Estimation Of An Adaptive Double-sparse High-dimensional Generalized Linear Model Based On Overlap Group Lasso

This paper mainly discusses the theoretical problems related to parameter estimation and variable selection of generalized linear models when the predictive variables have the form of structured sparsity.Structural sparsity(Structured Sparsity)is a natural extension of group structure,meaning that the support set in the model is not only sparse but also has some structural characteristics.When the variables in the data set have some complex structural characteristics,such as overlapping group structure or graph structure,how to effectively use the structural information between variables to improve model selection,estimation and prediction effect has become an increasingly concerned problem.The first chapter of this paper briefly introduces the research background and status quo of regularization methods commonly used in the field of high dimension at home and abroad as well as the research ideas and main innovations.The second chapter introduces the main model of structured sparse,from Group Lasso and applied to high dimensional generalized linear model,then introduces Latent Group Lasso,At last,a high dimensional generalized linear model with single sparse Overlap Group Lasso is introduced.The third chapter mainly introduces the adaptive double sparse Overlap Group Lasso high dimensional generalized linear model,and proves the finite sample property of its estimators and applies the corresponding theoretical results to the Logistic regression model.Group structure is a structure type often seen in structured sparse problems.When there is a certain group structure among variables,the appropriate use of group structure information can improve the prediction and interpretation ability of the model.However,when groups overlap each other,the traditional method of Group Lasso usually cannot fully utilize the group structure information.On the basis of single sparse Overlap Group Lasso solves this problem by copying overlapping variables,but the sparsity of intra-group variables selected by single sparse Overlap Group Lasso cannot be guaranteed.In order to solve the above problems,double sparse Overlap Group Lasso method is put forward in this paper,which ensures that after variables are selected by the whole Group in single sparse Overlap Group Lasso method,variables in the group are also sparse.In order to better balance penalty intensity,Adaptive weights are introduced.In this paper,the adaptive double sparse Overlap Group Lasso method is discussed and applied to high dimensional generalized linear model.Under very general conditions,the bounds of estimation error and prediction error are given.Finally,the corresponding theoretical results are applied to the Logistic regression model,and the expected results are obtained.