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  "notes": "The chart shows a Pareto frontier comparison between Muon and AdamW, illustrating better compute-time efficiency for Muon.",
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      "text": "Muon requires 10-15% fewer tokens than AdamW at large batch sizes (128K-16M), expanding the compute-time Pareto frontier.\nMuon works with maximal update parameterization (mUP) and telescoping, enabling efficient hyperparameter tuning at O(C log N) cost.\nThis could make second-order optimization economically viable.\nA recent study on optimizers validates these modest gains: when tested fairly with proper tuning, even the best optimizers (including Muon) achieve ~10% speedups over AdamW at scale.",
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