A debiased distributed estimation for sparse partially linear models in diverging dimensions

We consider a distributed estimation of the double-penalized least squares approach for high dimensional partial linear models, where the sample with a total of $N$ data points is randomly distributed among $m$ machines and the parameters of interest are calculated by merging their $m$ individual estimators. This paper primarily focuses on the investigation of the high dimensional linear components in partial linear models, which is often of more interest. We propose a new debiased averaging estimator of parametric coefficients on the basis of each individual estimator, and establish new non-asymptotic oracle results in high dimensional and distributed settings, provided that $m\leq \sqrt{N/\log p}$ and other mild conditions are satisfied, where $p$ is the linear coefficient dimension. We also provide an experimental evaluation of the proposed method, indicating the numerical effectiveness on simulated data. Even under the classical non-distributed setting, we give the optimal rates of the parametric estimator with a looser tuning parameter limitation, which is required for our error analysis.