  Pairwise learning usually refers to a learning task which involves a loss
function depending on pairs of examples, among which most notable ones include
ranking, metric learning and AUC maximization. In this paper, we study an
online algorithm for pairwise learning with a least-square loss function in an
unconstrained setting of a reproducing kernel Hilbert space (RKHS), which we
refer to as the Online Pairwise lEaRning Algorithm (OPERA). In contrast to
existing works \cite{Kar,Wang} which require that the iterates are restricted
to a bounded domain or the loss function is strongly-convex, OPERA is
associated with a non-strongly convex objective function and learns the target
function in an unconstrained RKHS. Specifically, we establish a general theorem
which guarantees the almost surely convergence for the last iterate of OPERA
without any assumptions on the underlying distribution. Explicit convergence
rates are derived under the condition of polynomially decaying step sizes. We
also establish an interesting property for a family of widely-used kernels in
the setting of pairwise learning and illustrate the above convergence results
using such kernels. Our methodology mainly depends on the characterization of
RKHSs using its associated integral operators and probability inequalities for
random variables with values in a Hilbert space.
