In supervised learning, the selection of sample points and models is crucial for acquiring a higher level of the generalization capability. So far, the problems of active learning and model selection have been independently studied. If sample points and models are simultaneously optimized, then a higher level of the generalization capability is expected. We call this problem active learning with model selection. However, active learning with model selection can not be generally solved by simply combining existing active learning and model selection techniques because of the active learning/model selection dilemma: the model should be fixed for selecting sample points and conversely the sample points should be fixed for selecting models. In this paper, we show that the dilemma can be dissolved if there is a set of sample points that is optimal for all models in consideration. Based on this idea, we give a practical procedure for active learning with model selection in trigonometric polynomial models. The effectiveness of the proposed procedure is demonstrated through computer simulations.

In this paper, we consider the problem of active learning, and give a necessary and sufficient condition of sample points for the optimal generalization capability. By utilizing the properties of pseudo orthogonal bases, we clarify the mechanism of achieving the optimal generalization capability. We also show that the condition does not only provide the optimal generalization capability but also reduces the computational complexity and memory required to calculate learning result functions. Based on the optimality condition, we give design methods of optimal sample points for trigonometric polynomial models. Finally, the effectiveness of the proposed active learning method is demonstrated through computer simulations.