Model selection of polynomial kernel regression

Polynomial kernel regression is one of the standard and state-of-the-art learning strategies. However, as is well known, the choices of the degree of polynomial kernel and the regularization parameter are still open in the realm of model selection. The first aim of this paper is to develop a strategy to select these parameters. On one hand, based on the worst-case learning rate analysis, we show that the regularization term in polynomial kernel regression is not necessary. In other words, the regularization parameter can decrease arbitrarily fast when the degree of the polynomial kernel is suitable tuned. On the other hand,taking account of the implementation of the algorithm, the regularization term is required. Summarily, the effect of the regularization term in polynomial kernel regression is only to circumvent the " ill-condition" of the kernel matrix. Based on this, the second purpose of this paper is to propose a new model selection strategy, and then design an efficient learning algorithm. Both theoretical and experimental analysis show that the new strategy outperforms the previous one. Theoretically, we prove that the new learning strategy is almost optimal if the regression function is smooth. Experimentally, it is shown that the new strategy can significantly reduce the computational burden without loss of generalization capability.Comment: 29 pages, 4 figure

Learning with sample dependent hypothesis spaces

AbstractMany learning algorithms use hypothesis spaces which are trained from samples, but little theoretical work has been devoted to the study of these algorithms. In this paper we show that mathematical analysis for these algorithms is essentially different from that for algorithms with hypothesis spaces independent of the sample or depending only on the sample size. The difficulty lies in the lack of a proper characterization of approximation error. To overcome this difficulty, we propose an idea of using a larger function class (not necessarily linear space) containing the union of all possible hypothesis spaces (varying with the sample) to measure the approximation ability of the algorithm. We show how this idea provides error analysis for two particular classes of learning algorithms in kernel methods: learning the kernel via regularization and coefficient based regularization. We demonstrate the power of this approach by its wide applicability

Fast Polynomial Kernel Classification for Massive Data

In the era of big data, it is highly desired to develop efficient machine learning algorithms to tackle massive data challenges such as storage bottleneck, algorithmic scalability, and interpretability. In this paper, we develop a novel efficient classification algorithm, called fast polynomial kernel classification (FPC), to conquer the scalability and storage challenges. Our main tools are a suitable selected feature mapping based on polynomial kernels and an alternating direction method of multipliers (ADMM) algorithm for a related non-smooth convex optimization problem. Fast learning rates as well as feasibility verifications including the convergence of ADMM and the selection of center points are established to justify theoretical behaviors of FPC. Our theoretical assertions are verified by a series of simulations and real data applications. The numerical results demonstrate that FPC significantly reduces the computational burden and storage memory of the existing learning schemes such as support vector machines and boosting, without sacrificing their generalization abilities much.Comment: arXiv admin note: text overlap with arXiv:1402.4735 by other author