FedMuon: Federated Matrix Optimization

• FedMuon is a family of federated optimization methods that adapts Muon’s matrix orthogonalization to improve updates for matrix-valued parameters while mitigating client drift.
• It employs mechanisms such as momentum aggregation, local-global alignment, and control variates to counter aggregation bias and stabilize training in non-IID environments.
• Empirical studies show FedMuon outperforms conventional methods in image classification, language tasks, and federated minimax settings, demonstrating faster convergence and better accuracy.

Federated Muon Optimizer, usually abbreviated FedMuon, denotes a family of federated optimization methods that adapt Muon—a matrix-structured optimizer based on orthogonalized momentum updates—to client-server training. In the recent literature, the label does not identify a single canonical algorithm. Instead, it covers several related constructions: a communication-efficient federated learner with momentum aggregation and local-global alignment, a bias-corrected linear-minimization-oracle formulation, a full-participation local-momentum method with periodic averaging, and the minimax variant FedMuon-DA for federated stochastic minimax optimization under heavy-tailed noise (Liu et al., 31 Oct 2025, Takezawa et al., 30 Sep 2025, Zhang et al., 4 Oct 2025, Zhang et al., 6 Nov 2025). What unifies these methods is the attempt to preserve Muon’s matrix geometry while controlling client drift, aggregation bias, or heavy-tailed instability.

1. Muon as the underlying matrix optimizer

Muon is defined for parameters represented as matrices. In its centralized form, for a matrix parameter $\mathbf W_{t-1}$, momentum coefficient $\mu$, and learning rate $\eta_t$, it updates

$$\mathbf M_t = \mu \mathbf M_{t-1} + \nabla \mathcal L_t(\mathbf W_{t-1}), \qquad \mathbf O_t = \text{Newton-Schulz}(\mathbf M_t), \qquad \mathbf W_t = \mathbf W_{t-1} - \eta_t \mathbf O_t.$$

If $$\mathbf M_t = \mathbf U \mathbf \Sigma \mathbf V^\top$$ is the singular value decomposition, then the target orthogonalized direction is $\mathbf U \mathbf V^\top$, i.e. the polar factor of the momentum matrix (Liu et al., 24 Feb 2025).

A complementary formulation treats Muon as a linear minimization oracle over a spectral-norm unit ball. In that view,

$$\operatorname{lmo}(X) = \arg\min_{Y:\|Y\|\le 1}\langle X,Y\rangle,$$

and when the norm is the spectral norm and $X=U\Sigma V^\top$, the exact Muon oracle is $-UV^\top$ (Takezawa et al., 30 Sep 2025). This makes Muon a geometry-changing optimizer rather than a coordinatewise adaptive one: it discards singular values and preserves singular subspaces.

That matrix geometry is the reason Muon is attractive for federated learning. The common motivation across FedMuon papers is that standard local optimizers such as SGD, Adam, or AdamW act elementwise, whereas many neural-network parameters—linear layers, attention projections, MLP blocks, LoRA factors, and related components—are naturally matrix-valued. The FedMuon literature therefore asks whether matrix orthogonalization can make each local client step more effective without destabilizing global aggregation (Liu et al., 31 Oct 2025, Liu et al., 24 Feb 2025).

2. Distinct meanings of “FedMuon” in the literature

By 2025, several non-identical federated methods were published under the name “FedMuon” or closely related terminology.

Method	Optimization setting	Defining mechanism
FedMuon (Liu et al., 31 Oct 2025)	Standard federated minimization with partial participation	Momentum aggregation and local-global alignment
FedMuon (Takezawa et al., 30 Sep 2025)	Federated LMO-based optimization	SCAFFOLD-style bias correction before applying the LMO
FedMuon (Zhang et al., 4 Oct 2025)	Full-participation nonconvex federated optimization over matrix variables	Local Muon updates with periodic averaging of parameters and momentum
FedMuon-DA (Zhang et al., 6 Nov 2025)	Federated stochastic minimax optimization under heavy-tailed noise	Primal-dual Muon updates with control variates and server drift aggregation

The underlying objectives differ substantially. In the cross-device setting of (Liu et al., 31 Oct 2025) and (Takezawa et al., 30 Sep 2025), the goal is the standard federated objective

$$f(\boldsymbol{x})=\frac{1}{N}\sum_{i=1}^N f_i(\boldsymbol{x}), \qquad f_i(\boldsymbol{x})=\mathbb E_{\xi_i\sim \mathcal D_i}[F_i(\boldsymbol{x};\xi_i)].$$

In contrast, (Zhang et al., 6 Nov 2025) studies a federated stochastic minimax problem

$\mu$0

with a nonconvex-PL structure in which the primal variable is nonconvex and the dual maximization satisfies a Polyak–Łojasiewicz condition. That paper is explicit that the setting is not standard ERM minimization, and that its symbol $\mu$1 denotes the number of local update steps per round rather than a participation probability (Zhang et al., 6 Nov 2025).

