  Recently, some mixture algorithms of pointwise and pairwise learning (PPL)
have been formulated by employing the hybrid error metric of "pointwise loss +
pairwise loss" and have shown empirical effectiveness on feature selection,
ranking and recommendation tasks. However, to the best of our knowledge, the
learning theory foundation of PPL has not been touched in the existing works.
In this paper, we try to fill this theoretical gap by investigating the
generalization properties of PPL. After extending the definitions of
algorithmic stability to the PPL setting, we establish the high-probability
generalization bounds for uniformly stable PPL algorithms. Moreover, explicit
convergence rates of stochastic gradient descent (SGD) and regularized risk
minimization (RRM) for PPL are stated by developing the stability analysis
technique of pairwise learning. In addition, the refined generalization bounds
of PPL are obtained by replacing uniform stability with on-average stability.
