Monarch-Muon: Structure-Aware Muon for Fast and Memory-Efficient Pretraining

Training large language models is constrained by both wall-clock time and memory footprint, and recent matrix-aware optimizers sharpen this tradeoff: Muon can improve optimization but its dense polar update remains expensive. We propose Monarch-Muon, a structured matrix-aware optimizer for weights parameterized by Monarch factors. The resulting structure reduces weights, gradients, and optimizer state, and exposes small independent update subproblems in place of a single dense orthogonalization; we formalize this through a blockwise trust-region interpretation. In quality runs, Monarch-Muon improves over Monarch-AdamW at fixed structure: 3.06 vs. 3.09 in the dense 257M model and 2.68 vs. 2.77 in the 1B MoE model; the two-block MoE variant also matches dense AdamW while using 15% fewer active and 25.4% fewer total parameters. In 6.89B-parameter systems measurements, Monarch-Muon reduces peak memory from 68.9 to 52.9 GB, Muon optimizer state from 14.60 to 6.77 GB, and the Muon optimizer step from 1288 to 304 ms.