## 1 Introduction  
The numerous innovations in data analysis in the last decade have dramatically affected the technologies used in our daily lives. Datasets with unprecedented sizes and complexities are rapidly generated and collected from a great variety of resources [1]. While big datasets bring us incredible opportunities for new discoveries, many traditional methods that perform well for moderate sample sizes are no longer realistic for analyzing massive amounts of data [2]. To address the big data challenges, many traditional statistical tools have been reinvented or adapted to deal with the gigantic volume or size of big data, which is a major goal of our work. Comprehensive reviews, such as [3; 4], have been provided for algorithmic solutions to big data problems, including divide-and-conquer, subsampling-based approaches, stochastic gradient descent, and online updating. Divide-and-conquer approaches split the big data into manageable blocks, extract local summaries from each data block, and then generate overall insight. Various types of algorithms have been proposed to adapt classic statistical tools to big data problems [5; 6; 7; 8; 9; 10; 11; 12]. Subsampling-based approaches draw subsamples from the original data by elaborately designed sampling mechanisms and then perform downstream inference and prediction based on the subsamples. For example, leveraging for big data regression, [13] constructed nonuniform sampling probabilities so that influential data points were sampled with high probabilities, and [14] proposed a novel method called information-based optimal sub-data selection (IBOSS), which selects samples from a big dataset based on the D-optimality criterion; it has been extended to various statistical models, such as logistic regression [15]. When multiple subsamples of the original data are available, different approaches have been proposed to aggregate the estimates obtained from different subsamples, including bagging or bootstrap aggregating [16], stacking [17; 18], magging or maximin aggregating [19], neagging or normalized entropy aggregating [20; 21]. These aggregation approaches are especially useful for inhomogeneous large-scale data under regression analysis. Stochastic gradient descent (SGD) algorithms [22; 23; 24] update in a sequential manner based on a noisy gradient. Local SGD is the key building block of different federated learning algorithms [25; 26], which were discussed for both homogeneous nodes [27; 28; 29; 30; 31; 32] and heterogeneous nodes [29; 33; 34; 35; 36; 37; 38; 39; 40; 41; 42]. As stated in [43], new challenges in the big data era involve data security and localization legislation. According to [44; 45], legislation on data localization has emerged as a global trend, with more nations developing laws that prohibit the transfer of sensitive data. Data localization may not kill global cooperation in data science, but it will undoubtedly create unpleasant barriers to data communication. A novel approach called the score-matching representative (SMR) was proposed by [43] for analyzing big data under communication restraints. Given an existing data partition, such as data blocks labeled by countries, regions, or sources, SMR constructs model-specified data representative(s) for each block and performs downstream analysis on the constructed representatives. Unlike the divide-and-conquer or subsampling approaches, SMR does not require communicating the actual data but rather their representatives, which are not part of the original data and, thus, avoid their transfer. Comprehensive studies show that the accuracy of the estimated model parameters based on SMR can be comparable to the full data estimates [43]. According to [43], the computational complexity of SMR is \(O(Np)\) given the sample size \(N\) and the number \(p\) of model parameters.