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.

翻译：近期，一些采用"逐点损失+成对损失"混合误差度量的逐点与成对学习(PPL)混合算法已被提出，并在特征选择、排序及推荐任务中展现出实证有效性。然而，据我们所知，现有研究尚未触及PPL的学习理论基础。本文通过探究PPL的泛化特性，力图填补这一理论空白。在将算法稳定性定义拓展至PPL设定后，我们为一致稳定的PPL算法建立了高概率泛化边界。此外，通过发展成对学习稳定性分析技术，我们给出了PPL中随机梯度下降(SGD)与正则化风险最小化(RRM)的显式收敛速率。进一步地，通过将一致稳定性替换为平均稳定性，获得了PPL的精炼泛化界。