aiqbench analysis
This research investigates the theoretical limits of Muon, a matrix-aware optimizer that orthogonalizes momentum matrices, specifically addressing its behavior in non-smooth optimization landscapes. The authors demonstrate that fixed momentum factors can prevent convergence to global optima but prove that adaptive momentum strategies restore asymptotic convergence even for convex Lipschitz objectives. By proposing MAGD, the study offers a practical alternative that combines orthogonalized momentum with gradient weighting to achieve optimal dimension-dependent rates across synthetic and LLM pretraining tasks.
Nonsmooth Optimization via Orthogonalized Momentum
arXiv:2609.13677v1 Announce Type: cross Abstract: Modern real application problems involve matrix-valued parameters, yet conventional optimizers treat them as vectors, thereby motivating matrix-aware methods that exploit input-output geometry, such as Muon which orthogonalizes the momentum matrices before parameter updates. Its empirical success raises a conceptual question: can orthogonalized momentum remain effective beyond smooth optimization? This paper studies this question for locally Lipschitz functions using a generalized derivative framework compatible with backpropagation. Our first contribution is to identify a key limitation: for every fixed momentum factor $\beta\in[0,1)$, Muon can fail to approach the global optimal solution of a convex Lipschitz objective from almost every initialization, when step sizes adapt to the full gradient history. The failure can occur even along bounded iterates. Our example is inspired by the one of Parshakova et al. which only covers $\beta\in[0,\frac{1}{2})$. Then, we show that the obstruction lies in fixed momentum rather than orthogonalization. Indeed, when the momentum factor is adaptive and approaches 1 together with a vanishing step size, Muon recovers asymptotic convergence for nonconvex nonsmooth optimization under the boundedness and regularity conditions. Moreover, we propose MAGD, which combines orthogonalized momentum with gradient, weighted based on their relative progress. MAGD retains asymptotic convergence in nonconvex settings and achieves an $O(\min\{m,n\}\epsilon^{-2})$ rate in convex settings. A lower bound shows the optimal dimension dependence. Experiments on synthetic problems, image classification, and LLM pretraining show MAGD is a simple and practical alternative to Muon. Together, our results characterize when orthogonalized momentum fails without smoothness and how it can be made reliable and we hope that the analysis may be useful more broadly.
Abstract:Modern real application problems involve matrix-valued parameters, yet conventional optimizers treat them as vectors, thereby motivating matrix-aware methods that exploit input-output geometry, such as Muon which orthogonalizes the momentum matrices before parameter updates. Its empirical success raises a conceptual question: can orthogonalized momentum remain effective beyond smooth optimization? This paper studies this question for locally Lipschitz functions using a generalized derivative framework compatible with backpropagation. Our first contribution is to identify a key limitation: for every fixed momentum factor $\beta\in[0,1)$, Muon can fail to approach the global optimal solution of a convex Lipschitz objective from almost every initialization, when step sizes adapt to the full gradient history. The failure can occur even along bounded iterates. Our example is inspired by the one of Parshakova et al. which only covers $\beta\in[0,\frac{1}{2})$. Then, we show that the obstruction lies in fixed momentum rather than orthogonalization. Indeed, when the momentum factor is adaptive and approaches 1 together with a vanishing step size, Muon recovers asymptotic convergence for nonconvex nonsmooth optimization under the boundedness and regularity conditions. Moreover, we propose MAGD, which combines orthogonalized momentum with gradient, weighted based on their relative progress. MAGD retains asymptotic convergence in nonconvex settings and achieves an $O(\min\{m,n\}\epsilon^{-2})$ rate in convex settings. A lower bound shows the optimal dimension dependence. Experiments on synthetic problems, image classification, and LLM pretraining show MAGD is a simple and practical alternative to Muon. Together, our results characterize when orthogonalized momentum fails without smoothness and how it can be made reliable and we hope that the analysis may be useful more broadly.
Optimization and Control (math.OC); Machine Learning (cs.LG)
arXiv:2609.13677 [math.OC]
(or arXiv:2609.13677v1 [math.OC] for this version)
https://doi.org/10.48550/arXiv.2609.13677 Focus to learn more arXiv-issued DOI via DataCite
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