随着大型计算与信息技术的飞速发展，我们越来越多地面对如何分析和处理高维海量数据的挑战，而学习理论则是对这一实际问题作出研究的数学理论基础。Wu 2008提出样本依赖假设空间及其相关算法的基本分析以来，由于其更符合实际应用中的需要，各种不同形式的算法蓬勃发展。本项目考虑的就是其中常用的一种系数正则化算法，使用最小二乘损失函数进行回归问题的分析。在引进矩的递增界的条件之下，我们的问题推广到了更一般的无界情形。以构造一个中间过渡函数为出发点，我们希望改变传统的误差分解，使$l^p$系数正则化算法所得到的函数产生的过泛化误差分解为样本误差、逼近误差和假设误差三部分，对三部分采取不同分析方法以得到学习率。本项目计划采用的是构造性的方法，只要能符合误差分解各部分界的要求，即可作为我们的中间函数。因此，本项目的方法比较灵活，也很可能具有一定的普遍适用的意义，能够推广到其他算法的分析之上。

As the development of large scale computation and information technology, we face more and more challenge of analyzing and handling with mass of data. Learning theory is a mathematical foundation for studying this practical problem. Traditional analysis in learning theory is based on algorithms in which hypothesis space is reproducing kernel Hilbert space (RKHS). However, Wu 2008 proposed a basic analysis of sample dependent hypothesis space and the relevant algorithms. As it fits the practical problems more than RKHS, various algorithms of different kinds arise. Such algorithms become a research area in learning theory these days..In this project we consider a common algorithm - coefficient regularization algorithm. Using least squares loss function, the analysis of regression problem can be conducted. By introducing moment incremental condition, our problem extends to be a more general unbounded problem. Begin with constructing a stepping stone function, we expect to improve the classical error decomposition. The excess generalization error, produced by $l^p$ coefficient regularization algorithm, can be decomposed to three parts: sample error, approximation error and hypothesis error. Sample error can be bounded by some concentration inequality, error estimates for approximation error and hypothesis error can be implemented via vector-valued concentration inequality and operator analysis. Finally we get learning rates (convergence rates) using iteration technique. Different form former analysis, we plan to apply construction method in this project. A function meets the needs of all parts in the error decomposition can be taken to be a stepping stone function, i.e., the infinite norm satisfies some condition according to sample error, the convergence condition after action of a integral operator according to approximation error. Therefore, our method is flexible while choosing stepping stone functions, and may be with general significance, which is probably applied to other algorithm analysis.