Exact law reveals stability boundary in normalized neural network optimization
Researchers have identified an exact mathematical law governing how learning-rate schedules and weight decay interact in normalized neural networks, revealing a sharp stability boundary that separates effective optimization regimes. The work exposes a hidden feedback loop where parameter normalization couples these two hyperparameters through norm dynamics, creating a self-quenching geometric effect that opposes learning-rate forcing. This theoretical advance matters because practitioners often tune these knobs independently, unaware they operate as a coupled system. The analysis, grounded in a fully solved normalized regression model, provides actionable insight into why certain schedule-decay combinations destabilize training and offers a principled framework for future optimizer design.
Modelwire context
ExplainerThe paper's core contribution is identifying that learning-rate schedules and weight decay don't operate independently in normalized networks, but instead form a coupled feedback loop through norm dynamics. This explains why certain combinations fail even when each hyperparameter seems reasonable in isolation.
This connects directly to the 'Silver Rate' paper from the same day, which established near-optimal convergence bounds for predetermined stepsizes in smooth convex settings. Where that work closed the theoretical gap on schedule optimality, this paper reveals a hidden constraint: the schedule's effectiveness depends on weight decay in ways current theory doesn't capture. Together, they suggest that convergence proofs may be incomplete without accounting for normalization-induced coupling. The credit-assignment reframing in the RNN paper also parallels this work's insight that hidden feedback loops (norm dynamics here, credit flow there) are often the real bottleneck, not the surface-level mechanism.
If practitioners applying this framework to modern optimizers (AdamW, Lion) report measurable training stability improvements within six months, the theory has crossed into actionable territory. Conversely, if the exact schedule law only holds in the simplified regression setting and breaks down on realistic architectures, the contribution remains primarily pedagogical rather than prescriptive.
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Modelwire summarizes, we don’t republish. arXiv cs.LG originally reported this story as “When Does Scale-Invariant Optimization Become Unstable? An Exact Schedule Law with Weight Decay”. The full content lives on arxiv.org. If you’re a publisher and want a different summarization policy for your work, see our takedown page.