  Pairwise learning is receiving increasing attention since it covers many
important machine learning tasks, e.g., metric learning, AUC maximization, and
ranking. Investigating the generalization behavior of pairwise learning is thus
of significance. However, existing generalization analysis mainly focuses on
the convex objective functions, leaving the nonconvex learning far less
explored. Moreover, the current learning rates derived for generalization
performance of pairwise learning are mostly of slower order. Motivated by these
problems, we study the generalization performance of nonconvex pairwise
learning and provide improved learning rates. Specifically, we develop
different uniform convergence of gradients for pairwise learning under
different assumptions, based on which we analyze empirical risk minimizer,
gradient descent, and stochastic gradient descent pairwise learning. We first
successfully establish learning rates for these algorithms in a general
nonconvex setting, where the analysis sheds insights on the trade-off between
optimization and generalization and the role of early-stopping. We then
investigate the generalization performance of nonconvex learning with a
gradient dominance curvature condition. In this setting, we derive faster
learning rates of order $\mathcal{O}(1/n)$, where $n$ is the sample size.
Provided that the optimal population risk is small, we further improve the
learning 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, no
matter of convex or nonconvex learning. Overall, we systematically analyzed the
generalization performance of nonconvex pairwise learning.
