Convergence of Spectral Descent for Non-smooth Optimization

Muon has emerged as a highly effective optimizer for large language model training, often achieving superior convergence and performance compared with the widely adopted Adam and AdamW optimizers. Nevertheless, Muon is prone to training instability due to its spectral flattening, manifested by loss spikes and unbounded growth of model weights. Existing approaches primarily rely on weight or attention-logit clipping, which require architecture-specific modifications and do not directly address instability across all model components. We propose MomentUm SpEctral Clipping (Musec), which replaces Muon's spectral flattening with spectral clipping: rather than setting all singular values of the momentum matrix to approximately one, Musec clips singular values that exceed a threshold while preserving the underlying spectral structure of the momentum. Our strategy provides an optimizer-level, architecture-agnostic mechanism for stabilizing Muon training. Theoretically, we establish convergence guarantees for Musec in nonconvex nonsmooth stochastic optimization. Practically, we develop Soft Musec, an efficient implementation that uses a smooth spectral saturation function approximated by coupled Newton-Schulz iterations. Empirically, Soft Musec consistently improves training stability over existing Muon variants across a wide range of learning rates and model sizes, remaining stable in settings where existing Muon variants diverge while matching their performance under well-tuned configurations.

Muon can be interpreted as optimizing a linear local objective over a spectral-norm ball. This gives a matrix-sign update that preserves the singular directions of the gradient and assigns the same magnitude to all active singular modes. We ask whether these two properties remain optimal when local curvature is taken into account. To answer this question, we keep Muon's spectral-norm constraint unchanged and replace the linear local model with a quadratic one. We call the resulting method \emph{Quadratic Spectral Descent} (QSD). We show that curvature can change both the singular values and the singular directions of the optimal update. To make QSD practical, we approximate curvature with Kronecker-factored statistics and solve the constrained quadratic with a small number of Frank--Wolfe steps, each of which has a closed-form matrix-sign subproblem. We further provide an optimality certificate, a comparison with Muon under the same quadratic surrogate, and an $O(1/K)$ convergence rate for the inner solver. Experiments on GPT pre-training show that QSD consistently improves validation loss over Muon and recent Muon variants, and reduces wall-clock training time by up to $8.49\%$ at matched validation loss.

Muon orthogonalizes the momentum buffer before each update, replacing its singular values with ones via Newton-Schulz iterations. This simple change lets Muon tolerate far larger learning rates and converge faster than other optimizers, but why? We show that the mechanism is spectral flattening, and develop two results around it. First, we prove that Muon's maximal stable step size scales with the average singular value of the gradient rather than the largest, which bottlenecks standard gradient descent. Second, we recast Muon as a preconditioned gradient method and show, under a Kronecker-factored curvature model, that it improves the effective convergence factor, with the improvement controlled by the spectrum of the gradient covariance. Extensive experiments validate both results: Muon remains stable at learning rates that cause SGD to diverge within the first few iterations, and reaches accuracy milestones several epochs earlier even at identical step sizes. Taken together, our results offer a principled, geometric explanation for Muon's empirical success.