Beyond the Ideal: Analyzing the Inexact Muon Update (2510.19933v1)

Abstract: The Muon optimizer has rapidly emerged as a powerful, geometry-aware alternative to AdamW, demonstrating strong performance in large-scale training of neural networks. However, a critical theory-practice disconnect exists: Muon's efficiency relies on fast, approximate orthogonalization, yet all prior theoretical work analyzes an idealized, computationally intractable version assuming exact SVD-based updates. This work moves beyond the ideal by providing the first analysis of the inexact orthogonalized update at Muon's core. We develop our analysis within the general framework of Linear Minimization Oracle (LMO)-based optimization, introducing a realistic additive error model to capture the inexactness of practical approximation schemes. Our analysis yields explicit bounds that quantify performance degradation as a function of the LMO inexactness/error. We reveal a fundamental coupling between this inexactness and the optimal step size and momentum: lower oracle precision requires a smaller step size but larger momentum parameter. These findings elevate the approximation procedure (e.g., the number of Newton-Schulz steps) from an implementation detail to a critical parameter that must be co-tuned with the learning schedule. NanoGPT experiments directly confirm the predicted coupling, with optimal learning rates clearly shifting as approximation precision changes.

• The paper's main contribution is the first formal convergence analysis of the Muon optimizer under inexact orthogonalization, quantifying convergence degradation.
• It introduces a rigorous LMO framework with an additive error model, extending the analysis to stochastic settings with momentum for practical hyperparameter tuning.
• Empirical results show that increasing iterative orthogonalization precision (e.g., using Newton-Schulz or PolarExpress) accelerates convergence and improves training stability.

Formal Analysis of Inexact Muon Updates in Geometry-Aware Optimization

The paper "Beyond the Ideal: Analyzing the Inexact Muon Update" (2510.19933) presents a rigorous theoretical and empirical investigation into the practical implementation of the Muon optimizer, a geometry-aware alternative to AdamW. The central contribution is the first formal convergence analysis of Muon under realistic, inexact orthogonalization, bridging the gap between idealized theory and practical deployment. The work is situated within the Linear Minimization Oracle (LMO) framework, introducing an additive error model to capture the inexactness of iterative matrix orthogonalization schemes such as Newton-Schulz and PolarExpress.

Motivation and Problem Statement

Muon leverages matrix geometry by applying momentum and orthogonalization to weight updates, theoretically requiring exact SVD-based polar decomposition. However, practical implementations rely on fast, iterative approximations (e.g., Newton-Schulz), introducing non-negligible error. Prior analyses assumed access to an exact LMO, which is computationally infeasible for large-scale neural networks. This disconnect leaves the real-world behavior of Muon unexplained and unoptimized.

The paper formalizes the inexact update as:

$\|\hat{d}^k - d^k\| \leq \delta_k$

where $d^k$ is the exact LMO solution and $\hat{d}^k$ is the inexact direction produced by iterative approximation. The error $\delta_k$ is directly linked to the number of iterations in the approximation algorithm.

Theoretical Framework and Main Results

Deterministic Setting

The analysis begins with unconstrained optimization in a normed space, assuming $L$-smoothness. The update rule is:

$x^{k+1} = x^k + \gamma_k \hat{d}^k$

with $\hat{d}^k$ satisfying the additive error bound. The main convergence result (Theorem 1) quantifies the degradation in convergence rate as a function of $\delta_k$:

$$\min_{0 \leq k < K} \|\nabla f(x^k)\|_* \leq \frac{\Delta^0 + \frac{L}{2} \sum_{k=0}^{K-1} \gamma_k^2 (1+\delta_k)^2}{\sum_{k=0}^{K-1} \gamma_k (1-\delta_k)}$$

where $\Delta^0 = f(x^0) - f^*$. The optimal constant step size is shown to scale as $\gamma^* \propto 1/(1+\delta)$, and the best achievable rate is degraded by a factor of $(1+\delta)/(1-\delta)$.

