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Kernel smoothing is a widely used nonparametric method in modern statistical analysis. The problem of efficiently conducting kernel smoothing for a massive dataset on a distributed system is a problem of great importance. In this work, we find that the popularly used one-shot type estimator is highly inefficient for prediction purposes. To this end, we propose a novel grid point approximation (GPA) method, which has the following advantages. First, the resulting GPA estimator is as statistically efficient as the global estimator under mild conditions. Second, it requires no communication and is extremely efficient in terms of computation for prediction. Third, it is applicable to the case where the data are not randomly distributed across different machines. To select a suitable bandwidth, two novel bandwidth selectors are further developed and theoretically supported. Extensive numerical studies are conducted to corroborate our theoretical findings. Two real data examples are also provided to demonstrate the usefulness of our GPA method.

Federated Learning (FL) has emerged as an excellent solution for performing deep learning on different data owners without exchanging raw data. However, statistical heterogeneity in FL presents a key challenge, leading to a phenomenon of skewness in local model parameter distributions that researchers have largely overlooked. In this work, we propose the concept of parameter skew to describe the phenomenon that can substantially affect the accuracy of global model parameter estimation. Additionally, we introduce FedSA, an aggregation strategy to obtain a high-quality global model, to address the implication from parameter skew. Specifically, we categorize parameters into high-dispersion and low-dispersion groups based on the coefficient of variation. For high-dispersion parameters, Micro-Classes (MIC) and Macro-Classes (MAC) represent the dispersion at the micro and macro levels, respectively, forming the foundation of FedSA. To evaluate the effectiveness of FedSA, we conduct extensive experiments with different FL algorithms on three computer vision datasets. FedSA outperforms eight state-of-the-art baselines by about 4.7% in test accuracy.

Bagging is a useful method for large-scale statistical analysis, especially when the computing resources are very limited. We study here the asymptotic properties of bagging estimators for $M$-estimation problems but with massive datasets. We theoretically prove that the resulting estimator is consistent and asymptotically normal under appropriate conditions. The results show that the bagging estimator can achieve the optimal statistical efficiency, provided that the bagging subsample size and the number of subsamples are sufficiently large. Moreover, we derive a variance estimator for valid asymptotic inference. All theoretical findings are further verified by extensive simulation studies. Finally, we apply the bagging method to the US Airline Dataset to demonstrate its practical usefulness.

The rapid emergence of massive datasets in various fields poses a serious challenge to traditional statistical methods. Meanwhile, it provides opportunities for researchers to develop novel algorithms. Inspired by the idea of divide-and-conquer, various distributed frameworks for statistical estimation and inference have been proposed. They were developed to deal with large-scale statistical optimization problems. This paper aims to provide a comprehensive review for related literature. It includes parametric models, nonparametric models, and other frequently used models. Their key ideas and theoretical properties are summarized. The trade-off between communication cost and estimate precision together with other concerns are discussed.

Momentum methods have been shown to accelerate the convergence of the standard gradient descent algorithm in practice and theory. In particular, the minibatch-based gradient descent methods with momentum (MGDM) are widely used to solve large-scale optimization problems with massive datasets. Despite the success of the MGDM methods in practice, their theoretical properties are still underexplored. To this end, we investigate the theoretical properties of MGDM methods based on the linear regression models. We first study the numerical convergence properties of the MGDM algorithm and further provide the theoretically optimal tuning parameters specification to achieve faster convergence rate. In addition, we explore the relationship between the statistical properties of the resulting MGDM estimator and the tuning parameters. Based on these theoretical findings, we give the conditions for the resulting estimator to achieve the optimal statistical efficiency. Finally, extensive numerical experiments are conducted to verify our theoretical results., Comment: 45 pages, 5 figures