Algorithmic Foundation of Deep X-Risk Optimization

X-risk is a term introduced to represent a family of compositional measures or objectives, in which each data point is compared with a set of data points explicitly or implicitly for defining a risk function. It includes many widely used measures or objectives, e.g., AUROC, AUPRC, partial AUROC, NDCG, MAP, top-$K$ NDCG, top-$K$ MAP, listwise losses, p-norm push, top push, precision/recall at top $K$ positions, precision at a certain recall level, contrastive objectives, etc. While these measures/objectives and their optimization algorithms have been studied in the literature of machine learning, computer vision, information retrieval, and etc, optimizing these measures/objectives has encountered some unique challenges for deep learning. In this technical report, we survey our recent rigorous efforts for deep X-risk optimization (DXO) by focusing on its algorithmic foundation. We introduce a class of techniques for optimizing X-risk for deep learning. We formulate DXO into three special families of non-convex optimization problems belonging to non-convex min-max optimization, non-convex compositional optimization, and non-convex bilevel optimization, respectively. For each family of problems, we present some strong baseline algorithms and their complexities, which will motivate further research for improving the existing results. Discussions about the presented results and future studies are given at the end. Efficient algorithms for optimizing a variety of X-risks are implemented in the LibAUC library at www.libauc.org.