Learning rates of $l^q$ coefficient regularization learning with Gaussian kernel

Regularization is a well recognized powerful strategy to improve the performance of a learning machine and $l^q$ regularization schemes with $0<q<\infty$ are central in use. It is known that different $q$ leads to different properties of the deduced estimators, say, $l^2$ regularization leads to smooth estimators while $l^1$ regularization leads to sparse estimators. Then, how does the generalization capabilities of $l^q$ regularization learning vary with $q$? In this paper, we study this problem in the framework of statistical learning theory and show that implementing $l^q$ coefficient regularization schemes in the sample dependent hypothesis space associated with Gaussian kernel can attain the same almost optimal learning rates for all $0<q<\infty$. That is, the upper and lower bounds of learning rates for $l^q$ regularization learning are asymptotically identical for all $0<q<\infty$. Our finding tentatively reveals that, in some modeling contexts, the choice of $q$ might not have a strong impact with respect to the generalization capability. From this perspective, $q$ can be arbitrarily specified, or specified merely by other no generalization criteria like smoothness, computational complexity, sparsity, etc..