Inexact Muon Update: Optimization & Uncertainty

• The update leverages approximate orthogonalization in the Muon optimizer, balancing fast iterative matrix methods with controlled additive error models.
• It quantifies inexactness through explicit convergence bounds, linking Newton–Schulz iterations to tuning learning rates and momentum for robust optimization.
• Empirical NanoGPT experiments show that increasing matrix iterations improves training speed and reduces validation loss, mirroring challenges in muonic magnetic moment predictions.

The inexact Muon update encompasses both the theoretical and practical ramifications of using approximate orthogonalization within the Muon optimizer, a geometry-aware alternative to AdamW, and finds analogy with the persistent uncertainty in precision predictions of the muonic anomalous magnetic moment due to inexact hadronic contributions and related numerical procedures.

1. Orthogonalization in Muon Optimization and the Inexact Update

The Muon optimizer is designed for large-scale neural network training, leveraging matrix geometry by performing steepest descent steps in the spectral norm. Each weight matrix $W \in \mathbb{R}^{n \times m}$ is updated by combining momentum smoothing and orthogonal projections via

$$M^k = (1-\alpha) M^{k-1} + \alpha G^k, \quad D^k = \text{polar}(-M^k),\quad W^{k+1} = W^k + \gamma D^k,$$

where $G^k$ is the gradient and $D^k$ is ideally the polar factor. In practice, exact computation using singular value decomposition (SVD) is replaced by fast approximations, typically a bounded number of Newton–Schulz or PolarExpress iterations, resulting in inexact orthogonalization (Shulgin et al., 22 Oct 2025).

The “inexact Muon update” refers precisely to employing an approximate $\hat D^k$ in place of the exact $D^k$, inducing error at each iteration.

2. LMO Perspective and Additive Error Modeling

Inexactness is formalized under the Linear Minimization Oracle (LMO) framework: $d^k = \arg\min_{\|d\|\le1}\langle m^k, d \rangle$ but in practice yields an approximation $\hat d^k$ satisfying the additive error assumption: $\|\hat d^k-d^k\| \le \delta_k < 1$ where $\delta_k$ is a precision parameter controlled by the number of matrix iterations in the algorithm. Crucially, $\hat d^k$ need not satisfy $\|\hat d^k\| \le 1$ (Shulgin et al., 22 Oct 2025).

This model quantifies the deviation arising from fast, iterative matrix procedures underpinning Muon’s computational efficiency, differentiating idealized and practical optimization behaviors.

3. Explicit Convergence Bounds and Parameter Coupling

Rigorous bounds are established that relate optimization convergence rates to the inexactness $\delta_k$:

• Deterministic Setting: For a $C^1$ function $f$ with $L$-Lipschitz gradients,

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

where performance degrades as $\delta_k$ increases.

• Stochastic Setting with Momentum: With unbiased stochastic gradients and Polyak momentum,

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

Optimal step size and momentum are shown to scale as

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

demonstrating a fundamental coupling: as approximation error increases, the optimal step size decreases while the optimal momentum increases (Shulgin et al., 22 Oct 2025).

4. Empirical Verification and Hyperparameter Coupling

Extensive NanoGPT experiments confirm that increasing the number of matrix iterations (reducing $\delta$) accelerates training and lowers validation loss. Key findings:

• Fewer iterations (higher $\delta$) require smaller learning rates and larger momentum coefficients for optimal convergence.
• Heatmaps reveal narrowing stable regions for hyperparameters as precision drops.
• There are diminishing returns beyond 5–8 matrix iterations; low precision (e.g., $p=2$) necessitates $\gamma \approx 0.03$, while higher precision ($p=5$) admits $\gamma \approx 0.05$ and broader stable ranges.

# PolarExp steps $p$	1	3	5	8
val-loss	3.0675	3.0109	3.0036	3.0023
time (min)	652.2	660.4	655.7	675.0

A plausible implication is that computational budgets may be strategically allocated between matrix approximation precision and training duration, using degradation bounds to guide trade-offs (Shulgin et al., 22 Oct 2025).

5. Practical Co-Tuning Guidelines

The inexact Muon update requires explicit co-tuning of hyperparameters based on $\delta$:

• Decrease the learning rate by $(1+\delta)^{-1/4}$ when reducing matrix approximation fidelity.
• Increase momentum coefficient by $\sqrt{1+\delta}$ correspondingly.
• For robust training, employ $p \ge 3$–5 iterations, then conduct a coarse sweep around predicted optimal parameters.

For example, after tuning at high precision $(\delta \approx 0)$, if switching to lower precision: $$\gamma' \approx \gamma_0/(1+\delta)^{1/4},\quad \alpha' \approx \min(0.999,\,\alpha_0\sqrt{1+\delta})$$

This decouples implementation fidelity from theoretical guarantees, making the approximation step a first-class parameter in optimizer scheduling (Shulgin et al., 22 Oct 2025).

6. Relation to Hadronic Uncertainties and Broader Interpretation

"Inexact Muon update" derives analogous significance in computational physics and particle theory. Precision calculations of the muon anomalous magnetic moment $a_\mu$ within the Standard Model are limited by theoretical inexactness—stemming chiefly from hadronic vacuum-polarization and light-by-light scattering contributions. These uncertainties manifest in dispersion-relations and data-driven fits, with error reductions arising from incremental improvements in low-energy channel measurements and lattice QCD cross-checks (Teubner et al., 2010, Jegerlehner, 2018).

The persistent $4\sigma$ discrepancy between $a_\mu^{\text{EXP}}$ and $a_\mu^{\text{SM}}$ is dominated by inexact theoretical contributions: $$\Delta a_\mu = (31.6 \pm 7.9) \times 10^{-10} \quad [1001.5401]$$ and forecasts suggest future experiments could raise this to $6$–$10\sigma$ if theoretical inexactness is halved (Jegerlehner, 2018). This highlights the critical need for precise modeling of error propagation—whether in numerical optimization or quantum field theory predictions.

7. Outlook

Both in neural optimization and high-energy phenomenology, inexactness—be it from fast matrix approximations or hadronic physics—requires systematic quantification and purposeful parameter adaptation. The inexact Muon update transitions implementation details into theory-level considerations, demanding explicit tuning for robust convergence and statistical significance. In both domains, ongoing methodological advances focus on reducing inexactness, enhancing reliability, and informing higher-order corrections or scheduling strategies (Shulgin et al., 22 Oct 2025, Teubner et al., 2010, Jegerlehner, 2018).