A Stochastic First-Order Method for Ordered Empirical Risk Minimization

We propose a new stochastic first-order method for empirical risk minimization problems such as those that arise in machine learning. The traditional approaches, such as (mini-batch) stochastic gradient descent (SGD), utilize an unbiased gradient estimator of the empirical average loss. In contrast, we develop a computationally efficient method to construct a gradient estimator that is purposely biased toward those observations with higher current losses, and that itself is an unbiased gradient estimator of an ordered modification of the empirical average loss. On the theory side, we show that the proposed algorithm is guaranteed to converge at a sublinear rate to a global optimum for convex loss and to a critical point for non-convex loss. Furthermore, we prove a new generalization bound for the proposed algorithm. On the empirical side, we present extensive numerical experiments, in which our proposed method consistently improves the test errors compared with the standard mini-batch SGD in various models including SVM, logistic regression, and (non-convex) deep learning problems.