Poisson Subsampling Algorithms for Large Sample Linear Regression in Massive Data

Large sample size brings the computation bottleneck for modern data analysis. Subsampling is one of efficient strategies to handle this problem. In previous studies, researchers make more fo- cus on subsampling with replacement (SSR) than on subsampling without replacement (SSWR). In this paper we investigate a kind of SSWR, poisson subsampling (PSS), for fast algorithm in ordinary least-square problem. We establish non-asymptotic property, i.e, the error bound of the correspond- ing subsample estimator, which provide a tradeoff between computation cost and approximation efficiency. Besides the non-asymptotic result, we provide asymptotic consistency and normality of the subsample estimator. Methodologically, we propose a two-step subsampling algorithm, which is efficient with respect to a statistical objective and independent on the linear model assumption.. Synthetic and real data are used to empirically study our proposed subsampling strategies. We argue by these empirical studies that, (1) our proposed two-step algorithm has obvious advantage when the assumed linear model does not accurate, and (2) the PSS strategy performs obviously better than SSR when the subsampling ratio increases.