The Quantile Regression Learning And Related Problems

Quantile regression learning which was first proposed by Koenker and Hallock is one of the research fields of statistical learning theory. It aims at estimating conditional quantile function f?,?. Existing literatures mainly adopt reproducing kernel Hilbert spaces associated with Mercer kernels as the hypothesis space. This circumstance prompts us to put the idea of the coefficient regularization in quantile regression learning. Coefficient regularized learning algorithm was first proposed by Vapnik for designing the linear programming support vector machine. The so-called coefficient regularized learning algorithm use the coefficients of the linear expansion of function in hypothesis space to make up the penalty term. The hypothesis space of this kind algorithm is related to samples and the generalized kernel function (continuous and bounded function).In this paper, we mainly study the quantile regression learning algorithm based on (?)2, (?)1,(?)Q(1?Q? 2) regularization. In our setting, we take the generalized kernels, which are continuous and uniformly bounded, not need to be symmetric and positive semi-definite. By the techniques of integral operator, we decompose the generalization error into approximation error, hypothesis error and sample error, through the introduction of regularization function f? and the function fz,? which belongs to the hypothesis space. Employing the covering number method and the estimate of operator norm, we bound the three parts respectively. Then, we can deduce the error bounds and learning rates. This thesis is organized as follows:In Chapter 1, the basic framework of statistical learning theory is introduced.In Chapter 2, the regularized regression learning is introduced. We not only introduce what is regression problem, but also give the research progress of regularized least-square regression learning algorithms and least-square coefficient regularized learning algorithms.In Chapter 3, the quantile regression learning is introduced. We present the research progress of the learning algorithm. This chapter is devoted to study the quantile regression coefficient regularized learning algorithm based on (?)2,(?)1,(?)Q (1?Q?2) regularization. By the techniques of integral operator and the method of error iteration, satisfied error bounds and learning rates are proved. In Chapter 4, we summarize the whole thesis and look forward to the future work.