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We cooked up Gram Newton-Schulz: a drop-in replacement of Muon’s Newton-Schulz that is up to 2x faster. Building this requires synthesizing ideas from linear algebra, numerical analysis, and kernel design. This makes for a great story and an even better optimizer! Amazing collaborators: @jcz42, @noahamsel, @tri_dao Blog: https://t.co/EyysR6p4Bs Code: https://t.co/5DjwIg7SAc
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"text": "We made Muon run up to 2x faster for free!\n\nIntroducing Gram Newton-Schulz: a mathematically equivalent but computationally faster Newton-Schulz algorithm for polar decomposition.\n\nGram Newton-Schulz rewrites Newton-Schulz such that instead of iterating on the expensive rectangular X matrix, we iterate on the small, square, symmetric XX^T Gram matrix to reduce FLOPs. This allows us to make more use of fast symmetric GEMM kernels on Hopper and Blackwell, halving the FLOPs of each of those GEMMs.\n\nGram Newton-Schulz is a drop-in replacement of Newton-Schulz for your Muon use case: we see validation perplexity preserved within 0.01, and share our (long!) journey stabilizing this algorithm and ensuring that training quality is preserved above all else.\n\nThis was a super fun project with @noahamsel, @berlinchen, and @tri_dao that spanned theory, numerical analysis, and ML systems! Blog and codebase linked below 🧵",
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