When Does Scale-Invariant Optimization Become Unstable? An Exact Schedule Law with Weight Decay

Hasan Amin; Wei-Kai Chang; Rajiv Khanna

cs.LGSep 8, 2026

When Does Scale-Invariant Optimization Become Unstable? An Exact Schedule Law with Weight Decay

Authors: Hasan Amin, Wei-Kai Chang, Rajiv Khanna

Organizations: Purdue University

Abstract

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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Jul 14, 2026cs.LG

Same Loss, Same Noise, Opposite Schedules: Noise Structure and Optimizer Normalization Jointly Determine Whether Learning-Rate Cooldown Helps

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Jul 26, 2026cs.LG

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\mathbf{30\%}

faster at our largest scale across models from

72 - 930

million parameters trained at

\sim 600

tokens per active parameter. If this trend continues to hold, the method promises to substantially accelerate the pre-training of frontier models while requiring only a few lines of code to implement.

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