This terminological multiplicity is important. “FedMuon” can refer either to a federated Muon method for ordinary nonconvex minimization, to an LMO-corrected federated method whose matrix specialization is Muon, or to a primal-dual minimax optimizer that uses Muon only in local matrix updates.

3. Core algorithmic mechanisms

A recurring misconception is that FedMuon is simply FedAvg with Muon as the local optimizer. The literature argues that this is not sufficient. Two failure modes recur: first, Muon’s orthogonalization is nonlinear, so local directions do not average in the same way as gradients; second, non-IID clients can induce geometry-specific drift because each client orthogonalizes a momentum matrix fitted to its own local distribution (Takezawa et al., 30 Sep 2025, Liu et al., 31 Oct 2025).

In “FedMuon: Accelerating Federated Learning with Matrix Orthogonalization” (Liu et al., 31 Oct 2025), the solution is built around two mechanisms. The first is momentum aggregation: each participating client initializes its local momentum from a server-aggregated momentum,

$\mu$2

The second is local-global alignment: instead of stepping only along the local Muon direction $\mu$3, the client uses a convex combination with a server-estimated global direction $\mu$4,

$\mu$5

The server then aggregates model deltas, updates $\mu$6 from the average drift, and broadcasts the new model, momentum, and global direction.

In the LMO-based formulation of (Takezawa et al., 30 Sep 2025), the central issue is different. The paper shows that straightforward local Muon inside FedAvg, called LocalMuon, is biased because

$\mu$7

Its FedMuon therefore introduces client and server control variates $\mu$8 and $\mu$9, and applies the oracle to a corrected quantity,

$\eta_t$0

The paper explicitly notes that if the LMO is removed and $\eta_t$1, the method reduces to vanilla SCAFFOLD.

The FedMuon of (Zhang et al., 4 Oct 2025) is structurally simpler. It uses local momentum matrices $\eta_t$2, computes their SVDs,

$\eta_t$3

forms the orthonormalized direction

$\eta_t$4

and updates

$\eta_t$5

Every $\eta_t$6 iterations, it synchronizes both parameters and momentum: $\eta_t$7 This version is closest to a local-momentum FedAvg-M analogue with Muon-style local directions.

FedMuon-DA (Zhang et al., 6 Nov 2025) extends the pattern to saddle-point optimization. Each client maintains local primal and dual variables $\eta_t$8, local momenta corrected by global-minus-local control variates, orthonormalizes the resulting momentum matrices by SVD, and performs descent in $\eta_t$9 and ascent in $$\mathbf M_t = \mu \mathbf M_{t-1} + \nabla \mathcal L_t(\mathbf W_{t-1}), \qquad \mathbf O_t = \text{Newton-Schulz}(\mathbf M_t), \qquad \mathbf W_t = \mathbf W_{t-1} - \eta_t \mathbf O_t.$$0: $$\mathbf M_t = \mu \mathbf M_{t-1} + \nabla \mathcal L_t(\mathbf W_{t-1}), \qquad \mathbf O_t = \text{Newton-Schulz}(\mathbf M_t), \qquad \mathbf W_t = \mathbf W_{t-1} - \eta_t \mathbf O_t.$$1 The server does not perform ordinary FedAvg-style final-model averaging. Instead, it updates global variables using scaled averages of local drifts and separately updates global momentum and global control variates.

4. Convergence theory and robustness claims

The theoretical literature on FedMuon emphasizes that orthonormalized matrix directions alter both the geometry of descent and the way federated drift is controlled.

