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Life sciences · Preprint
arXiv · September 8, 2026
Raises a question worth testing. It does not answer one.
This is an unreviewed theoretical preprint proposing an exact scalar law governing the interaction between learning-rate schedules and weight decay in normalized neural networks. The authors provide exact mathematical analysis of a simplified regression model, a unified homogeneous-optimizer framework, and computational validation across several architectures and datasets, but do not report empirical performance gains, clinical outcomes, or peer-reviewed confirmation.
Preprint.
A single scalar quantity captures all schedule and decay forcing in scale-invariant optimization Norm growth induces a geometric self-quenching effect that opposes parameter expansion A sharp boundary separates contraction- and expansion-dominated effective learning rate regimes
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The source did not state who this applies to in practice.
This is a theoretical and computational study proposing a mechanistic framework for scale-invariant optimization dynamics; it lacks empirical validation on real clinical or practical outcomes and presents mathematical analysis with illustrative experiments rather than confirmatory evidence of practical utility.
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Normalization renders large parts of neural networks effectively scale invariant, inducing a hidden feedback loop in which learning-rate schedules and weight decay interact through the parameter norm to control the effective step taken by the optimizer. We show that this interaction is governed by an exact discrete-time law: a single scalar quantity captures all schedule and decay forcing, while norm growth induces an opposing geometric self-quenching effect. This yields a sharp boundary that cleanly separates contraction- and expansion-dominated effective learning rate regimes. To understand the underlying mechanism, we provide exact analysis of a fully solved normalized regression model where the dynamics reduce to two dimensions and show that the balance point is intrinsically unstable, implying that constant learning rate with weight decay cannot stably maintain an interior equilibrium and instead produces recurrent behavior driven by discrete-time Jacobian structure. We further extend this perspective across optimizers through unified homogeneous-optimizer framework that reveals a structural dichotomy in self-quenching strength, providing a first-principles explanation for why adaptive methods exhibit systematically weaker stabilization under normalization. Across dynamical systems and neural networks (MLP, CNN, GPT2 / MNIST, CIFAR, wikiText, OpenWebText), the predicted law holds with high precision and enables direct control of training via the identified scalar, with performance peaking sharply at the predicted boundary. Together, these results isolate a single governing quantity for scale-invariant optimization, providing a precise and actionable lens on training dynamics, optimizer behavior, and schedule design in modern deep learning. Code is available in https://github.com/shasanamin/normalized-optimization-dynamics.
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