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arxiv: 2606.04662 · v1 · pith:DCXPIFM3new · submitted 2026-06-03 · 💻 cs.LG · cs.AI

Why Muon Outperforms Adam: A Curvature Perspective

Shuche Wang , Fengzhuo Zhang , Jiaxiang Li , Dirk Bergemann , Zhuoran Yang This is my paper

Pith reviewed 2026-06-28 07:02 UTC · model grok-4.3

classification 💻 cs.LG cs.AI
keywords Muon optimizerAdam optimizernormalized directional sharpnesscurvature analysisLLM trainingquadratic problemsdata imbalancesecond-order Taylor expansion
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The pith

Muon outperforms Adam by achieving smaller curvature penalty through lower normalized directional sharpness.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that Muon's training advantage over Adam stems from a smaller second-order curvature penalty in the loss landscape. Although both optimizers deliver similar first-order improvements, Muon reduces the normalized directional sharpness component of that penalty. This reduction is strengthened by imbalanced training data and occurs mainly through lower within-layer curvature in later training phases. Analysis of quadratic problems with heterogeneous curvature proves that Muon lowers average NDS compared to gradient descent by spreading update energy evenly, which produces a smaller local loss when curvature differences are large enough.

Core claim

Applying a second-order Taylor expansion, the paper demonstrates that Muon produces a larger one-step loss decrease than Adam at the same validation loss because its curvature penalty is smaller. This advantage traces to lower Normalized Directional Sharpness rather than differences in update norm. In controlled experiments with Zipf-PCFG data, data imbalance widens the gap, and layer decomposition shows the effect concentrates in within-layer curvature during middle and late training. For stylized quadratics with heterogeneous curvature and gradient alignment to high-curvature modes, Muon is proven to achieve smaller average NDS than gradient descent by balancing update energy, resulting in

What carries the argument

Normalized Directional Sharpness (NDS), the curvature term in the second-order loss change normalized by the squared update norm, which Muon reduces by balancing update energy across curvature groups.

If this is right

  • Muon achieves larger one-step loss decrease than Adam at matched validation loss.
  • Data imbalance amplifies Muon's NDS advantage over Adam.
  • Muon's NDS reduction in middle and late stages is driven by smaller within-layer curvature.
  • In quadratic settings with strong curvature heterogeneity, balancing update energy yields lower local loss after equal steps.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The balancing mechanism may extend to other first-order methods that adjust updates according to curvature variation.
  • The result implies that ignoring curvature heterogeneity could limit performance in data regimes with strong imbalance.
  • Architecture modifications that reduce within-layer curvature differences could interact with or amplify such optimizer effects.

Load-bearing premise

The second-order Taylor approximation accurately captures the one-step loss decrease in the high-dimensional non-convex landscape of large language model training.

What would settle it

If measurements during actual LLM training show that Muon does not produce a larger one-step loss reduction than Adam when validation losses are matched, or that its NDS is not lower, the curvature explanation would not hold.

Figures

Figures reproduced from arXiv: 2606.04662 by Dirk Bergemann, Fengzhuo Zhang, Jiaxiang Li, Shuche Wang, Zhuoran Yang.

