Tilde Research Introduces Aurora: A Leverage-Aware Optimizer That Fixes a Hidden Neuron Death Problem in Muon
Researchers at Tilde Research have launched Aurora, a new optimizer for coaching neural networks that addresses a structural flaw in the widely-used Muon optimizer. The flaw quietly kills off a important fraction of MLP neurons throughout coaching and retains them completely useless. Aurora comes with a 1.1B parameter pretraining experiment, a new state-of-the-art end result on the modded-nanoGPT speedrun benchmark, and open codes.
What is Muon?
To perceive Aurora, it helps to first perceive Muon. The Muon optimizer attracted consideration in the ML group after outperforming AdamW in wall-clock time to convergence on the nanoGPT speedrun competitors — a group benchmark that measures how briskly you possibly can practice a GPT-style mannequin to a goal validation loss. Since then, Muon has been adopted in frontier-scale mannequin coaching by a number of analysis teams.
Muon’s key algorithmic step is computing the polar issue of the gradient matrix. For a gradient matrix G with skinny Singular Value Decomposition (SVD) G = UΣVᵀ, Muon computes polar(G) = UVᵀ, which is the closest semi-orthogonal matrix to G in the Frobenius norm. This orthogonalized gradient is then used to replace the weights: W ← W − η UVᵀ for a studying fee η. The use of matmul-only iterative algorithms to compute the polar issue is what makes Muon sensible at scale.
The NorMuon Puzzle: Row Normalization Helps, But Why?
Before Aurora, NorMuon led the modded-nanoGPT speedrun. It launched a row-normalization step—much like Adam’s per-parameter scaling—that adjusted the polar issue by its inverse RMS norm. While this usually pulls the replace away from a strictly orthogonal gradient, NorMuon nonetheless yields spectacular outcomes. The Tilde workforce got down to perceive precisely what hole in Muon’s formulation NorMuon was addressing.
The Core Problem: Row-Norm Anisotropy and Neuron Death in Tall Matrices
The analysis workforce found that the Muon optimizer unintentionally “kills” a massive portion of neurons in tall weight matrices, akin to these discovered in SwiGLU-based MLP layers. Because it’s mathematically unattainable for these particular matrix shapes to remain completely orthogonal whereas protecting row updates even, the optimizer finally ends up giving large updates to some neurons whereas just about ignoring others. This outcomes in a “dying spiral” the place under-performing neurons obtain much less sign over time, finally changing into completely inactive.
The analysis examine revealed that by the five hundredth coaching step, a couple of in 4 neurons are successfully useless. This isn’t simply a native difficulty; the dearth of exercise in these neurons starves subsequent layers of vital knowledge, spreading the inefficiency all through the mannequin. Aurora solves this through the use of a new mathematical strategy that enforces uniform updates throughout all neurons with out sacrificing the advantages of orthogonalization.
The Intermediate Step: U-NorMuon
Before arriving at Aurora, the analysis introduces an intermediate repair referred to as U-NorMuon. The key statement is that NorMuon normalizes every row to unit norm (norm = 1), however that is truly the fallacious goal for a tall matrix. For a column-orthogonal tall matrix, the mathematically appropriate common row norm is √(n/m), not 1. U-NorMuon corrects this by normalizing tall matrix rows to have norm √(n/m) as a substitute of 1.
In experiments at 340M scale, U-NorMuon outperforms each Muon and commonplace NorMuon and utterly eliminates the neuron dying phenomenon — leverage scores turn out to be roughly isotropic all through coaching. Crucially, U-NorMuon propagates this profit to layers it doesn’t immediately contact: maintaining/gate rows alive ensures isotropic gradient move into the down-projection, stabilizing its column leverage with none direct intervention.
However, U-NorMuon nonetheless has a drawback: it forcefully overrides the polar issue with uniform row norms, sacrificing polar issue precision, which is each theoretically undesirable and empirically expensive in the Muon framework (the paper reveals that Muon achieves monotonically decrease loss with extra exact orthogonalization). This is the motivation for Aurora.
Aurora: Steepest Descent Under Two Joint Constraints
Aurora reformulates the update-selection drawback from scratch. Rather than working orthogonalization after which patching it with row normalization, Aurora asks: what’s the optimum replace below the joint constraint of left semi-orthogonality and uniform row norms?
Formally, for tall matrices, Aurora solves:
The analysis reveals that these two constraints collectively pressure all singular values of U to precisely equal 1. This means the joint constraint nonetheless produces a legitimate left semi-orthogonal replace, not a compromised one. This is the important thing perception that separates Aurora from NorMuon and U-NorMuon: it achieves row-norm uniformity and orthogonality concurrently slightly than buying and selling one off towards the opposite.
The analysis additionally offers two algorithmic implementations of Aurora’s resolution. The Riemannian Aurora makes use of a gradient projection strategy restricted to the joint Stiefel/equal-row-leverage manifold. The vanilla Aurora is a easier, extra sensible implementation. Both are open-sourced. For non-tall (extensive and sq.) matrices, row-norm uniformity is already implied by orthogonality, so Aurora leaves these parameters unchanged.
Results
Aurora was used to coach a 1.1B mannequin that achieves 100x knowledge effectivity on open-source web knowledge and outperforms bigger fashions on normal evals like HellaSwag. At 1B scale, Aurora achieves massive features over each Muon and NorMuon. On the modded-nanoGPT optimization speedrun, Aurora’s submitted run outperforms the prior state-of-the-art (which was NorMuon). Untuned Aurora carries solely a 6% compute overhead over conventional Muon and is designed as a drop-in substitute.
The analysis workforce additionally discovered that Aurora’s efficiency features scale with MLP width, suggesting it’s notably efficient for networks with massive MLP growth components — which is per the neuron dying speculation, since wider MLPs have extra tall matrices and extra alternative for leverage anisotropy to compound.
Key Takeaways
- Muon’s polar issue replace inherits row-norm anisotropy on tall matrices, inflicting over 25% of MLP neurons to completely die as early as step 500 of coaching.
- Aurora solves this by discovering the optimum replace below a joint constraint of left semi-orthogonality and uniform row norms — reaching each concurrently slightly than buying and selling one off towards the opposite.
- At 1.1B scale, Aurora achieves 100x knowledge effectivity on open-source web knowledge, outperforms bigger fashions on HellaSwag, and units a new SoTA on the modded-nanoGPT speedrun.
- Aurora is a near-drop-in substitute for Muon with solely 6% compute overhead, and its features scale with MLP width.
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