# Stability and Optimization Error of Stochastic Gradient Descent for Pairwise Learning

arXiv: https://arxiv.org/abs/1904.11316 · HTML: https://mlsanity.com/paper/1904.11316/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 Apr 2019.

## Built on by (most cited first, 8 of 8)

- [Simple Stochastic and Online Gradient Descent Algorithms for Pairwise Learning](https://mlsanity.com/paper/2111.12050/) (2021)
- [Stability and Generalization for Markov Chain Stochastic Gradient Methods](https://mlsanity.com/paper/2209.08005/) (2022)
- [Stability and Generalization of Stochastic Compositional Gradient Descent Algorithms](https://mlsanity.com/paper/2307.03357/) (2023)
- [Fine-grained Analysis of Stability and Generalization for Stochastic Bilevel Optimization](https://mlsanity.com/paper/2604.04090/) (2026)
- [Pairwise Learning via Stagewise Training in Proximal Setting](https://mlsanity.com/paper/2208.04075/) (2022)
- [On the Stability and Generalization of Triplet Learning](https://mlsanity.com/paper/2302.09815/) (2023)
- [Stability beyond Bounded Differences: Sharp Generalization Bounds under Finite Lp Moments](https://mlsanity.com/paper/2606.06855/) (2026)
- [Learning Rates for Nonconvex Pairwise Learning](https://mlsanity.com/paper/2111.05232/) (2021)

## Abstract (arXiv)

In this paper we study the stability and its trade-off with optimization error for stochastic gradient descent (SGD) algorithms in the pairwise learning setting. Pairwise learning refers to a learning task which involves a loss function depending on pairs of instances among which notable examples are bipartite ranking, metric learning, area under ROC (AUC) maximization and minimum error entropy (MEE) principle. Our contribution is twofold. Firstly, we establish the stability results of SGD for pairwise learning in the convex, strongly convex and non-convex settings, from which generalization bounds can be naturally derived. Secondly, we establish the trade-off between stability and optimization error of SGD algorithms for pairwise learning. This is achieved by lower-bounding the sum of stability and optimization error by the minimax statistical error over a prescribed class of pairwise loss functions. From this fundamental trade-off, we obtain lower bounds for the optimization error of SGD algorithms and the excess expected risk over a class of pairwise losses. In addition, we illustrate our stability results by giving some specific examples of AUC maximization, metric learning and MEE.