The strongest negative result appears in (Takezawa et al., 30 Sep 2025). There, LocalMuon is shown not to converge to a stationary point in general: the paper proves that there exist convex client objectives such that, for any number of rounds, LocalMuon remains at the initial point while

$$\mathbf M_t = \mu \mathbf M_{t-1} + \nabla \mathcal L_t(\mathbf W_{t-1}), \qquad \mathbf O_t = \text{Newton-Schulz}(\mathbf M_t), \qquad \mathbf W_t = \mathbf W_{t-1} - \eta_t \mathbf O_t.$$2

For the corrected FedMuon, the paper then gives stationarity guarantees for both exact and approximate Muon. With exact LMO and full participation, the dominant term in the average stationarity bound has the form

$$\mathbf M_t = \mu \mathbf M_{t-1} + \nabla \mathcal L_t(\mathbf W_{t-1}), \qquad \mathbf O_t = \text{Newton-Schulz}(\mathbf M_t), \qquad \mathbf W_t = \mathbf W_{t-1} - \eta_t \mathbf O_t.$$3

while with Newton–Schulz approximation the rate expression retains the same form but the guaranteed stationarity is measured in a Schatten-$$\mathbf M_t = \mu \mathbf M_{t-1} + \nabla \mathcal L_t(\mathbf W_{t-1}), \qquad \mathbf O_t = \text{Newton-Schulz}(\mathbf M_t), \qquad \mathbf W_t = \mathbf W_{t-1} - \eta_t \mathbf O_t.$$4 norm whose exponent depends on the number of Newton–Schulz iterations. A distinctive conclusion is that FedMuon converges for any number of Newton–Schulz iterations $$\mathbf M_t = \mu \mathbf M_{t-1} + \nabla \mathcal L_t(\mathbf W_{t-1}), \qquad \mathbf O_t = \text{Newton-Schulz}(\mathbf M_t), \qquad \mathbf W_t = \mathbf W_{t-1} - \eta_t \mathbf O_t.$$5, while larger $$\mathbf M_t = \mu \mathbf M_{t-1} + \nabla \mathcal L_t(\mathbf W_{t-1}), \qquad \mathbf O_t = \text{Newton-Schulz}(\mathbf M_t), \qquad \mathbf W_t = \mathbf W_{t-1} - \eta_t \mathbf O_t.$$6 sharpens the descent geometry from Frobenius-type toward trace-norm behavior (Takezawa et al., 30 Sep 2025).

The FedMuon of (Liu et al., 31 Oct 2025) targets standard nonconvex federated learning and emphasizes removal of the heterogeneity term from the dominant stochastic bound. The paper compares Local SGD and Local Muon with a rate involving both $$\mathbf M_t = \mu \mathbf M_{t-1} + \nabla \mathcal L_t(\mathbf W_{t-1}), \qquad \mathbf O_t = \text{Newton-Schulz}(\mathbf M_t), \qquad \mathbf W_t = \mathbf W_{t-1} - \eta_t \mathbf O_t.$$7 and $$\mathbf M_t = \mu \mathbf M_{t-1} + \nabla \mathcal L_t(\mathbf W_{t-1}), \qquad \mathbf O_t = \text{Newton-Schulz}(\mathbf M_t), \qquad \mathbf W_t = \mathbf W_{t-1} - \eta_t \mathbf O_t.$$8, whereas FedMuon achieves

$$\mathbf M_t = \mu \mathbf M_{t-1} + \nabla \mathcal L_t(\mathbf W_{t-1}), \qquad \mathbf O_t = \text{Newton-Schulz}(\mathbf M_t), \qquad \mathbf W_t = \mathbf W_{t-1} - \eta_t \mathbf O_t.$$9

without requiring the heterogeneity term $$\mathbf M_t = \mathbf U \mathbf \Sigma \mathbf V^\top$$0 in the final rate statement. The dominant stochastic term therefore exhibits linear speedup in the number of participating clients $$\mathbf M_t = \mathbf U \mathbf \Sigma \mathbf V^\top$$1 and the number of local steps $$\mathbf M_t = \mathbf U \mathbf \Sigma \mathbf V^\top$$2 (Liu et al., 31 Oct 2025).

In “On Provable Benefits of Muon in Federated Learning” (Zhang et al., 4 Oct 2025), the analysis is for full participation and uses a bounded heterogeneity assumption

$$\mathbf M_t = \mathbf U \mathbf \Sigma \mathbf V^\top$$3

Under bounded variance, the average stationarity measure satisfies an $$\mathbf M_t = \mathbf U \mathbf \Sigma \mathbf V^\top$$4 rate, and the corresponding communication complexity is $$\mathbf M_t = \mathbf U \mathbf \Sigma \mathbf V^\top$$5. Under heavy-tailed noise with only a bounded $$\mathbf M_t = \mathbf U \mathbf \Sigma \mathbf V^\top$$6-th moment for $$\mathbf M_t = \mathbf U \mathbf \Sigma \mathbf V^\top$$7, the rate becomes

$$\mathbf M_t = \mathbf U \mathbf \Sigma \mathbf V^\top$$8

with communication complexity

$$\mathbf M_t = \mathbf U \mathbf \Sigma \mathbf V^\top$$9

The paper’s central interpretation is that the local orthonormalized direction has bounded Frobenius norm,

