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R. Tyrell Rockafellar and collaborators introduced, in a series of works, newregression modeling methods based on the notion of superquantile (orconditional value-at-risk). These methods have been influential in economics,finance, management science, and operations research in general. Recently, theyhave been the subject of a renewed interest in machine learning, to addressissues of distributional robustness and fair allocation. In this paper, wereview some of these new applications of the superquantile, with references torecent developments. These applications involve nonsmooth superquantile-basedobjective functions that admit explicit subgradient calculations. To make thesesuperquantile-based functions amenable to the gradient-based algorithms popularin machine learning, we show how to smooth them by infimal convolution anddescribe numerical procedures to compute the gradients of the smoothapproximations. We put the approach into perspective by comparing it to othersmoothing techniques and by illustrating it on toy examples.

Superquantiles at Work: Machine Learning Applications and Efficient Subgradient Computation