Handling Covariate Mismatch in Collaborative Linear Prediction

Training predictive models across multiple centers typically assumes that all centers collect the same set of covariates. In practice, however, they may record different features of their observations, a setting we refer to as covariate mismatch. We study linear prediction under this challenging setting, assuming center-wise MCAR missingness patterns, and develop estimators that exploit information across centers despite heterogeneous feature sets. In the low-dimensional regime, we propose a plug-in estimator of the oracle linear predictor based on component-wise aggregation of covariance and cross-moment estimates. In higher dimensions, we study an impute-then-regress strategy that first completes the missing covariates using an exchangeability-preserving imputation procedure and then fits a ridge-regularized linear model. All proposed estimators are compatible with federated learning constraints: individual-level data remain local to each center, and only aggregated quantities are exchanged. We provide asymptotic and finite-sample learning rates for our predictors, explicitly characterizing their behaviour with the global dimension, the center-specific feature partition, and the distribution of samples across centers, and validate our approach through numerical experiments.