Balancing Early Performance Sacrifices with Long-Term Gains: Scaling Learning-Rate Warmup Duration Across Training Horizons

Learning-rate warmup is a standard technique in language-model training, yet its duration remains largely heuristic. Common approaches use either a fixed number of updates or a fixed fraction of the training horizon, two choices that imply very different scaling as training gets longer. When should warmup stay fixed, and when should it grow with the horizon? We address this question with a quadratic model whose modes respond differently to the peak learning rate. Warmup slows progress in directions that already contract well at the peak rate, but can remove persistent error in directions near the stability edge, with higher peak rates shifting the balance toward longer warmup durations. This yields a compact horizon scaling law that captures regimes ranging from essentially no warmup, through fixed-duration warmup, to durations that grow with the training horizon, and explains how the preferred regime changes with peak learning rate. Because the law captures the tradeoff between giving up early progress and improving the trajectory that follows, it can be fit using shorter runs and used to predict warmup at substantially longer horizons. Together, our results explain several familiar properties of warmup through a single tradeoff and suggest treating warmup duration as a horizon-dependent hyperparameter rather than a fixed training heuristic.