Quadruply Stochastic Gradient Method for Large Scale Nonlinear Semi-Supervised Ordinal Regression AUC Optimization

Abstract

Semi-supervised ordinal regression (S²OR) problems are ubiquitous in real-world applications, where only a few ordered instances are labeled and massive instances remain unlabeled. Recent researches have shown that directly optimizing concordance index or AUC can impose a better ranking on the data than optimizing the traditional error rate in ordinal regression (OR) problems. In this paper, we propose an unbiased objective function for S²OR AUC optimization based on ordinal binary decomposition approach. Besides, to handle the large-scale kernelized learning problems, we propose a scalable algorithm called QS³ORAO using the doubly stochastic gradients (DSG) framework for functional optimization. Theoretically, we prove that our method can converge to the optimal solution at the rate of O(1/t), where t is the number of iterations for stochastic data sampling. Extensive experimental results on various benchmark and real-world datasets also demonstrate that our method is efficient and effective while retaining similar generalization performance.