# Fast Convergence of Online Pairwise Learning Algorithms

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Martin Boissier,
Siwei Lyu,
Yiming Ying,
Ding-Xuan Zhou
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Proceedings of the 19th International Conference on Artificial Intelligence and Statistics, PMLR 51:204-212, 2016.

#### Abstract

Pairwise learning usually refers to a learning task which involves a loss function depending on pairs of examples, among which most notable ones are bipartite ranking, metric learning and AUC maximization. In this paper, we focus on online learning algorithms for pairwise learning problems without strong convexity, for which all previously known algorithms achieve a convergence rate of \mathcalO(1/\sqrtT) after T iterations. In particular, we study an online learning algorithm for pairwise learning with a least-square loss function in an unconstrained setting. We prove that the convergence of its last iterate can converge to the desired minimizer at a rate arbitrarily close to \mathcalO(1/T) up to logarithmic factor. The rates for this algorithm are established in high probability under the assumptions of polynomially decaying step sizes.

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