CAST: History-Adaptive Spectral Clipping for LLM Pretraining

While spectral-based optimizers like Muon operate directly on the spectrum of updates, standard adaptive methods such as AdamW do not account for the spectral structure of weights and gradients, leaving them vulnerable to two empirical issues in large language model (LLM) training: (i) the optimizer updates can have large spectral norms, potentially destabilizing training and degrading generalization; (ii) stochastic gradient noise can exhibit sparse spectral spikes, with a few dominant singular values much larger than the rest. We propose SPECTRA, a general framework addressing these by (i) post-spectral clipping of updates to enforce spectral-norm constraints (ii) optional pre-spectral clipping of gradients to suppress spectral noise spikes. We prove that post-clipping constitutes a Composite Frank-Wolfe method with spectral-norm constraints and weight regularization. We further analyze how pre-clipping mitigates sparse spectral spikes. We propose efficient soft spectral clipping via Newton-Schulz iterations, avoiding expensive SVD. Experiments on LLM pretraining show SPECTRA uniformly improves validation loss for various optimizers, including AdamW, Signum, Mars, and AdEMAMix, with the best-performing variants achieving state-of-the-art results. Models trained with SPECTRA exhibit smaller weight norms, confirming the link between spectral clipping and regularization.

Orthogonal optimisers such as Muon can substantially accelerate large language model pretraining relative to Adam, yet the mechanism remains incompletely understood. We investigate this through an out-of-sample spectral probing analysis of Transformer loss landscapes. At checkpoints along real training trajectories, we decompose each momentum buffer into its singular directions and estimate the loss-optimal step size along each direction on held-out data. The resulting spectral profile is anisotropic yet stable across batches and training stages, and consistent across the optimisers and model scales: a volatile head operating at the Edge-of-Stability supports a much smaller step size than the tolerant bulk, which permits substantially larger steps. This profile provides a unified spectral allocation account of why Muon outperforms Adam, which outperforms SGD. It also exposes a limitation of Muon's uniform scaling: it still underutilises the bulk. Guided by this finding, we introduce Spectral-Aware Muon (SAMuon), which holds the head at the Muon scale and amplifies the bulk using a static spectral prior. We provide two variants: the complete SAMuon follows the measured profile using a low-rank randomised SVD and the simplified SAMuon-lite uses a two-level approximation via rank-one power iteration. Neither method adds persistent optimiser state or notable extra FLOPs beyond Muon at scale, and the idealised exact-whitening versions of both retain Muon's asymptotic convergence rate under standard assumptions. Across "modded-nanogpt" models from 124M to 1B parameters, both variants outperform tuned AdamW and Muon (Scion implementation) baselines in all evaluated model-scale and batch-size configurations. SAMuon requires 13.3% to 24.0% fewer training tokens to reach the same validation loss as Muon, while SAMuon-lite retains most of this gain with near-zero wall-clock overhead.

Modern LLM optimizers such as Muon often produce weight matrices with higher effective rank than Adam, yet further spectral control has delivered only modest gains. We identify a tension behind this result: concentrated spectra can suppress gradient directions in coupled weight matrices and slow optimization, while constraints maintained throughout training can limit task-specific adaptation and raise the attainable loss floor. We introduce ORCA (Orthogonal Regularization, Cooled After), a minimal optimizer intervention that applies strong but temporary soft orthogonality regularization early in training, then removes it. This allows the weights to benefit from a broader spectrum early on and adapt freely afterward. Across LLaMA, Qwen3, and fine-grained mixture-of-experts models ranging from 130M to 8B parameters, ORCA achieves lower final validation loss than Muon. Its loss reduction relative to Muon matches or exceeds Muon's reduction relative to Adam. Ablations support the early-shaping, later-release design. Further, ORCA requires no architectural changes and adds minimal overhead.