We also introduce anew on-average stability measure to develop optimistic bounds in a low noise setting.

Pairwise learning refers to learning tasks with loss functions depending on a pair of training examples, which includes ranking and metric learning as specific examples. Recently, there has been an increasing amount of attention on the generalization analysis of pairwise learning to understand its practical behavior. However, the existing stability analysis provides suboptimal high-probability generalization bounds. In this paper, we provide a refined stability analysis by developing generalization bounds which can be $\sqrt{n}$-times faster than the existing results, where $n$ is the sample size. This implies excess risk bounds of the order $O(n^{-1/2})$ (up to a logarithmic factor) for both regularized risk minimization and stochastic gradient descent. We also introduce a new on-average stability measure to develop optimistic bounds in a low noise setting. We apply our results to ranking and metric learning, and clearly show the advantage of our generalization bounds over the existing analysis.

Pairwise learning refers to learning tasks where the loss function depends on a pair of instances.

Multiple kernel clustering (MKC) is an important research topic that has been widely studied for decades.

We collect in Table A.1 the notations of performance measures used in this paper.X input space Y output space Z sample space S training dataset n sample size z To this aim, we require the following lemma on the self-bounding property of smooth loss functions. We only consider Part (b). We can plug the above inequality back into (B.1), and get E[F ( A (S)) F In this section, we prove Theorem 2. To this aim, we first introduce some lemmas. Lemma C.3 is motivated by a recent Let p 2 be any number. C.1 to show that 1 null The stated bound then follows by combining the above two inequalities together.

In ranking, we aim to find a function to predict the ordering of examples.

Pairwise learning refers to learning tasks where the loss function depends on a pair of instances.