Learning as Minimum Structural Repair: A Variational Principle for Adaptive Systems

Abstract

Learning systems are described by the error they reduce, yet post-training adaptation is often governed by a different quantity: how much of a useful system must change before a new requirement becomes true. We formalize this quantity as repair complexity, the minimum admissible structural cost required to reach a satisfaction locus. Building on Closure Complexity, the framework separates behavioral defect, feasibility, and repair cost; proves target monotonicity and repair-interaction bounds; recovers steepest-gradient and natural-gradient directions as minimum-repair solutions under Euclidean and Fisher geometries; establishes covariance of quadratic repair under invertible linear reparameterization; and derives a closed-form multi-constraint metric projection. This yields Minimum Structural Repair Projection (MSR), which prices parameter motion by preservation sensitivity rather than raw Euclidean distance alone. Across 27 matched neural-editing settings on Digits, Wine, and Breast Cancer, every method achieves the requested edits, while MSR preserves 71.8% of non-edited predictions versus 68.1% for an identical Euclidean projection and 45.8–54.1% for gradient baselines. MSR lowers common structural cost against Euclidean repair in all 27 matched settings. Sensitivity and sequential-repair experiments further expose geometry-dependent trade-offs. The results support minimum structural repair as a testable variational primitive for adaptive systems, not a universal learning law.