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.

翻译：多项式核回归是标准且最前沿的学习策略之一。然而众所周知，多项式核的次数和正则化参数的选择在模型选择领域仍是一个开放问题。本文的首要目标是提出一种参数选择策略。一方面，基于最坏情况学习率分析，我们证明多项式核回归中的正则化项并非必要。换言之，当多项式核次数得到适当调整时，正则化参数可以任意快速地减小。另一方面，考虑到算法的实现，正则化项又是必需的。总结而言，多项式核回归中正则化项的作用仅在于规避核矩阵的"病态"问题。基于此，本文的第二个目的是提出一种新的模型选择策略，并据此设计高效的学习算法。理论与实验分析均表明，新策略优于已有方法。在理论上，我们证明了当回归函数光滑时，新学习策略几乎是最优的；在实验上，新策略能在不损失泛化能力的情况下显著降低计算负担。