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

The paper investigates how normalization makes neural networks scale‑invariant, creating a feedback loop between learning‑rate schedules and weight decay that controls the effective step size of the optimizer. It derives an exact discrete‑time law showing that a single scalar quantity captures all schedule and decay effects, with norm growth providing a self‑quenching counter‑force that defines a sharp boundary between contraction‑ and expansion‑dominated regimes. Through exact analysis of a normalized regression model and experiments on MLPs, CNNs, GPT‑2, and various datasets, the authors demonstrate that constant learning rates with weight decay are intrinsically unstable, leading to recurrent dynamics, and that adaptive optimizers exhibit weaker stabilization under normalization. "whyItMatters":"The study provides a precise, actionable rule for controlling training dynamics and schedule design in modern deep learning by isolating a single governing quantity for scale‑invariant optimization."

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