A Note on Stability for Orthogonalized Matrix Momentum with Client Sampling
2026-06-01 • Machine Learning
Machine Learning
AI summaryⓘ
The authors analyze how well a distributed optimization method works when only some clients participate each round, focusing on models represented by matrices and using momentum updates that keep certain mathematical properties (orthogonality). They provide a mathematical guarantee on how close the learned model's performance is to the ideal (population) case, considering uneven data and client participation. Their results depend on a technical condition for the updates (Lipschitz continuity) and explain when smoothing or extra assumptions are needed. They also show a simple example illustrating why these conditions matter.
distributed optimizationmatrix-valued parametersmomentum updatesfinite-sample generalizationLipschitz continuitypolar decompositionNewton--Schulz iterationspectral separationGaussian smoothingclient sampling
Authors
Da Chang, Qiankun Shi, Lvgang Zhang, Yu Li, Ruijie Zhang
Abstract
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.