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We consider a formulation of supervised learning that endows models withrobustness to distributional shifts from training to testing. The formulationhinges upon the superquantile risk measure, also known as the conditionalvalue-at-risk, which has shown promise in recent applications of machinelearning and signal processing. We show that, thanks to a direct smoothing ofthe superquantile function, a superquantile-based learning objective isamenable to gradient-based optimization, using batch optimization algorithmssuch as gradient descent or quasi-Newton algorithms, or using stochasticoptimization algorithms such as stochastic gradient algorithms. A companionsoftware SPQR implements in Python the algorithms described and allowspractitioners to experiment with superquantile-based supervised learning.

Superquantile-based learning: a direct approach using gradient-based optimization