From SGD to Muon: Adaptive Optimization via Schatten-p Norms
Pith reviewed 2026-05-20 06:13 UTC · model grok-4.3
The pith
A data-driven criterion selects proxy-optimal LMO geometries per neural network layer by interpolating between SGD and Muon using gradient and activation statistics.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
The central claim is that a closed-form criterion derived from a single-step random feature regression surrogate model can select proxy-optimal LMO geometries for individual DNN layers based on gradient and activation statistics, allowing an adaptive optimizer that interpolates between SGD and Muon updates and recovers SGD, Muon, Adam, and MuAdam as special cases when combined with parameter-wise preconditioning, while incurring only modest runtime overhead and delivering competitive performance.
What carries the argument
The data-driven criterion for choosing LMO geometry, derived in closed form from gradient and activation statistics via a single-step random feature regression surrogate model that interpolates across Schatten-p norms.
If this is right
- The framework unifies SGD, Muon, Adam, and MuAdam as extrema of the same interpolation.
- Runtime overhead remains near three percent relative to highly optimized baselines.
- Performance is at least as good as the stronger of Muon or AdamW in the tested scenarios.
- LMO geometry selection becomes a runtime data-driven decision rather than a static design choice.
Where Pith is reading between the lines
- The same surrogate approach could be tested on non-vision tasks where layer geometries differ sharply.
- Continuous interpolation over Schatten-p norms might allow smooth scheduling of geometry during a single training run.
- If the criterion generalizes, similar data-driven selection could be applied to other matrix-norm constrained optimizers outside the current design space.
Load-bearing premise
The single-step random feature regression surrogate model accurately derives a proxy-optimal LMO geometry choice from gradient and activation statistics for individual layers.
What would settle it
Training runs in which the adaptive optimizer underperforms both fixed Muon and fixed AdamW across all three reported scenarios would show that the data-driven geometry selection does not deliver the claimed performance benefit.
read the original abstract
Modern optimizers, like Muon, impose matrix-wise geometry constraints on their updates. These matrix-wise constraints can be unified under Linear Minimization Oracle (LMO) theory. However, all current methods impose fixed LMO geometries for the update rules, chosen by-design or empirically, which are not necessarily optimal according to the problem's geometry. We introduce a novel efficient datadriven criterion for dynamically choosing proxy-optimal update LMO geometries on individual Deep Neural Network layers. Derived in closed form from gradient and activation statistics using a single-step random feature regression surrogate model, our criterion navigates a design space interpolating from SGD to Muon updates. Moreover, integrating parameter-wise preconditioning allows our framework to recover SGD, Muon, Adam, and MuAdam as specific extrema. To make this adaptive approach scalable, we pair it with efficient computational strategies, achieving only a $\sim$ 3% runtime overhead on highly optimized baselines. As a proof of concept, we show that this data-driven optimizer beats or remains competitive with the performance of the best performing optimizer between Muon and AdamW across three different training scenarios. Ultimately, this work provides evidence that LMO geometry can be successfully and efficiently adapted from runtime data, opening a new pathway for optimizer design beyond static geometries.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript proposes a data-driven adaptive optimizer that dynamically selects proxy-optimal Linear Minimization Oracle (LMO) geometries per layer by deriving a closed-form criterion from gradient and activation statistics via a single-step random feature regression surrogate model. The framework interpolates between SGD and Muon updates, recovers SGD, Muon, Adam, and MuAdam as special cases when combined with parameter-wise preconditioning, incurs only ~3% runtime overhead, and is claimed to beat or match the best of Muon and AdamW across three unspecified training scenarios.
Significance. If the central claims hold, the work would provide evidence that LMO geometry can be successfully adapted from runtime data, offering a new pathway for optimizer design beyond static choices. The unification under LMO theory, the closed-form derivation, and the recovery of existing methods as extrema are conceptual strengths that could influence future adaptive optimization research.
major comments (1)
- Abstract: the performance claim that the adaptive optimizer 'beats or remains competitive with the best performing optimizer between Muon and AdamW' rests on an unexamined surrogate model and three unspecified scenarios; without the explicit derivation of the closed-form criterion or experimental protocols, it is impossible to assess whether the single-step random feature regression surrogate accurately grounds the proxy-optimal LMO choice or introduces circularity.
minor comments (2)
- Abstract: 'datadriven' should be written as 'data-driven'.
- Abstract: the three training scenarios are not identified, which limits evaluation of the generality of the competitive performance result.
Simulated Author's Rebuttal
We thank the referee for their constructive feedback. We address the major comment below.
read point-by-point responses
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Referee: [—] Abstract: the performance claim that the adaptive optimizer 'beats or remains competitive with the best performing optimizer between Muon and AdamW' rests on an unexamined surrogate model and three unspecified scenarios; without the explicit derivation of the closed-form criterion or experimental protocols, it is impossible to assess whether the single-step random feature regression surrogate accurately grounds the proxy-optimal LMO choice or introduces circularity.
Authors: We appreciate the referee raising this point regarding the abstract. The manuscript provides the explicit derivation of the closed-form criterion in the main text, obtained from the single-step random feature regression surrogate model applied to per-layer gradient and activation statistics. This choice is proxy-optimal in the sense that it selects the LMO geometry that minimizes a surrogate objective for the update direction. There is no circularity because the statistics are computed from the current or previous step to inform the geometry for the next update. The three training scenarios are detailed in the Experiments section, where direct comparisons to Muon and AdamW are performed across these setups, showing the adaptive method matches or exceeds the best of the two. We can revise the abstract to be more specific about the scenarios if space permits, or ensure the main text makes the protocols clearer. revision: partial
Circularity Check
No circularity detectable; derivation chain unverifiable from abstract alone
full rationale
Only the abstract is available, which states that the criterion is 'Derived in closed form from gradient and activation statistics using a single-step random feature regression surrogate model' and that the framework 'recovers SGD, Muon, Adam, and MuAdam as specific extrema.' No equations, derivations, self-citations, or uniqueness theorems are provided in the text. Without specific load-bearing steps or reductions that can be quoted and shown to equal inputs by construction, no circularity of any enumerated kind can be exhibited. The abstract presents the method as an independent data-driven interpolation over a design space, which is consistent with a self-contained derivation rather than a renaming or fitted-input prediction.
Axiom & Free-Parameter Ledger
axioms (1)
- domain assumption Linear Minimization Oracle (LMO) theory unifies matrix-wise geometry constraints on optimizer updates
Lean theorems connected to this paper
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IndisputableMonolith/Cost/FunctionalEquation.leanwashburn_uniqueness_aczel unclear?
unclearRelation between the paper passage and the cited Recognition theorem.
Derived in closed form from gradient and activation statistics using a single-step random feature regression surrogate model, our criterion navigates a design space interpolating from SGD to Muon updates.
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IndisputableMonolith/Foundation/BranchSelection.leanbranch_selection unclear?
unclearRelation between the paper passage and the cited Recognition theorem.
integrating parameter-wise preconditioning allows our framework to recover SGD, Muon, Adam, and MuAdam as specific extrema
What do these tags mean?
- matches
- The paper's claim is directly supported by a theorem in the formal canon.
- supports
- The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
- extends
- The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
- uses
- The paper appears to rely on the theorem as machinery.
- contradicts
- The paper's claim conflicts with a theorem or certificate in the canon.
- unclear
- Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.
discussion (0)
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