$\mathbf U \mathbf V^\top$0

which allows the recommended hyperparameters to be chosen without depending on problem-specific constants such as the smoothness $\mathbf U \mathbf V^\top$1, the noise scale $\mathbf U \mathbf V^\top$2, or the heterogeneity level $\mathbf U \mathbf V^\top$3 (Zhang et al., 4 Oct 2025).

FedMuon-DA (Zhang et al., 6 Nov 2025) extends rigorous analysis to heavy-tailed federated minimax optimization. The stochastic gradient noise is assumed unbiased with only an $\mathbf U \mathbf V^\top$4-th moment bound,

$\mathbf U \mathbf V^\top$5

Under a nonconvex-PL structure and without assuming bounded client heterogeneity, the paper proves a convergence rate

$\mathbf U \mathbf V^\top$6

with communication complexity

$\mathbf U \mathbf V^\top$7

It also gives a key drift lemma: $\mathbf U \mathbf V^\top$8 showing how Muon orthonormalization bounds actual update magnitudes even when stochastic gradients themselves are heavy-tailed. The paper positions these guarantees as the first rigorous federated minimax results under heavy-tailed noise (Zhang et al., 6 Nov 2025).

5. Empirical evidence across tasks and architectures

The empirical record is distributed across several application domains rather than a single benchmark suite.

The most extensive standard FL study appears in (Liu et al., 31 Oct 2025). On CIFAR-100 with Dirichlet heterogeneity $\mathbf U \mathbf V^\top$9 and 300 communication rounds, FedMuon improves over both Local Muon and conventional federated baselines. For ResNet-18, the reported accuracies are $$\operatorname{lmo}(X) = \arg\min_{Y:\|Y\|\le 1}\langle X,Y\rangle,$$0 for FedAvg, $$\operatorname{lmo}(X) = \arg\min_{Y:\|Y\|\le 1}\langle X,Y\rangle,$$1 for Local Muon, and $$\operatorname{lmo}(X) = \arg\min_{Y:\|Y\|\le 1}\langle X,Y\rangle,$$2 for FedMuon; for ViT-Tiny, they are $$\operatorname{lmo}(X) = \arg\min_{Y:\|Y\|\le 1}\langle X,Y\rangle,$$3, $$\operatorname{lmo}(X) = \arg\min_{Y:\|Y\|\le 1}\langle X,Y\rangle,$$4, and $$\operatorname{lmo}(X) = \arg\min_{Y:\|Y\|\le 1}\langle X,Y\rangle,$$5, respectively. On the milder $$\operatorname{lmo}(X) = \arg\min_{Y:\|Y\|\le 1}\langle X,Y\rangle,$$6 split, FedMuon still exceeds Local Muon: $$\operatorname{lmo}(X) = \arg\min_{Y:\|Y\|\le 1}\langle X,Y\rangle,$$7 versus $$\operatorname{lmo}(X) = \arg\min_{Y:\|Y\|\le 1}\langle X,Y\rangle,$$8 on ResNet-18 and $$\operatorname{lmo}(X) = \arg\min_{Y:\|Y\|\le 1}\langle X,Y\rangle,$$9 versus $X=U\Sigma V^\top$0 on ViT-Tiny. The same paper reports $X=U\Sigma V^\top$1 on CIFAR-100 and $X=U\Sigma V^\top$2 on Tiny ImageNet for fine-tuned Swin Transformer under Dir-0.1, and best reported results across all listed GLUE tasks for RoBERTa-Base with LoRA (Liu et al., 31 Oct 2025).

The ablations in that study are especially informative because they isolate the two proposed federated corrections. On CIFAR-100 Dir-0.1, removing momentum aggregation degrades performance from $X=U\Sigma V^\top$3 to $X=U\Sigma V^\top$4 on ResNet-18 and from $X=U\Sigma V^\top$5 to $X=U\Sigma V^\top$6 on ViT-Tiny. Removing local-global alignment also reduces accuracy, to $X=U\Sigma V^\top$7 and $X=U\Sigma V^\top$8, respectively. The paper therefore attributes the gains to the interaction between matrix orthogonalization and federated drift correction rather than to local Muon alone (Liu et al., 31 Oct 2025).

