FedMIX-P: Mixing Local and Global Preconditioners for Federated Vision and Language Model Training

Abstract: Adaptive preconditioners accelerate model training, but heterogeneous client geometries can bias federated updates even when gradients are evaluated at the same model. Round-start synchronization alone cannot prevent this mismatch from reappearing during local training. We propose \texttt{FedMIX-P}, which mixes shared and local preconditioners at every local step, retaining local adaptation while reducing mean-squared operator mismatch by a factor of $λ<sup>2$. For smooth nonconvex objectives with stochastic gradients and partial participation, we establish an $O(R<sup>{-1/2})$ stationarity bound using suitable stepsizes and a horizon-dependent mixing weight, without requiring local preconditioners to converge to one another. A two-client counterexample shows that fixed positive mixing can preserve a nonstationary fixed point. The theory covers bounded linear symmetric positive-definite preconditioners. Experiments with SOAP, Sophia, and Muon variants across vision and language tasks show improvements over corresponding local optimizers, including accuracy gains of up to $19.47$ percentage points and lower validation loss for 60M--350M LLMs. Full nonlinear and momentum-based updates require separate analysis.