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1. Optimal Tuning for Divide-and-conquer Kernel Ridge Regression with Massive Data

   Xu, G.; Shang, Z.; Cheng, G. (January 2018, The Thirty-fifth International Conference on Machine Learning)

   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. 

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2. Distributed Generalized Cross-Validation for Divide-and-Conquer Kernel Ridge Regression and Its Asymptotic Optimality

   Xu, G.; Shang, Z.; Cheng, G. (May 2019, Journal of computational and graphical statistics)

   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. 

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3. Distributed Generalized Cross-Validation for Divide-and-Conquer Kernel Ridge Regression and Its Asymptotic Optimality

   Xu, G. (January 2019, Journal of computational and graphical statistics)

   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. 

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4. Distributed Bayesian Varying Coefficient Modeling Using a Gaussian Process Prior

   Guhaniyogi, R.; Li, C.; Savitsky, T.D.; Srivastava, S. (May 2022, Journal of machine learning research)

   McCulloch, R. (Ed.)

   Varying coefficient models (VCMs) are widely used for estimating nonlinear regression functions for functional data. Their Bayesian variants using Gaussian process priors on the functional coefficients, however, have received limited attention in massive data applications, mainly due to the prohibitively slow posterior computations using Markov chain Monte Carlo (MCMC) algorithms. We address this problem using a divide-and-conquer Bayesian approach. We first create a large number of data subsamples with much smaller sizes. Then, we formulate the VCM as a linear mixed-effects model and develop a data augmentation algorithm for obtaining MCMC draws on all the subsets in parallel. Finally, we aggregate the MCMC-based estimates of subset posteriors into a single Aggregated Monte Carlo (AMC) posterior, which is used as a computationally efficient alternative to the true posterior distribution. Theoretically, we derive minimax optimal posterior convergence rates for the AMC posteriors of both the varying coefficients and the mean regression function. We provide quantification on the orders of subset sample sizes and the number of subsets. The empirical results show that the combination schemes that satisfy our theoretical assumptions, including the AMC posterior, have better estimation performance than their main competitors across diverse simulations and in a real data analysis. 

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5. Performance Analysis of Divide-and-Conquer strategies for Large scale Simulations in R

   https://doi.org/10.1109/BigData.2018.8622068  

   Subramanian, Ranjini; Zhang, Hui (December 2018, 2018 IEEE International Conference on Big Data (Big Data))

   As the volume of data and technical complexity of large-scale analysis increases, many domain experts desire powerful computational and familiar analysis interface to fully participate in the analysis workflow by just focusing on individual datasets, leaving the large-scale computation to the system. Towards this goal, we investigate and benchmark a family of Divide-and-Conquer strategies that can help domain experts perform large-scale simulations by scaling up their analysis code written in R, the most popular data science and interactive analysis language. We implement the Divide-and-Conquer strategies that use R as the analysis (and computing) language, allowing advanced users to provide custom R scripts and variables to be fully embedded into the large-scale analysis workflow in R. The whole process will divide large-scale simulations tasks and conquer tasks with Slurm array jobs and R. Simulations and final aggregations are scheduled as array jobs in parallel means to accelerate the knowledge discovery process. The objective is to provide a new analytics workflow for performing similar large-scale analysis loops where expert users only need to focus on the Divide-and-Conquer tasks with the domain knowledge. 

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