Figure 1
Figure 1. Figure 1: Decomposition of one-step optimization progress along the update directions of Muon and Adam. [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: NDS and update-norm comparisons between Muon and Adam. Panel (a) plots NDS for Muon and Adam. Panel (b) plots the update norms of Muon and Adam. Panel (c) reports the Adam-to-Muon ratios of the curvature penalty, NDS, and the squared Frobenius norm of the update. Muon and Adam have similar update norms, whereas Muon has smaller NDS than Adam. Moreover, the Adam-to-Muon ratio of NDS closely tracks that of t… view at source ↗
Figure 3
Figure 3. Figure 3: Effect of different levels of imbalance ( [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Within-layer and cross-layer decomposition of directional sharpness over training. Panel (a) [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Empirical support for Assumption 5.1. Panel (a) reports the fraction of Frobenius energy explained by low-rank Kronecker approximations to the Hessians of the four attention matrices. Panel (b) visualizes the WV Hessian, its rank-4 Kronecker approximation, and the residual error. The results show that the Hessians of attention matrices can be well approximated by low-rank Kronecker products. Y ∈ R d1×d2 : … view at source ↗
Figure 6
Figure 6. Figure 6: Empirical support for Assumption 5.2– 5.4. Panel (a) shows the average values of the simultaneous￾diagonalization score ηsd for {Ak} r k=1 and {Bk} r k=1. Panel (b) shows the value of positive curvatures. Panel (c) shows the cumulative gradient energy ratio ζ(i). These results support the approximate simultaneous diagonalization of the matrices in the Hessian decomposition, as well as the alignment of grad… view at source ↗
Figure 7
Figure 7. Figure 7: Panel (a) reports the NDS ratio, and Panel (b) reports the loss-decrease ratio. The results show that GD and Adam exhibit similar behavior in both NDS and loss decrease on quadratic problems satisfying Assumptions 5.1–5.4. Assumption 5.4 (Gradient Alignment). The gradient G is in the subspace spanned by {Mi} q i=1, i.e., G = Pq i=1 σiMi, where σi = ⟨G, Mi⟩. In addition, the coefficients σi have the same tw… view at source ↗
Figure 8
Figure 8. Figure 8: NDS and the corresponding ratio comparison along the training steps of Muon and Adam. Panel [PITH_FULL_IMAGE:figures/full_fig_p023_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Layerwise localization of the Adam–Muon within-layer sharpness gap across the 12 Transformer [PITH_FULL_IMAGE:figures/full_fig_p024_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: WQ Hessian, rank-4 Kronecker approximation, and residual. 24 [PITH_FULL_IMAGE:figures/full_fig_p024_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: WK Hessian, rank-4 Kronecker approximation, and residual. 0 100 200 row index i 0 0 50 100 150 200 250 r o w i n d e x i mat(H) 0 100 200 row index i 0 Hr (r = 4) 0 100 200 row index i 0 mat(H) ¡ Hr 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1e 6 [PITH_FULL_IMAGE:figures/full_fig_p025_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: WO Hessian, rank-4 Kronecker approximation, and residual. D.2 Simultaneous Diagonalization (Assumption 5.2) We verify this assumption on the attention block, using Kronecker rank r = 4. This choice balances approximation quality and computational tractability where at r = 4, the Kronecker approximation already captures ξ(4) ≥ 0.87 of the Frobenius energy for three of the four attention matrices (Appendix … view at source ↗
read the original abstract

Muon improves training efficiency over Adam in large language-model training by about two times, but the local geometric source of this advantage remains unclear. Our work takes a first step toward demystifying Muon's superiority over Adam from a curvature perspective. First, we apply a second-order Taylor approximation to the training landscape and show that Muon achieves a larger one-step loss decrease than Adam at matched validation loss. The two optimizers have comparable first-order gains, but Muon consistently incurs a smaller second-order curvature penalty. Second, we decompose this curvature penalty into the squared update norm and Normalized Directional Sharpness (NDS). We find that Muon and Adam have comparable update norms, so Muon's smaller curvature penalty is driven by lower NDS, not update scale. Third, we study how training data and model structure shape Muon's NDS advantage. Using Zipf-Probabilistic Context-Free Grammar (PCFG) data with controlled imbalance, we show that data imbalance amplifies Muon's NDS advantage over Adam. A within-/cross-layer decomposition further shows that, in the middle and late stages of training, Muon's lower NDS is mainly sustained by smaller within-layer curvature. Beyond empirical evidence, we analyze stylized quadratic problems with heterogeneous curvature and gradient alignment toward high-curvature modes. We prove that Muon attains a smaller average NDS than GD by balancing update energy across curvature groups; when curvature heterogeneity is sufficiently strong, this also yields lower local quadratic loss after the same number of steps.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper claims Muon outperforms Adam in LLM training by achieving a larger one-step loss decrease under a second-order Taylor approximation of the loss landscape, driven by a smaller curvature penalty from lower Normalized Directional Sharpness (NDS) rather than comparable update norms. Empirical decompositions on real training runs and Zipf-PCFG data with controlled imbalance show data imbalance amplifies the NDS advantage, with within-layer curvature sustaining it in middle/late stages. A proof on stylized quadratic problems with heterogeneous curvature and gradient alignment proves Muon attains smaller average NDS than GD by balancing update energy across curvature groups, yielding lower local quadratic loss when heterogeneity is strong.

