Anytime Training with Schedule-Free Spectral Optimization

arXiv:2605.23061v2 Announce Type: replace Abstract: 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 $1$--$8\times$ Chinchilla horizons, and trails horizon-aware, cosine-scheduled NorMuon by only $\sim 0.03$ 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.

Abstract: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 $1$--$8\times$ Chinchilla horizons, and trails horizon-aware, cosine-scheduled NorMuon by only $\sim 0.03$ 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.

Machine Learning (cs.LG); Artificial Intelligence (cs.AI); Optimization and Control (math.OC); Machine Learning (stat.ML)

arXiv:2605.23061 [cs.LG]

(or arXiv:2605.23061v2 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2605.23061 Focus to learn more arXiv-issued DOI via DataCite

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Anytime Training with Schedule-Free Spectral Optimization

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