研究分布式优化中客户端采样与正交动量的泛化稳定性，给出可证明的误差上界。
A Note on Stability for Orthogonalized Matrix Momentum with Client Sampling
- 基于耦合邻居稳定性递推和加权集中性分析，推导有限轮次上界。
- 在均匀全参与时，误差达到 $ ilde{ m O}(n^{-1} + n^{-1/2})$ 阶。
- 适用于带正则化的矩阵正交化方法，对非正则情况需额外条件。
我们研究了具有矩阵参数和正交化动量更新的客户端采样分布式优化方案的有限样本泛化性能。核心关注点是：当每轮仅部分客户端参与时，返回模型在总体目标与经验目标之间的差距。在客户端数据独立异构、局部样本数不等、聚合权重固定的前提下，通过耦合邻居稳定性递推和加权集中性步骤，推导出有限轮次的上尾概率保证。该上界保留了客户端选择次数通过放大因子 $Y_i(/mathcal C)$；在均匀全参与全批次情况下，只要控制与时间相关的放大项，即可得到 $ ilde{ m O}(n^{-1}+n^{-1/2})$ 的收敛阶。矩阵正交化规则需沿配对轨迹满足Lipschitz条件，这由正则化极坐标型映射和归一化有限步牛顿-舒尔兹正交化器满足。对于未正则化的矩阵符号函数，该论证需依赖耦合谱分离；而高斯平滑则提供有限轮次的平滑变体。一维反例表明，间隙、平滑或正则性条件必不可少。
原文摘要 · Abstract (English)
We study finite-sample generalization for a client-sampled distributed optimization scheme with matrix-valued parameters and orthogonalized momentum updates. The central quantity is the gap between the population and empirical objectives at the returned model when only a subset of clients participates in each round. Under independent heterogeneous client data, unequal local sample counts, and fixed aggregation weights, we derive a finite-round upper-tail guarantee from a coupled-neighbor stability recursion and a weighted concentration step. The bound keeps the client-selection counts through the amplification factor \(Y_i(\mathcal C)\); in the uniform full-participation full-batch regime, it yields \(\widetilde{\mathcal O}(n^{-1}+n^{-1/2})\) scaling whenever the horizon-dependent amplification terms are controlled. The matrix-orthogonalization rule is required to be Lipschitz along paired trajectories, a condition satisfied by regularized polar-type maps and normalized finite-step Newton--Schulz orthogonalizers. For the unregularized matrix sign, the same argument requires coupled spectral separation, whereas Gaussian smoothing gives a finite-round smoothed variant. A one-dimensional counterexample shows why a gap, smoothing, or regularity condition is necessary.
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