In this paper, we study stochastic optimization of areas under precision-recall curves (AUPRC), which is widely used for combating imbalanced classification tasks. Although a few methods have been proposed for maximizing AUPRC, stochastic optimization of AUPRC with convergence guarantee remains an undeveloped territory. A recent work [42] has proposed a promising approach towards AUPRC based on maximizing a surrogate loss for the average precision, and proved an O(1/ǫ 5 ) complexity for finding an ǫ-stationary solution of the non-convex objective. In this paper, we further improve the stochastic optimization of AURPC by (i) developing novel stochastic momentum methods with a better iteration complexity of O(1/ǫ 4 ) for finding an ǫ-stationary solution; and (ii) designing a novel family of stochastic adaptive methods with the same iteration complexity of O(1/ǫ 4 ), which enjoy faster convergence in practice. To this end, we propose two innovative techniques that are critical for improving the convergence: (i) the biased estimators for tracking individual ranking scores are updated in a randomized coordinate-wise manner; and (ii) a momentum update is used on top of the stochastic gradient estimator for tracking the gradient of the objective. Extensive experiments on various data sets demonstrate the effectiveness of the proposed algorithms. Of independent interest, the proposed stochastic momentum and adaptive algorithms are also applicable to a class of two-level stochastic dependent compositional optimization problems.

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