---
title: 'Inexact Muon Update: Optimization & Uncertainty'
url: https://www.emergentmind.com/topics/inexact-muon-update
type: topic
---

# Inexact Muon Update: Optimization & Uncertainty

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 [2510.19933].

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 $\|\hat d^k\| \le 1$ [2510.19933].

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 [2510.19933].

## 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 [2510.19933].

## 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 [2510.19933].

## 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 [1001.5401, 1804.07409].

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 [1804.07409]. 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 [2510.19933, 1001.5401, 1804.07409].

Source: https://www.emergentmind.com/topics/inexact-muon-update