Optimal granularity selection based on algorithm stability with application to attribute reduction in rough set theory

Optimal granularity selection is a key issue in rough set, by which decision rules can be generated to assign corresponding decision labels for new samples. The current evaluation criteria of optimal granularity mainly focus on the experimental effect of prediction algorithms, thus ignoring theoretical relationship between granularity and the generalization ability of algorithms. this paper, we study the issue overlooked above and propose a novel framework for optimal granularity selection with theoretical guarantee. In our framework, a rough set-based prediction algorithm that incorporates a global confidence as scoring function is introduced, which characterized by the granularity-based loss function. On the bias, the relationship between granularity and the generalization ability of algorithm is studied by introducing the stability theory in machine learning. The generalization error bound of algorithm is derived as a theoretical guarantee for optimal granularity selection. Finally, a novel optimal granularity selection strategy with theoretical guarantee is proposed, which is combined with existing attribute reduction methods to design the attribute re-reduction algorithms for generate optimal granularity reducing the relatively unimportant attributes. Numerical experiments verify the rationality optimal granularity selection framework and prove the effectiveness of the optimal granularity generated by the attribute re-reduction algorithms.