Error Bounds for l^p‐Norm Multiple Kernel Learning with Least Square Loss

Abstract

The problem of learning the kernel function with linear combinations of multiple kernels has attracted considerable attention recently in machine learning. Specially, by imposing anlp-norm penalty on the kernel combination coefficient, multiple kernel learning (MKL) was proved useful and effective for theoretical analysis and practical applications (Kloft et al., 2009, 2011). In this paper, we present a theoretical analysis on the approximation error and learning ability of thelp-norm MKL. Our analysis shows explicit learning rates forlp-norm MKL and demonstrates some notable advantages compared with traditional kernel-based learning algorithms where the kernel is fixed.

Multiple Kernel SVM Based on Two-Stage Learning

“…Remark 1: (i) From Theorem 2, we have that when β = 1 and s tend to 0, R(sign(π(f z ))) − R(f c ) is arbitrarily close to O(n −1 ), which is same as that obtained in [23], [24] and [39] for the case of randomly independent samples. This means that the leaning rate obtained in Theorem 2 is optimal for the u.e.M.c.…”

“…samples. In other words, Theorem 2 extended the previously known works on MKSVM algorithm (2) from randomly independent samples [23], [24], [39] to u.e.M.c. samples.…”

“…Kp is regularization term and λ is regularization parameter [23]- [24]. The corresponding MKSVM classifier is defined as sign(f z ), where the sign function is defined as sign(f (x)) = 1 for f (x) ≥ 0 and sign(f (x)) = −1 for f (x) < 0.…”

An Efficient Kernel Learning Algorithm for Semisupervised Regression Problems

“…However, these multiple kernel learning algorithms do not have better performance than traditional nonweighted kernel = ∑ in SVM sometimes, and Cortes [8] questioned that "can learning kernels help performance?". Recently Kloft and Blanchard [9] introduced a multikernel learning withnorm ( ≥ 1) approach, which has been shown effective in both theory and practice [10,11]. Essentially, -norm is a kind of minimizing empirical risk algorithm with kernel candidate set { = ∑ =1 | ‖ ‖ ≤ 1, ≥ 0}.…”

The Learning Rates of Regularized Regression Based on Reproducing Kernel Banach Spaces

“…Recently, there is an increasing research interest in learning with abstract functional spaces, and considerable work has been done in [1][2][3] and so on.…”