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General Machine Learning: Theory for Learning Under Variable Regimes

Aomar Osmani

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.

General Machine Learning: Theory for Learning Under Variable Regimes

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.
Paper Structure (154 sections, 20 theorems, 52 equations, 1 figure, 2 tables)

This paper contains 154 sections, 20 theorems, 52 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Hadamard-style well-posedness as a foundational constraint.
  3. Relation to a broader structural programme.
  4. Canonical definition and scope.
  5. Contributions.
  6. Logical levels of the results.
  7. Roadmap and reading guide.
  8. Preliminaries and Conventions
  9. Why a mathematical grammar is needed.
  10. Typed perspective.
  11. Three structural objects governing regime change.
  12. Axiomatic Core of GML
  13. Canonical definition.
  14. Reading of the canonical definition.
  15. Reading discipline for the expanded specification.
  16. ...and 139 more sections

Key Result

Proposition 6.2

Under Assumption ass:classical-degeneration, GML reduces to the statement that a program learns from experience $E$ with respect to task family $T$ and performance measure $P$ if its performance on $T$, as measured by $P$, improves with $E$.

Figures (1)

  • Figure 1: Architecture of GML under the four standing structural requirements of Section \ref{['sec:requirements']}. 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.

Theorems & Definitions (60)

  • Definition 3.1: GML, canonical definition
  • Definition 3.2: Expanded learning-specific specification
  • Definition 3.3: Typed regime transformation
  • Definition 3.4: Transition grammar and partial composition
  • Definition 3.5: Protected evaluative core
  • Definition 3.6: Protected equivalence
  • Definition 3.7: Compatibility between transformation and evaluator
  • Definition 3.8: Admissibility certificate
  • Definition 3.9: Mitchell-expressible process
  • Definition 3.10: Expanded formal membership criterion for GML
  • ...and 50 more