Anytime Training with Schedule-Free Spectral Optimization

Standard neural network training relies on learning-rate schedules tied to a fixed horizon, leading to strong path dependence and costly re-tuning as data availability changes. Schedule-Free (SF) methods address this by removing explicit schedules, yet SF-AdamW, the current state-of-the-art anytime optimizer, consistently underperforms well-tuned AdamW baselines. We propose SF-NorMuon, a schedule-free spectral optimizer that closes this gap: with a single hyperparameter configuration, SF-NorMuon matches or exceeds tuned AdamW on 125M and 772M parameter language models across 1--8× Chinchilla horizons, and trails horizon-aware, cosine-scheduled NorMuon by only ~ 0.03 nats, substantially narrowing the gap between anytime and scheduled spectral optimizers. On the theoretical side, we prove a stationarity guarantee for schedule-free spectral dynamics and identify weight decay at the fast iterate as essential for long-horizon stability. SF-NorMuon enables practitioners to obtain high-quality checkpoints at any point during training without committing to a horizon in advance. Fully matching horizon-aware spectral optimizers without a schedule remains an open problem, which we highlight as a direction for future work. By closing the performance gap with tuned AdamW baselines, SF-NorMuon makes horizon-free optimization more practical, taking a step towards truly open-ended, continual learning.