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Regularization is a well-recognized powerful strategy to improve the performance of a learning machine and l(q) regularization schemes with 0 &lt; q &lt; ∞ are central in use. It is known that different q leads to different properties of the deduced estimators, say, l(2) regularization leads to a smooth estimator, while l(1) regularization leads to a sparse estimator. Then how the generalization capability of l(q) regularization learning varies with q is worthy of investigation. In this letter, we study this problem in the framework of statistical learning theory. Our main results show that implementing l(q) coefficient regularization schemes in the sample-dependent hypothesis space associated with a gaussian kernel can attain the same almost optimal learning rates for all 0 &lt; q &lt; ∞. That is, the upper and lower bounds of learning rates for l(q) regularization learning are asymptotically identical for all 0 &lt; q &lt; ∞. Our finding tentatively reveals that in some modeling contexts, the choice of q might not have a strong impact on the generalization capability. From this perspective, q can be arbitrarily specified, or specified merely by other nongeneralization criteria like smoothness, computational complexity or sparsity.

Learning Rates of lq Coefficient Regularization Learning with Gaussian Kernel

Learning Rates of l^q Coefficient Regularization Learning with Gaussian Kernel