On the Stability and Generalization of Triplet Learning

TL;DR

This paper provides the first theoretical generalization bounds for triplet learning algorithms, analyzing their stability and risk, and applies these results to triplet metric learning.

Contribution

It establishes high-probability and expectation-based generalization bounds for triplet learning, filling a key gap in theoretical understanding.

Findings

High-probability generalization bound of O(n^{-1/2} log n) for SGD and RRM.

An optimistic O(n^{-1}) bound in low noise scenarios for RRM.

Application of bounds to triplet metric learning.

Abstract

Triplet learning, i.e. learning from triplet data, has attracted much attention in computer vision tasks with an extremely large number of categories, e.g., face recognition and person re-identification. Albeit with rapid progress in designing and applying triplet learning algorithms, there is a lacking study on the theoretical understanding of their generalization performance. To fill this gap, this paper investigates the generalization guarantees of triplet learning by leveraging the stability analysis. Specifically, we establish the first general high-probability generalization bound for the triplet learning algorithm satisfying the uniform stability, and then obtain the excess risk bounds of the order $O(n^{-\frac{1}{2}} \mathrm{log}n)$ for both stochastic gradient descent (SGD) and regularized risk minimization (RRM), where $2n$ is approximately equal to the number of training samples. Moreover, an optimistic generalization bound in expectation as fast as $O(n^{-1})$ is derived for RRM in a low noise case via the on-average stability analysis. Finally, our results are applied to triplet metric learning to characterize its theoretical underpinning.