  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.
