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

arXiv: https://arxiv.org/abs/2201.00505 · HTML: https://mlsanity.com/paper/2201.00505/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 Jan 2022.

## Built on by (most cited first, 3 of 3)

- [Rank-based Decomposable Losses in Machine Learning: A Survey](https://mlsanity.com/paper/2207.08768/) (2022)
- [Fast Computation of Superquantile-Constrained Optimization Through Implicit Scenario Reduction](https://mlsanity.com/paper/2405.07965/) (2024)
- [Convergence of empirical subgradients for optimal transport-based objectives](https://mlsanity.com/paper/2605.28134/) (2026)

## Abstract (arXiv)

We consider a formulation of supervised learning that endows models with robustness to distributional shifts from training to testing. The formulation hinges upon the superquantile risk measure, also known as the conditional value-at-risk, which has shown promise in recent applications of machine learning and signal processing. We show that, thanks to a direct smoothing of the superquantile function, a superquantile-based learning objective is amenable to gradient-based optimization, using batch optimization algorithms such as gradient descent or quasi-Newton algorithms, or using stochastic optimization algorithms such as stochastic gradient algorithms. A companion software SPQR implements in Python the algorithms described and allows practitioners to experiment with superquantile-based supervised learning.