The LMO-based FedMuon of (Takezawa et al., 30 Sep 2025) is evaluated on FashionMNIST with LeNet and on CIFAR-10 with ResNet-18, using $X=U\Sigma V^\top$9 clients, $-UV^\top$0 sampled clients per round, and $-UV^\top$1 local steps. The paper reports that FedMuon attains the highest test accuracy in all settings considered. It also studies inexact Muon directly: FedMuon already works at Newton–Schulz iteration count $-UV^\top$2, moving from $-UV^\top$3 to $-UV^\top$4 yields a large gain, and the best observed values are $-UV^\top$5 in one homogeneous setting and $-UV^\top$6 in one heterogeneous setting (Takezawa et al., 30 Sep 2025).

The full-participation FedMuon of (Zhang et al., 4 Oct 2025) is tested on ResNet-18, ViT, and RNN models over CIFAR-10, CIFAR-100, and Sentiment140. The paper reports faster training-loss reduction and higher test accuracy than a broad set of baselines, with the advantage becoming larger when the communication period is increased from $-UV^\top$7 to $-UV^\top$8. It also states that the gains are especially pronounced on ViT and on Sentiment140, which the paper describes as naturally exhibiting heavy-tailed noise (Zhang et al., 4 Oct 2025).

FedMuon-DA (Zhang et al., 6 Nov 2025) is evaluated on deep AUC maximization for imbalanced text classification, using Sentiment140, SST2, and Yelp Review with a two-layer RNN of input dimension $-UV^\top$9, hidden dimension $$f(\boldsymbol{x})=\frac{1}{N}\sum_{i=1}^N f_i(\boldsymbol{x}), \qquad f_i(\boldsymbol{x})=\mathbb E_{\xi_i\sim \mathcal D_i}[F_i(\boldsymbol{x};\xi_i)].$$0, and output dimension $$f(\boldsymbol{x})=\frac{1}{N}\sum_{i=1}^N f_i(\boldsymbol{x}), \qquad f_i(\boldsymbol{x})=\mathbb E_{\xi_i\sim \mathcal D_i}[F_i(\boldsymbol{x};\xi_i)].$$1. The federated configuration uses $$f(\boldsymbol{x})=\frac{1}{N}\sum_{i=1}^N f_i(\boldsymbol{x}), \qquad f_i(\boldsymbol{x})=\mathbb E_{\xi_i\sim \mathcal D_i}[F_i(\boldsymbol{x};\xi_i)].$$2 clients, batch size $$f(\boldsymbol{x})=\frac{1}{N}\sum_{i=1}^N f_i(\boldsymbol{x}), \qquad f_i(\boldsymbol{x})=\mathbb E_{\xi_i\sim \mathcal D_i}[F_i(\boldsymbol{x};\xi_i)].$$3, and both homogeneous and heterogeneous label-ratio splits. The paper reports that FedMuon-DA and its normalized-gradient counterpart Fed-NSGDA-M consistently outperform LocalSGDAM, FedSGDA-M, F-SAGDA, FESS-GDA, and a clipping baseline FedSGDA-Clip. The performance advantage is described as particularly strong in the heterogeneous regime, where heavy-tailed effects are amplified by client heterogeneity (Zhang et al., 6 Nov 2025).

6. Implementation issues, misconceptions, and later extensions

A practical FedMuon implementation is constrained by the same issues that govern centralized Muon. Large-scale Muon training uses decoupled weight decay and shape-aware update scaling,

$$f(\boldsymbol{x})=\frac{1}{N}\sum_{i=1}^N f_i(\boldsymbol{x}), \qquad f_i(\boldsymbol{x})=\mathbb E_{\xi_i\sim \mathcal D_i}[F_i(\boldsymbol{x};\xi_i)].$$4

