Muon: Structure-Aware Matrix Optimizer | Generated by AI
“Muon” in Kimi refers to an advanced optimizer—specifically a structure-aware, matrix-orthogonalizing optimizer used in training the Kimi K2 language model.
What Is the Muon Optimizer?
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Matrix-Aware Optimization Unlike AdamW or SGD, Muon treats weight matrices as geometric entities rather than independent scalar parameters. It applies Newton–Schulz iterations to orthogonalize the momentum-averaged gradient, yielding well-conditioned, balanced updates that respect both row and column structure of the matrix (Medium, kellerjordan.github.io).
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Orthogonalization via Newton–Schulz Rather than performing expensive Singular Value Decomposition (SVD), Muon uses a fast iterative method (Newton–Schulz) to approximate the nearest orthogonal matrix to the gradient. This keeps the update under spectral norm constraints, preserving energy and spreading learning across all directions equally (Medium, kellerjordan.github.io).
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Pipeline Adjustment The standard update flow—Gradient → Momentum → Parameter Update—is replaced by: Gradient → Momentum → Newton–Schulz Orthogonalization → Parameter Update. This modification enhances training efficiency and stability for 2D parameter matrices (Medium, kellerjordan.github.io).
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Efficient in Practice Despite adding a small computational overhead, Muon delivers significant speedups:
- Records in NanoGPT speedrunning, improving training time by ~35% (kellerjordan.github.io).
- Scales well in large language model training when combined with weight decay and per-parameter RMS adjustments (arXiv).
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Strong Theoretical Foundations Recent research supports Muon’s convergence properties, benefits of weight decay, and optimal batch sizing. Studies confirm tighter theoretical bounds and efficiency across practical scenarios (arXiv).
Role of Muon in Kimi K2
Kimi K2 is Moonshot AI’s Mixture-of-Experts (MoE) model, with 1 trillion total parameters (32B activated) optimized for agentic capabilities. It achieved exceptional performance on knowledge, reasoning, and coding benchmarks thanks in part to training with Muon (and a variant termed MuonClip) (Hugging Face, Poe).
- The MuonClip optimizer refers to applying Muon at massive scale while addressing stability challenges intrinsic to such large models (Hugging Face, Poe).
- The optimizer was key to enabling zero training instability over a large token dataset (~15.5 trillion tokens) and delivering state-of-the-art results (Hugging Face).
In Summary (No Table, as Requested)
- Muon = “MomentUm Orthogonalized by Newton–Schulz”—an optimizer that replaces raw gradient updates with orthogonalized updates, improving learning dynamics for matrix parameters.
- It preserves structure, ensures balanced updates, and remains efficient through iterative orthogonalization instead of costly decompositions.
- Kimi K2 leverages Muon (and MuonClip) to push the limits of large-scale MoE training, achieving top performance with stability and compute efficiency.
Let me know if you’d like a deeper dive—perhaps how Muon compares to AdamW in training trajectories, how MuonClip tweaks the algorithm for MoEs, or how the Newton–Schulz method works in code!