonical special-case relations: continual contains transfer, transfer contains supervised, and meta-learning contains supervised under aligned raw-example accounting. We further derive a non-exact episode-to-example reduction for episodic meta-learning and capture within-paradigm refinements such as replay memory and task identifiers. The framework thus provides a unified language for structuring learning paradigms and transferring complexity guarantees across them.

Policy models often assume that the relationship between a policy instrument and its outcome remains stable across institutional conditions. In adaptive socio-technical systems this assumption may fail: regulatory change can alter incentives, agents can respond strategically, and the mapping from policy variables to aggregate outcomes can change. This paper studies such regime change as a transfer-learning problem in adaptive multi-agent systems. A policy regime is represented as a learning problem induced by an observable input distribution and a target function mapping policy variables to outcomes. We compare a blank-slate learner that searches a flexible hypothesis class in the new regime with a transfer learner whose effective hypothesis class is restricted by structural knowledge from the previous regime. Transfer is beneficial when this restriction preserves the new target function while reducing effective complexity; it is harmful when the restriction excludes the new target and creates misspecification. A stylized emissions-regulation experimental environment and a dynamic ABM robustness experiment support the claim. When the target regime preserves an affine monotone tax-emissions relation, transfer improves empirical small-sample performance. When the target regime introduces a threshold break, the same transferred structure produces negative transfer: held-out error remains high, online prediction generates more mistakes, and repeated online streams show larger cumulative and final-window error under misspecification. The contribution is methodological: previous regulatory experience should be reused when it captures stable structural invariants, but treated cautiously when policy change alters the policy-outcome relationship.

Two of the most fundamental questions in statistical learning theory are the following: which prediction problems are learnable, and how should they be learned? For the former, elegant answers often take the form of combinatorial dimensions. The latter question, however, has proved considerably more elusive: all known general-purpose multiclass learners rely on intricate orientations of exponentially large one-inclusion structures, and familiar algorithmic principles such as proper learning and regularization remain poorly understood. Motivated by prior work, we ask whether learning reduces to proper learning---possibly over a larger hypothesis class---and whether proper or improper multiclass learning can ultimately be captured by suitable regularizers. Our primary results answer both questions negatively, resolving three open problems from prior work. First, we exhibit a learnable multiclass problem that cannot be embedded in any properly learnable class, meaning learning cannot be reduced to proper learning by enlarging the hypothesis class. Second, we demonstrate that proper learning can require training error and characterize this phenomenon precisely: every properly learnable class admits a proper learner making $o(m)$ errors on samples of size $m$, but every prescribed sublinear scale $a_m=o(m)$ is necessary for some properly learnable problem. Third, regularization is not a general learner: we exhibit a properly learnable class that cannot be learned by any Structural Risk Minimization (SRM) learner, and a learnable class that cannot be learned by any local regularizer. We complement these impossibility results with a positive theory that gives two sufficient conditions for SRM learnability and characterizes SRM representability through integrability of revealed preferences.