  The empirical loss, commonly referred to as the average loss, is extensively
utilized for training machine learning models. However, in order to address the
diverse performance requirements of machine learning models, the use of the
rank-based loss is prevalent, replacing the empirical loss in many cases. The
rank-based loss comprises a weighted sum of sorted individual losses,
encompassing both convex losses like the spectral risk, which includes the
empirical risk and conditional value-at-risk, and nonconvex losses such as the
human-aligned risk and the sum of the ranked range loss. In this paper, we
introduce a unified framework for the optimization of the rank-based loss
through the utilization of a proximal alternating direction method of
multipliers. We demonstrate the convergence and convergence rate of the
proposed algorithm under mild conditions. Experiments conducted on synthetic
and real datasets illustrate the effectiveness and efficiency of the proposed
algorithm.
