把 Muon 更新压成每参数 1 bit 的符号，实验里 sign-after-LMO 最强，尽管理论上会发散，结论很反直觉。
论文提出 SignMuon，将 Muon 优化器的更新逐参数取符号压缩为 1 bit。在 CIFAR-10 与 nanoGPT speedrun 实验中，SignMuon 优于 SignSGD，但作者构造线性函数实例证明它可能上升。把符号操作放在线性最小化 Oracle 之前或两侧同样无法保证下降。误差反馈用在 Muon 输出上对所有平滑常数、步长和动量都可能失效；用在梯度上则有效，EF21-MuonSign 在光滑非凸问题上达到 O(T^{-1/2}) 平方梯度范数收敛率。实验里最强的压缩方法是符号放在 Oracle 之后，尽管该配置理论上发散。
Sign compression for Muon: SignMuon, MuonSign, and the Limits of Error Feedback
SignMuon compresses the Muon update to one bit per parameter by taking its elementwise sign, providing the most direct way to run a matrix-aware optimizer under an extremely low communication budget. It outperforms SignSGD in practice, yet it can ascend even on a linear function. Signing the gradient before the Linear Minimization Oracle (LMO), rather than after, does not repair this: we construct a small explicit instance on which sign-before (MuonUSign) and sign-on-both-sides (MuonSign) ascend as well, so no placement of the sign around the oracle descends in general. Error feedback, the standard remedy for a biased compressor, does not rescue SignMuon: when applied to Muon's output, error feedback can fail for every smoothness constant, step size, and momentum. Applied to the gradient, error feedback does work, and EF21-MuonUSign and EF21-MuonSign attain the standard $\mathcal{O}(T^{-1/2})$ rate for the squared gradient norm on smooth nonconvex problems, the latter at one bit in each direction. Experiments then reverse the ordering: across centralized CIFAR-10, federated CIFAR-10, and the nanoGPT speedrun, the strongest compressed method is consistently sign-after-the-LMO, precisely the placement we prove divergent, with the provably convergent variants trailing it. Compressing after the LMO, a heuristic, matters more at these scales than the guarantee does.