This note corrects an error in the proof of corollary 1 of Li et al. ( 2014 ). The original claim of the contraction principle in appendix D of Li et al. no longer holds.

Preference learning has caused great attention in machining learning. In this letter we propose a learning framework for pairwise loss based on empirical risk minimization of U -processes via Rademacher complexity. We first establish a uniform version of Bernstein inequality of U -processes of degree 2 via the entropy methods. Then we estimate the bound of the excess risk by using the Bernstein inequality and peeling skills. Finally, we apply the excess risk bound to the pairwise preference and derive the convergence rates of pairwise preference learning algorithms with squared loss and indicator loss by using the empirical risk minimization with respect to U -processes.

In this letter, we consider a density-level detection (DLD) problem by a coefficient-based classification framework with -regularizer and data-dependent hypothesis spaces. Although the data-dependent characteristic of the algorithm provides flexibility and adaptivity for DLD, it leads to difficulty in generalization error analysis. To overcome this difficulty, an error decomposition is introduced from an established classification framework. On the basis of this decomposition, the estimate of the learning rate is obtained by using Rademacher average and stepping-stone techniques. In particular, the estimate is independent of the capacity assumption used in the previous literature.