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Pairwise learning is receiving increasing attention since it covers manyimportant machine learning tasks, e.g., metric learning, AUC maximization, andranking. Investigating the generalization behavior of pairwise learning is thusof significance. However, existing generalization analysis mainly focuses onthe convex objective functions, leaving the nonconvex learning far lessexplored. Moreover, the current learning rates derived for generalizationperformance of pairwise learning are mostly of slower order. Motivated by theseproblems, we study the generalization performance of nonconvex pairwiselearning and provide improved learning rates. Specifically, we developdifferent uniform convergence of gradients for pairwise learning underdifferent assumptions, based on which we analyze empirical risk minimizer,gradient descent, and stochastic gradient descent pairwise learning. We firstsuccessfully establish learning rates for these algorithms in a generalnonconvex setting, where the analysis sheds insights on the trade-off betweenoptimization and generalization and the role of early-stopping. We theninvestigate the generalization performance of nonconvex learning with agradient dominance curvature condition. In this setting, we derive fasterlearning rates of order $\mathcal{O}(1/n)$, where $n$ is the sample size.Provided that the optimal population risk is small, we further improve thelearning rates to $\mathcal{O}(1/n^2)$, which, to the best of our knowledge,are the first $\mathcal{O}(1/n^2)$-type of rates for pairwise learning, nomatter of convex or nonconvex learning. Overall, we systematically analyzed thegeneralization performance of nonconvex pairwise learning.

Learning Rates for Nonconvex Pairwise Learning