Robust Statistical Inference Of Communication-Efficient Distributed Linear Regression Model

With the advent of the era of big data,it is increasingly common that the data are distributed,the direct application of traditional statistical inference will bring high cost and privacy to communication.In this thesis,we discuss the distributed estimation of high-dimensional linear regression model with heavy-tailed noise.To deal with heavy-tailed noise whose variance can be infinite,we adopt the quantile regression loss function instead of the commonly square loss.However,the nonsmooth quantile loss brings double challenges to the distributed system in terms of computation and theoretical development.Therefore,we transform the response variable and transform the non-smooth quantile regression loss into a smooth square loss which optimizes the tedious calculation.In this thesis,two efficient iterative algorithms are proposed.In each iteration,node machines carry out computation in parallel and communicate with central processor,where only the gradient information is communicated.The algorithms fully adopt to the similarity among loss functions on node machines and converge rapidly when each node machine has large sample size.So the algorithm has a relatively robust mechanism,which realizes the distributed optimization of efficient communication and robust statistical inference.this thesis further studies the theoretical properties of the algorithm,and proves that the optimization error has a linear convergence rate under general conditions.We show that statistical efficiency can be achieved in finite steps in typical statistical applications.Finally,numerical simulations are used to verify the theoretical properties of our algorithms.