Introducing Compositional Muon, an optimizer that extends Muon from individual matrices to composed transformer circuits.
Modern optimizers usually draw trust regions around individual parameters. But in attention, the loss often sees compositions like QK^T and OV. Updating each factor independently can therefore control the wrong object. Compositional Muon closes this gap by deriving partner-whitened update rules. Each factor’s update is shaped by the spectral geometry of the matrix it is composed with, producing more stable composed updates and better effective learning-rate allocation across heads and layers.
For QK, this gives a head-local half-split rule. For OV, the circuit geometry selects a hybrid rule: (V) is optimized per-head, while (W_O) is optimized as the single matrix that aggregates all heads back into the residual stream.
CM improves over Muon at 340M and 1B scale, transfers to the modded-nanoGPT optimization benchmark, and can be approximated cheaply as partner-rescaled Muon via the isotropic rule.
The broader point is optimizer-architecture co-design: better optimizers should not only ask how to update a parameter, but what composed circuit that parameter participates in. CM is one step toward optimizers that respect the functional structure the loss actually sees.
