perf(optim): implement Gram Newton-Schulz for Muon#301
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- Fix fp16 stability: Using float32 exclusively for the initial spectral normalization step prevents instability, allowing the rest of the algorithm to safely execute in fp16. - Integrate Gram Newton-Schulz: Computes iterations on the smaller Gram matrix. - Benchmarks show up to a 42% time reduction for heavily rectangular matrices (e.g., 8192x1024 drops from 58ms to 33ms) with no performance penalty on square shapes.
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Description
This PR optimizes the orthogonalization step in the Muon optimizer by integrating the Gram Newton-Schulz (GramNS) algorithm. Additionally, it addresses a known training stability issue when using
float16.Key Changes
float32is sufficient to prevent gradient overflow/underflow. This allows the subsequent Newton-Schulz iterations to run stably infloat16without strictly requiringbfloat16.Benchmarks
Performance comparison between standard NS5 and GramNS (Batch Size = 8).
GramNS demonstrates massive efficiency gains on highly rectangular matrices (up to ~40% time reduction for
8192x1024), while maintaining parity on square matrices.