Distributed Learning of Single Index Model with Differential Privacy and Massive Data

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Distributed learning enables scalable data analysis by reducing storage and computational burdens, although the communication of local outputs in divide-and-conquer (DC) implementations may introduce privacy risks.We study distributed estimation and inference for the single-index model (SIM), a flexible semiparametric framework with an interpretable low-dimensional index structure.We propose a DC estimator for the SIM and establish convergence rates and error bounds for the nonparametric component, together with asymptotic normality of the index parameter estimator.Under mild conditions, the aggregated estimator attains the optimal nonparametric convergence rate. To provide rigorous privacy protection, we further develop differentially private (DP) algorithms for the SIM under the DC framework that satisfy (ε, δ)-DP.We establish a high-probability bound on the average squared gradient mapping, thereby characterizing how privacy noise and model complexity affect optimization accuracy.Simulations and a real-data application demonstrate accurate estimation and effective privacy protection.