Distributed spectral pairwise ranking algorithms

This paper considers spectral pairwise ranking algorithms in a reproducing kernel Hilbert space. The concerned algorithms include a large family of regularized pairwise learning algorithms. Motivated by regularization methods, spectral algorithms are proposed to solve ill-posed linear inverse problems, then developed in learning theory and inverse problems. Recently, pairwise learning tasks such as bipartite ranking, similarity metric learning, Minimum Error Entropy principle, and AUC maximization have received increasing attention due to their wide applications. However, the spectral algorithm acts on the spectrum of the empirical integral operator or kernel matrix, involving the singular value decomposition or the inverse of the matrix, which is time-consuming when the sample size is immense. Our contribution is twofold. First, under some general source conditions and capacity assumptions, we establish the first-ever mini-max optimal convergence rates for spectral pairwise ranking algorithms. Second, we consider the distributed version of the algorithms based on a divide-and-conquer approach and show that, as long as the partition of the data set is not too large, the distributed learning algorithm enjoys both computational efficiency and statistical optimality.