Significance. If the Taylor approximation reliably captures one-step dynamics, the work supplies a geometric account of optimizer differences via NDS and curvature heterogeneity, with credit due to the exact self-contained quadratic proof and the controlled empirical decomposition separating NDS from update norm. The introduction of NDS and the data-imbalance experiments provide mechanistic insight beyond standard optimizer comparisons.

major comments (1)
  1. [Abstract (first analysis step)] Abstract (first analysis step): The central claim that Muon achieves larger one-step loss decrease than Adam (at matched validation loss) because of smaller curvature penalty rests on the second-order Taylor expansion accurately predicting the actual loss change. In high-dimensional non-convex LLM landscapes, higher-order terms, movement across curvature regions, or step-size effects may dominate; no verification comparing the quadratic prediction to observed loss decrease is provided, so the subsequent NDS decomposition does not yet reliably explain performance gaps.
minor comments (2)
  1. The manuscript would benefit from an explicit early definition and formula for Normalized Directional Sharpness (NDS) before its use in the decomposition.
  2. Figure captions for the within-/cross-layer NDS plots should state the exact training stages (e.g., step ranges) and number of runs averaged.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed feedback on the reliability of the second-order Taylor approximation. We address the major comment below.

read point-by-point responses

  1. Referee: The central claim that Muon achieves larger one-step loss decrease than Adam (at matched validation loss) because of smaller curvature penalty rests on the second-order Taylor expansion accurately predicting the actual loss change. In high-dimensional non-convex LLM landscapes, higher-order terms, movement across curvature regions, or step-size effects may dominate; no verification comparing the quadratic prediction to observed loss decrease is provided, so the subsequent NDS decomposition does not yet reliably explain performance gaps.

    Authors: We agree that an explicit verification comparing the quadratic prediction to the observed one-step loss decrease would strengthen the central claim. While the NDS decomposition itself is performed using gradients and curvature estimates from actual training runs (rather than purely from the approximation), we did not include a direct side-by-side comparison of predicted versus realized loss change. In the revised manuscript we will add this verification: on the LLM training trajectories we will compute the actual loss after a single optimizer step on held-out batches and report its correlation with the second-order Taylor prediction, along with the relative contribution of higher-order terms. This addition will clarify the regime in which the approximation remains informative and thereby support the subsequent geometric interpretation via NDS. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained against external assumptions

full rationale

The paper's central chain begins with a second-order Taylor expansion applied to the loss (abstract, first analysis step), decomposes the curvature penalty into update norm and NDS (defined quantities), and then proves a comparison result for stylized quadratic problems with heterogeneous curvature. The quadratic proof is an exact derivation within its stated model and does not reduce to a fitted parameter or self-citation; NDS is computed from the Taylor term rather than reverse-engineered from performance gaps. No self-definitional loops, fitted-input predictions, or load-bearing self-citations appear in the provided derivation steps. The Taylor approximation's applicability to real LLM landscapes is a separate correctness question, not a circularity issue.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The analysis rests on the Taylor approximation as a domain assumption and introduces NDS without external validation; no free parameters are explicitly fitted in the abstract description.

axioms (1)
  • domain assumption Second-order Taylor approximation suffices to compare one-step loss decreases between optimizers.
    Invoked to establish Muon's larger loss decrease at matched validation loss.
invented entities (1)
  • Normalized Directional Sharpness (NDS) no independent evidence
    purpose: Decompose curvature penalty into directional alignment component separate from update norm.
    New quantity defined to explain Muon's advantage; no independent evidence outside the paper's derivations.

pith-pipeline@v0.9.1-grok · 5811 in / 1360 out tokens · 33734 ms · 2026-06-28T07:02:08.964393+00:00 · methodology

discussion (0)

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