一比特压缩优化器在低通信场景下表现优异，但存在发散风险。
Sign compression for Muon: SignMuon, MuonSign, and the Limits of Error Feedback
- 用符号压缩更新向量至每参数1比特，实现极低通信开销的矩阵感知优化。
- 在平滑非凸问题上，该方法可达到标准的$\mathcal{O}(T^{-1/2})$收敛率，且仅需1比特方向信息。
- 实验表明，压缩后于LMO之后进行更有效，尽管理论证明其可能发散，反超理论收敛的方法。
SignMuon通过取更新向量的逐元素符号，将每参数压缩至1比特，是极低通信预算下运行矩阵感知优化器最直接的方式。其实际表现优于SignSGD，但在线性函数上仍可能出现上升现象。无论在线性最小化算子（LMO）之前或之后取符号，均无法修复此问题：我们构造了小规模实例，显示sign-before（MuonUSign）与sign-on-both-sides（MuonSign）同样会发散，因此符号操作在算子前后的位置无法保证下降。误差反馈作为纠正有偏压缩的标准手段，在应用于Muon输出时，对任意光滑常数、步长和动量均可能失效；而将其应用于梯度，则可成功，使EF21-MuonUSign与EF21-MuonSign在平滑非凸问题上达到标准的$\mathcal{O}(T^{-1/2})$平方梯度范数收敛速率，后者仅需每个方向1比特。然而实验结果反转：在集中式与联邦式CIFAR-10及nanoGPT速度跑实验中，压缩后于LMO之后执行的方法始终最强，尽管理论上其可能发散，而理论上收敛的变体则落后。这表明，压缩后于算子之后执行这一启发式策略，在实际尺度下比理论保证更重要。
原文摘要 · Abstract (English)
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.
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