Federated learning is a novel framework for distributed learning, which aims to break isolated data islands, as well as protect data privacy. To further prevent privacy leakage by specially crafted attacks, differential privacy is often integrated. Although differential privacy effectively secures sensitive information, it can reduce the statistical efficiency of the resulting estimators. This leads to a trade-off relationship between statistical efficiency and privacy protection. To theoretically understand this relationship, we start with the classic linear regression model and a noise-adding federated gradient descent algorithm. Its numerical convergence properties and asymptotic properties are rigorously studied. This results in fruitful insights into the trade-off relationship between statistical efficiency and privacy protection. Guided by these theoretical understandings, we further develop a Polyak-Ruppert-type averaged estimator, which can achieve good statistical efficiency with guaranteed privacy protection. Extensive simulation studies are presented to corroborate our theoretical results. Finally, we illustrate the application of our proposed method on an enterprise community data set. History: Accepted by Ram Ramesh, Area Editor for Data Science & Machine Learning. Funding: Financial support from the National Natural Science Foundation of China [Grants 12401386, 72371241, 72495123, and 12271012], the Ministry of Education Project of Key Research Institute of Humanities and Social Sciences [Grant 22JJD910001], the Postdoctoral Fellowship Program of China Postdoctoral Science Foundation [Grant GZB20230070], and the Beijing Municipal Social Science Foundation [Grant 24GLC033] is gratefully acknowledged. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2024.0554 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2024.0554 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

We study in this work the problem of multi-class logistic regression with one major class and multiple rare classes, which is motivated by a real application in TikTok live stream data. The model is inspired by the two-class logistic regression model of Wang (2020) but with surprising theoretical findings, which in turn motivate new estimation methods with excellent statistical and computational efficiency. Specifically, since rigorous theoretical analysis suggests that the resulting maximum likelihood estimators of different rare classes should be asymptotically independent, we consider to solve multiple pairwise two-class logistic regression problems instead of optimizing the joint log-likelihood function with computational challenge in multi-class problem, which are computationally much easier and can be conducted in a fully parallel way. To further reduce the computation cost, a subsample-based pairwise likelihood estimator is developed by down-sampling the major class. We show rigorously that the resulting estimators could be as asymptotically efficient as the global maximum likelihood estimator under appropriate regularity conditions. Extensive simulation studies are presented to support our theoretical findings and a TikTok live stream dataset is analyzed for illustration purpose.

We study here a Gaussian mixture model (GMM) with rare events data. In this case, the commonly used Expectation-Maximization (EM) algorithm exhibits extremely slow numerical convergence rate. To theoretically understand this phenomenon, we formulate the numerical convergence problem of the EM algorithm with rare events data as a problem about a contraction operator. Theoretical analysis reveals that the spectral radius of the contraction operator in this case could be arbitrarily close to 1 asymptotically. This theoretical finding explains the empirical slow numerical convergence of the EM algorithm with rare events data. To overcome this challenge, a Mixed EM (MEM) algorithm is developed, which utilizes the information provided by partially labeled data. As compared with the standard EM algorithm, the key feature of the MEM algorithm is that it requires additionally labeled data. We find that MEM algorithm significantly improves the numerical convergence rate as compared with the standard EM algorithm. The finite sample performance of the proposed method is illustrated by both simulation studies and a real-world dataset of Swedish traffic signs.

The judgment of controversial cases has always been an important judicial issue, but it is not easy to discover them in practice.In this paper, based on 1,361,354 legal instruments data collected from China Judgments Online, we adopt a deep learning framework to classify 147 different kinds of crimes.The proposed method has three critical steps: 1) We adopt a deep learning model to predict crime categorization; 2) With the trained model, each case is given a score vector which represents the probability that it belongs to each crime; 3) With the probability score, we develop an entropy-based index to measure the controversy of each case.We find that the larger the entropy, the more inconsistent the result given by the model based on the first instance judgment.To verify the proposed entropy measure, we provide 1) two-sided evidence based on second instance judgments; 2) comparison with some baseline models.Both confirm the practical usefulness of the entropy measure.Our results indicate that the proposed framework has an ability to discover potentially controversial cases.It should be noted that the goal of this study is not to substitute the model result for the judge's decision, but to provide a guiding reference for the judicial practice of sentencing.

The emergence of massive data in recent years brings challenges to automatic statistical inference. This is particularly true if the data are too numerous to be read into memory as a whole. Accordingly, new sampling techniques are needed to sample data from a hard drive. In this paper, we propose a sequential addressing subsampling (SAS) method that can sample data directly from the hard drive. The newly proposed SAS method is time saving in terms of addressing cost compared to that of the random addressing subsampling (RAS) method. Estimators (e.g., the sample mean) based on the SAS subsamples are constructed, and their properties are studied. We conduct a series of simulation studies to verify the finite sample performance of the proposed SAS estimators. The time cost is also compared between the SAS and RAS methods. An analysis of the airline data is presented for illustration purpose.