Extreme compression (1 bit) of matrix-aware optimizers like Muon is possible but creates a theory-practice gap: the provably convergent variants underperform the theoretically divergent sign-after-LMO approach on real tasks like CIFAR-10 and language model training.
This paper studies SignMuon, which compresses the Muon optimizer to one bit per parameter by taking the sign of updates. While it outperforms SignSGD in practice, the authors prove it can diverge even on linear functions. They show error feedback—the standard fix for biased compression—fails when applied to Muon's output but works when applied to gradients, achieving convergence guarantees.