and retains AdamW for non-matrix parameters such as RMSNorm, the LM head, and embeddings. The same work uses five Newton–Schulz iterations in practice, stores only one momentum buffer, reports optimizer-state memory at half that of Distributed AdamW, and reports communication in $$f(\boldsymbol{x})=\frac{1}{N}\sum_{i=1}^N f_i(\boldsymbol{x}), \qquad f_i(\boldsymbol{x})=\mathbb E_{\xi_i\sim \mathcal D_i}[F_i(\boldsymbol{x};\xi_i)].$$5 times Distributed AdamW with only about $$f(\boldsymbol{x})=\frac{1}{N}\sum_{i=1}^N f_i(\boldsymbol{x}), \qquad f_i(\boldsymbol{x})=\mathbb E_{\xi_i\sim \mathcal D_i}[F_i(\boldsymbol{x};\xi_i)].$$6 to $$f(\boldsymbol{x})=\frac{1}{N}\sum_{i=1}^N f_i(\boldsymbol{x}), \qquad f_i(\boldsymbol{x})=\mathbb E_{\xi_i\sim \mathcal D_i}[F_i(\boldsymbol{x};\xi_i)].$$7 end-to-end latency overhead (Liu et al., 24 Feb 2025). These details matter because several FedMuon papers state that the method is for matrix-valued weights but do not specify a complete layerwise policy for biases, normalization parameters, embeddings, or higher-order tensors (Liu et al., 31 Oct 2025, Zhang et al., 4 Oct 2025).

Another recurring misconception is that local orthogonalization can simply be averaged. The LMO-based theory already shows that naive clientwise Muon is biased under averaging (Takezawa et al., 30 Sep 2025). A related centralized development, Newton–Muon, replaces Muon’s direction $$f(\boldsymbol{x})=\frac{1}{N}\sum_{i=1}^N f_i(\boldsymbol{x}), \qquad f_i(\boldsymbol{x})=\mathbb E_{\xi_i\sim \mathcal D_i}[F_i(\boldsymbol{x};\xi_i)].$$8 with

$$f(\boldsymbol{x})=\frac{1}{N}\sum_{i=1}^N f_i(\boldsymbol{x}), \qquad f_i(\boldsymbol{x})=\mathbb E_{\xi_i\sim \mathcal D_i}[F_i(\boldsymbol{x};\xi_i)].$$9

where $\mu$00 is an input second-moment matrix. That paper emphasizes that gradients and second moments are additive before inversion and orthogonalization, whereas the inverse and matrix sign are not additive (Du et al., 1 Apr 2026). This suggests that a federated Newton–Muon design would be most faithful to the centralized derivation if gradient and activation-second-moment statistics were aggregated before applying inversion and matrix sign.

Communication and participation assumptions also vary sharply across the literature. The FedMuon of (Liu et al., 31 Oct 2025) explicitly supports partial participation, with $\mu$01 or $\mu$02 participation in several vision experiments, and proposes FedMuon_SVD, which keeps only the top-$\mu$03 singular components of each momentum matrix with $\mu$04 set to $\mu$05 of rank. The reported effect is about 95\% reduction in momentum communication cost, with communication overhead reduced from $\mu$06 to about $\mu$07 relative to no momentum aggregation while preserving nearly the same accuracy (Liu et al., 31 Oct 2025). By contrast, the theory and pseudocode in (Zhang et al., 4 Oct 2025) and (Zhang et al., 6 Nov 2025) use full participation.

The assumptions behind the convergence theory are not uniform. The FedMuon of (Zhang et al., 4 Oct 2025) uses a bounded heterogeneity assumption, whereas (Liu et al., 31 Oct 2025) and (Zhang et al., 6 Nov 2025) emphasize guarantees without a heterogeneity term or without a bounded heterogeneity assumption. Theorem statements are not always internally pristine: in (Zhang et al., 6 Nov 2025), the theorem line gives $\mu$08, while a later remark states $\mu$09, so the manuscript is internally inconsistent on that specific ratio.

Subsequent centralized work points toward possible future generalizations of FedMuon rather than settled federated algorithms. FISMO reformulates Muon as orthogonalized momentum in a Fisher-structured geometry using Kronecker factors $\mu$10 and $\mu$11 (Xu et al., 29 Jan 2026). “From SGD to Muon” embeds Muon in a broader Schatten-$\mu$12 LMO family and derives a layerwise data-driven selector

$\mu$13

interpolating between SGD-like and Muon-like updates (Massena et al., 19 May 2026). These developments suggest, but do not establish, that future FedMuon variants may need to aggregate not only model or momentum states but also geometry-dependent statistics such as curvature surrogates, activation second moments, or layerwise LMO parameters.

Taken together, the current literature presents FedMuon less as a single optimizer than as a rapidly expanding class of federated matrix-geometry methods. Their shared premise is that Muon’s orthonormalized update direction can be useful in federated learning, but only when the resulting geometric nonlinearity is matched with federated-specific correction mechanisms—momentum aggregation, local-global alignment, control variates, periodic momentum synchronization, or primal-dual drift control—appropriate to the optimization setting at hand.