An Inexact ADMM Algorithm For Solving Regularized Logistic Regression Problems

With the rapid development of big data,the technology of collecting,storing,transmitting,and processing data has been greatly improved.In such highly information age,the connection between the Internet and human life is getting closer and closer.In the world,huge amounts of data are generated all the time.However,it is an important issue for people to make use of these data more effectively.Many fields such as engineering computing,investment portfolios,biomedicine,and artificial intelligence generate huge amounts of data.When dealing with such huge data,it is necessary to consider not only the complexity and high dimension of the data itself,but also the linear or non-linear and other complex relationships among the data.The logistic regression model with total variational regularization is widely used in the field of sparse optimization and machine learning.Due to the indifferentiability of the regular-ization term,under the framework of the Alternating Direction Method of Multipliers method（ADMM）,the original problem was decomposed into two subproblems by separating the vari-ables.For the x^k+1subproblem,in order to avoid the complex calculation of the inverse of Hessian matrix,a stochastic quasi-Newton algorithm is proposed to approximate the inverse of Hessian matrix by using the Taylor expansion of the inverse of a matrix and the random sam-pling technique.For the subproblem of z^k+1,the explicit expression of the solution can be obtained by using the property of proximal point operator.Moreover,to calculate that proximal point operator efficiently,Condat algorithm can be used to solve the corresponding Fenchel dual problem.Finally,the proposed algorithm is tested in Python language.Experimental results on three datasets of different sizes in UCI-Residential Building,MNIST-49,and UCI E2006 show that the proposed algorithm can converge to the optimal solution efficiently.