Stochastic Setting with Momentum

Extending to stochastic gradients and momentum, the analysis introduces additional terms for variance and momentum error. The main result (Theorem 2) provides an upper bound on the average expected gradient norm:

$$\frac{1}{K} \sum_{k=1}^K \mathbb{E}\|\nabla f(x^k)\|_* \leq \frac{1}{1-\delta} \left[ \frac{\Delta^0}{K\gamma} + 2\rho\sigma\left(\frac{1}{\alpha K} + \sqrt{\alpha}\right) + L\gamma\left(\frac{7+3\delta}{2} + \frac{2(1+\delta)}{\alpha}\right) \right]$$

where $\sigma^2$ is the variance bound and $\rho$ is the norm compatibility constant. The optimal parameters are derived as:

• $\gamma^* \propto 1/(1+\delta)^{1/4}$
• $\alpha^* \propto \sqrt{1+\delta}$

This reveals a fundamental coupling: lower LMO precision (higher $\delta$) requires a smaller step size and larger momentum.

Extensions

The framework is generalized to $(L^0, L^1)$-smoothness and layer-wise analysis, accommodating heterogeneous network structures and non-Euclidean geometries. The layer-wise extension demonstrates that the global convergence rate is bottlenecked by the layer with the worst LMO precision.

Empirical Validation

Experiments on NanoGPT (FineWeb) and CIFAR-10 CNNs validate the theoretical predictions:

• Convergence Degradation: Increasing the number of Newton-Schulz or PolarExpress iterations (i.e., reducing $\delta$) yields faster convergence and lower final loss.

Figure 1: Validation loss behavior; more iterations (smaller $\delta$) drive faster convergence.

• Hyperparameter Coupling: Hyperparameter sweeps show that the optimal learning rate shifts lower as LMO precision decreases, and the region of stable performance narrows.

Figure 2: Training loss drops faster as Newton-Schulz precision increases.

Figure 3: Improvements saturate after $\sim$5 PolarExp iterations.

• LMO Error Evolution: The LMO error remains nearly constant for a fixed number of iterations, with higher iteration counts maintaining lower error throughout training.

  Figure 4: LMO error evolution across training steps for different numbers of PolarExpress iterations.

• Hyperparameter Stability: Higher LMO precision increases optimizer robustness to step size and momentum choices, as shown by accuracy heatmaps.

  Figure 5: Full final test accuracy heatmaps on CIFAR-10 across a grid of step sizes and momentum values for different numbers of Newton-Schulz iterations.

Implementation Guidance

Practical Recommendations

• LMO Iteration Count: The number of Newton-Schulz or PolarExpress iterations should be treated as a core hyperparameter, not a mere implementation detail. It must be co-tuned with learning rate and momentum.
• Step Size and Momentum: For lower LMO precision, decrease the step size and increase the momentum parameter to maintain stability and convergence.
• Layer-wise Tuning: Consider allocating more computational resources to layers sensitive to LMO inexactness, as the global rate is bottlenecked by the worst-performing layer.
• Approximation Schemes: Employ advanced iterative schemes (e.g., PolarExpress, CANS) to minimize $\delta$ efficiently, directly improving convergence rates.

Computational Considerations

• Resource Trade-offs: Increasing LMO precision improves convergence but incurs higher computational cost per iteration. Empirical results show diminishing returns beyond 5–8 iterations.
• Scaling: The analysis holds for large-scale models and batch sizes, with empirical validation on 124M parameter NanoGPT and CIFAR-10 CNNs.

Implications and Future Directions

The explicit characterization of the coupling between LMO inexactness and optimizer hyperparameters provides a principled basis for tuning geometry-aware optimizers in practice. The results suggest that further research into adaptive LMO precision, layer-wise resource allocation, and improved approximation algorithms could yield substantial gains in both efficiency and stability for large-scale neural network training.

The theoretical framework generalizes to arbitrary norms and smoothness models, opening avenues for application in heterogeneous architectures and non-Euclidean optimization landscapes. The empirical findings highlight the necessity of moving beyond idealized analyses to fully understand and optimize real-world training dynamics.

Conclusion

This work closes a critical gap in the theory and practice of geometry-aware optimization by providing the first formal analysis of inexact Muon updates. The explicit dependence of convergence rates on LMO precision, step size, and momentum establishes a new paradigm for hyperparameter tuning and algorithm design in large-scale deep learning. The results are directly actionable for practitioners seeking to deploy Muon and related optimizers efficiently, and set the stage for future advances in scalable, geometry-aware optimization.