Discussion on ‘A review of distributed statistical inference’

We congratulate the authors on an impressive team effort to comprehensively review various statistical estimation and inference methods in distributed frameworks. This paper is an excellent resource for anyone wishing to understand why distributed inference is important in the era of big data, what the challenges of conducting distributed inference instead of centralized inference are, and how statisticians propose solutions to overcome these challenges. First, we notice that this paper focuses mainly on distributed estimation, and we would like to point out several other works on distributed inference. For smooth loss functions, Jordan et al. (2018) established asymptotic normality for their multi-round distributed estimator, which yields two communication-efficient approaches to constructing confidence regions using a sandwiched covariance matrix. For non-smooth loss functions, Chen et al. (2021) similarly proposed a sandwich-type confidence interval based on the asymptotic normality of their distributed estimator. More generic inference approaches, such as bootstrap, have also been studied in the massive data setting including the distributed framework. The authors reviewed the Bag of Little Bootstraps (BLB) method proposed by Kleiner et al. (2014), which is to repeatedly resample and refit the model at each local machine and finally aggregate the bootstrap statistics. Considering the huge computational cost of BLB, Sengupta et al. (2016) proposed the Subsampled Double Bootstrap (SDB) method, which has higher computational efficiency but requires a large number of local machines to maintain statistical accuracy. In addition to distributed samples, the dimensionality can also become large in the big data era, and in this case researchers may be more interested in simultaneous inference onmultiple parameters. In the centralized setting, bootstrap is one of the solutions to the simultaneous inference problems (Zhang & Cheng, 2017). In a distributed framework where the dimensionality grows, Yu et al. (2020) proposed distributed bootstrap methods for simultaneous inference, which not only are efficient in terms of both communication and