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Training Deep Learning Models with Norm-Constrained LMOs
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Proposes unconstrained and constrained stochastic conditional gradient (LMO-based) optimizers with operator-norm choices for deep nets (Scion), giving convergence rates and showing hyperparameter transfer and speedups on nanoGPT training without Adam.
Problem
Choosing a geometry (norm) a priori for training deep networks, instead of relying on on-the-fly adaptive methods like Adam, while retaining hyperparameter transfer across model sizes and convergence guarantees under stochastic nonconvex optimization.
Approach
Use the linear minimization oracle over layerwise operator-norm balls (spectral, sign, row/column norm) as the update direction, applied to unconstrained problems by summing LMO steps (uSCG), with an operator-norm layer-wise choice that yields width-independent learning rates; the resulting method Scion uses no Adam and stores only one set of weights and gradients.
Key results
- Scion is reported to give significant speedups over Adam on nanoGPT training (models up to 3B parameters) without relying on Adam.
- Scion's chosen operator-norm LMOs yield learning rates that transfer across model widths, consistent across MLPs, CNNs, ViT on ImageNet and nanoGPT.
- The paper establishes an O(n^{-1/4}) order convergence rate for the stochastic unconstrained/constrained LMO methods under nonconvexity, and gives the first provable rate for Muon with and without weight decay.
Surprising findings
- Conditional gradient style LMO updates can be applied to unconstrained problems, not only constrained ones.
Limitations
- Convergence guarantees rely on constant-momentum or vanishing-momentum analysis with Lipschitz assumptions in the chosen norm.
- Muon-style baselines are applied only to hidden layers, so comparisons differ in scope.
- Sign-type theoretical guarantees for signSGD-like variants rely on increasing batch size.
Experimental setup
Datasets: ImageNet CIFAR10 nanoGPT pretraining corpus
Benchmarks: ImageNet CIFAR10 nanoGPT validation loss
Metrics: validation loss accuracy
Baselines: Adam AdamW Muon SignSGD Shampoo
Builds on: Stochastic Conditional Gradient (SCG) Muon maximal update parametrization (muP) Lion-K Modular norm
Closest papers by meaning
- From SGD to Muon: Adaptive Optimization via Schatten-p Norms (NeurIPS 2026 · similarity 0.86)
- Tensorion: A Tensor-Aware Generalization of the Muon Optimizer (NeurIPS 2026 · similarity 0.85)
- Random Sparse Lifts: Construction, Analysis and Convergence of finite sparse networks (ICLR 2024 · similarity 0.85)
- Generalized Gradient Norm Clipping & Non-Euclidean $(L_0,L_1)$-Smoothness (NeurIPS 2025 · similarity 0.85)
- Learning in Compact Spaces with Approximately Normalized Transformer (NeurIPS 2025 · similarity 0.85)
- QuEST: Stable Training of LLMs with 1-Bit Weights and Activations (ICML 2025 · similarity 0.85)
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