无界抽样情形下不定核的系数正则化回归

Coefficient-based regularized regression with indefinite kernels by unbounded sampling

本文讨论样本依赖空间中无界抽样情形下最小二乘损失函数的系数正则化问题.这里的学习准则与之前再生核Hilbert空间的准则有着本质差异:核除了满足连续性和有界性之外,不需要再满足对称性和正定性;正则化子是函数关于样本展开系数的l2-范数;样本输出是无界的.上述差异给误差分析增加了额外难度.本文的目的是在样本输出不满足一致有界的情形下,通过l2-经验覆盖数给出误差的集中估计(concentration estimates).通过引入一个恰当的Hilbert空间以及l2-经验覆盖数的技巧,得到了与假设空间的容量以及与回归函数的正则性有关的较满意的学习速率.

We investigate the coefficient-based regularized least squares regression with unbounded sampling in a data dependent hypothesis space. The learning scheme is essentially different from the standard one in a reproducing kernel Hilbert space： we do not need the kernel to be symmetric or positive semi-definite except for continuity and boundedness, the regularizer is the l^2-norm of a function expansion involving samples and the unboudedness of the sampling output. This leads to additional difficulty in the error analysis. In this paper, the goal is to investigate some concentration estimates for the error based on/Z-empirical covering numbers without the assumption of uniform boundedness for sampling. By introducing a suitable reproducing kernel Hilbert space and applying concentration techniques with L^2-empirical covering numbers, we derive satisfactory learning rates in terms of regularity of the regression function and capacity of the hypothesis space.