A Statistical Viewpoint on Adam, Shampoo and Muon

Efficient stochastic optimization typically integrates an update direction that performs well in the deterministic regime with a mechanism adapting to stochastic perturbations. While Adam uses adaptive moment estimates to promote stability, Muon utilizes the weight layers' matrix structure via orthogonalized momentum, showing superior performance in large language model training. We propose a new optimizer and a diagonal extension, NAMO and NAMO-D, providing the first principled integration of orthogonalized momentum with norm-based Adam-type noise adaptation. NAMO scales orthogonalized momentum using a single adaptive stepsize, preserving orthogonality while improving upon Muon at negligible additional cost. NAMO-D instead right-multiplies orthogonalized momentum by a diagonal matrix with clamped entries. This design enables neuron-wise noise adaptation and aligns with the common near block-diagonal Hessian structure. Under standard assumptions, we establish optimal convergence rates for both algorithms in the deterministic setting and show that, in the stochastic setting, their convergence guarantees adapt to the noise level of stochastic gradients. Experiments on pretraining GPT-2 models demonstrate improved performance of both NAMO and NAMO-D compared to the AdamW and Muon baselines, with NAMO-D achieving further gains over NAMO via an additional clamping hyperparameter that balances the competing goals of maintaining a well-conditioned update direction and leveraging fine-grained noise adaptation.

Muon is emerging as a promising alternative to AdamW for large-scale neural network training, yet theoretical understanding of its practical implementation remains incomplete, as existing analyses often simplify or omit two key components: (i) practical Newton--Schulz iterations with empirically tuned polynomial coefficients $(3.4445,-4.7750,2.0315)$; and (ii) decoupled weight decay for regularization. In this paper, we provide an optimization interpretation and establish convergence for practical Muon, jointly accounting for both components. Specifically, we interpret practical Muon as right-preconditioned optimization of the original loss with a dynamic weighted $\ell_2$ regularizer that vanishes as stationarity is approached, so that the optimization target remains the original objective. We then establish, to our best knowledge, the first convergence guarantee for practical Muon in the stochastic nonconvex setting, with an $\mathcal{O}(T^{-1/4})$ convergence rate in terms of the expected Frobenius norm of the gradient, improving the dimension dependence of the best known AdamW's convergence rate by a factor of $\sqrt{d}$, where $T$ is the iteration horizon and $d$ is the parameter dimension. Experiments further support the theoretical convergence results.

Orthogonalized momentum updates, as used in Muon-style optimizers, have recently shown strong empirical stability in large-scale deep learning. However, most current orthogonalized methods are still paired with fixed, externally scheduled, or otherwise open-loop magnitude rules, so their scale is not directly calibrated from the realized optimization trajectory. Motivated by the closed-loop perspective behind Lipschitz-free and noise-adaptive methods, we propose OptMuon, a family of adaptive momentum orthogonalization methods for stochastic nonconvex optimization. OptMuon combines Muon-style polar-factor directions with a trajectory-dependent AdaGrad-Norm-type coefficient schedule, so that the update magnitude is determined by the observed gradient and momentum history rather than by a prescribed Lipschitz-dependent rule. The schedule does not use the smoothness constant, the variance level, or the bounded-gradient constant in parameter selection, and its running-maximum correction prevents isolated gradient spikes from causing excessive coefficient collapse. Under lower-boundedness, unbiased stochastic gradients with bounded variance, smoothness, and an almost-sure bounded stochastic-gradient condition, we prove two complementary expected-stationarity guarantees. OptMuon-A achieves the noise-adaptive rate $\tilde{\mathcal O}(T^{-1/2}+σ^{1/2}T^{-1/4})$ under average smoothness, while OptMuon-I achieves $\tilde{\mathcal O}(T^{-1/2}+σ^{1/3}T^{-1/3})$ under individual smoothness. In the zero-noise regime, both bounds automatically reduce to a nearly optimal deterministic first-order rate $\tilde{\mathcal O}(T^{-1/2})$ without manual hyperparameter retuning. These results show that closed-loop scalar adaptation can be combined with Muon-style momentum orthogonalization while retaining noise adaptivity and zero-noise optimality up to logarithmic factors.