Anytime Training with Schedule-Free Spectral Optimization

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. 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. By closing the performance gap with tuned baselines, SF-NorMuon makes horizon-free optimization more practical, taking a step towards truly open-ended, continual learning.

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BibTeX

@article{apte2026anytime,
  title = {Anytime Training with Schedule-Free Spectral Optimization},
  author = {Anuj Apte and Pranav Deshpande and Niraj Kumar and Shouvanik Chakrabarti and Junhyung Lyle Kim},
  year = {2026},
  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},
  url = {https://arxiv.org/abs/2605.23061},
  keywords = {cs.LG, cs.AI, math.OC, stat.ML},
  eprint = {2605.23061},
  archiveprefix = {arXiv},
}

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