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Tuning parameter selection is of critical importance for kernel ridge regression. To date, a data-driven tuning method for divide-and-conquer kernel ridge regression (d-KRR) has been lacking in the literature, which limits the applicability of d-KRR for large datasets. In this article, by modifying the generalized cross-validation (GCV) score, we propose a distributed generalized cross-validation (dGCV) as a data-driven tool for selecting the tuning parameters in d-KRR. Not only the proposed dGCV is computationally scalable for massive datasets, it is also shown, under mild conditions, to be asymptotically optimal in the sense that minimizing the dGCV score is equivalent to minimizing the true global conditional empirical loss of the averaged function estimator, extending the existing optimality results of GCV to the divide-and-conquer framework. Supplemental materials for this article are available online. 

Tuning parameter selection is of critical importance for kernel ridge regression. To this date, data driven tuning method for divide-and-conquer kernel ridge regression (d-KRR) has been lacking in the literature, which limits the applicability of d-KRR for large datasets. In this article, by modifying the generalized crossvalidation (GCV) score, we propose a distributed generalized cross-validation (dGCV) as a data-driven tool for selecting the tuning parameters in d-KRR. Not only the proposed dGCV is computationally scalable for massive datasets, it is also shown, under mild conditions, to be asymptotically optimal in the sense that minimizing the dGCV score is equivalent to minimizing the true global conditional empirical loss of the averaged function estimator, extending the existing optimality results of GCV to the divide-and-conquer framework. Supplemental materials for this article are available online. 

Divide-and-conquer is a powerful approach for large and massive data analysis. In the nonparameteric regression setting, although various theoretical frameworks have been established to achieve optimality in estimation or hypothesis testing, how to choose the tuning parameter in a practically effective way is still an open problem. In this paper, we propose a data-driven procedure based on divide-and-conquer for selecting the tuning parameters in kernel ridge regression by modifying the popular Generalized Cross-validation (GCV, Wahba, 1990). While the proposed criterion is computationally scalable for massive data sets, it is also shown under mild conditions to be asymptotically optimal in the sense that minimizing the proposed distributed-GCV (dGCV) criterion is equivalent to minimizing the true global conditional empirical loss of the averaged function estimator, extending the existing optimality results of GCV to the divide-and-conquer framework. 

Divide-and-conquer is a powerful approach for large and massive data analysis. In the nonparameteric regression setting, although various theoretical frameworks have been established to achieve optimality in estimation or hypothesis testing, how to choose the tuning parameter in a practically effective way is still an open problem. In this paper, we propose a data-driven procedure based on divide-and-conquer for selecting the tuning parameters in kernel ridge regression by modifying the popular Generalized Cross-validation (GCV, Wahba, 1990). While the proposed criterion is computationally scalable for massive data sets, it is also shown under mild conditions to be asymptotically optimal in the sense that minimizing the proposed distributed-GCV (dGCV) criterion is equivalent to minimizing the true global conditional empirical loss of the averaged function estimator, extending the existing optimality results of GCV to the divide-and-conquer framework.