General Machine Learning: Theory for Learning Under Variable Regimes

Aomar Osmani

cs.LGMar 24, 2026

General Machine Learning: Theory for Learning Under Variable Regimes

Authors: Aomar Osmani

Organizations: INSA-Rouen, France

Abstract

We study learning under regime variation, where the learner, its memory state, and the evaluative conditions may evolve over time. This paper is a foundational and structural contribution: its goal is to define the core learning-theoretic objects required for such settings and to establish their first theorem-supporting consequences. The paper develops a regime-varying framework centered on admissible transport, protected-core preservation, and evaluator-aware learning evolution. It records the immediate closure consequences of admissibility, develops a structural obstruction argument for faithful fixed-ontology reduction in genuinely multi-regime settings, and introduces a protected-stability template together with explicit numerical and symbolic witnesses on controlled subclasses, including convex and deductive settings. It also establishes theorem-layer results on evaluator factorization, morphisms, composition, and partial kernel-level alignment across semantically commensurable layers. A worked two-regime example makes the admissibility certificate, protected evaluative core, and regime-variation cost explicit on a controlled subclass. The symbolic component is deliberately restricted in scope: the paper establishes a first kernel-level compatibility result together with a controlled monotonic deductive witness. The manuscript should therefore be read as introducing a structured learning-theoretic framework for regime-varying learning together with its first theorem-supporting layer, not as a complete quantitative theory of all learning systems.

Figures & tables

Earlier symbol	Present GML object	Role in this paper
$r$	$\mathcal{R}$ , regime-indexed carriers	regime carrier
$\mathcal{H}$	$\mathcal{S}_{r}$	local learning-state space
$\mathcal{M}$	$\mathcal{M}_{r}$	memory carrier
$\mathcal{E}$	$V_{r}$ , $\Phi$ , $\sim_{\Phi}$	evaluator and protected core
$\mathcal{L}$ , $T_{\theta}$	$\mathcal{U}$ , transport grammar	update and transport
$\mathcal{T}$ -certification	$\Gamma$ , protected comparison	admissibility certificate

Table 1: Historical correspondence between earlier structural notation and the present GML notation.

Panel of Figure 1 — Figure 1: Architecture of GML under the four standing structural requirements of Section 4 . The figure distinguishes the canonical GML definition from its expanded learning-specific specification, the instantiation architecture used in the paper, the plurality of legitimate working theorem layers, and the future meta-level left open. The present paper develops only the first explicit theorem-supporting instantiation.

Tradition	Correctly formalizes	Where GML begins	Relation to GML
Mitchell	Fixed-regime evaluative improvement	Evaluator transport; memory admissibility	Boundary degeneration
PAC / VC / SRM	Learnability; sample complexity	Regime-indexed risk semantics	Fixed-regime subclass
SLT / PAC-Bayes	Generalization; fixed risk	Guarantee transport across regimes	Regime-local subclass
Online / Regret	Sequential adaptation; regret	Comparator legitimacy certification	Neighborhood; overlap
Dataset shift	Distributional variation	Evaluator and memory semantic evolution	Statistical subclass
Domain adaptation	Domain-indexed distribution shift	Evaluative legitimacy transport	Partial overlap

Table 2: GML relative to existing learning-theoretic traditions. Each entry records what the tradition correctly formalizes, where the specifically GML boundary begins, and the exact structural relation to GML.

Statistical Learning Theory

Statistical learning theory studies when data-driven algorithms generalize, how many samples they need, and what statistical or computational limits apply.

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Structuring Relations Among Learning Paradigms via Protocol--Objective--Resource Reductions

Junwei Su, Changjie Wang, Dongyang Chang

University of Science and Technology of China

Statistical Learning Theory Meta-Learning Sample Complexity

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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.

Abstract

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