A Convex Loss Function for Set Prediction with Optimal Trade-offs Between Size and Conditional Coverage

TL;DR

This paper introduces a convex loss function for set prediction that balances coverage and set size, with efficient algorithms and improved performance demonstrated on synthetic classification and regression datasets.

Contribution

It proposes a novel convex loss function based on Choquet integrals for set prediction, enabling optimal trade-offs between coverage and set size, with practical algorithms and experimental validation.

Findings

Improved set prediction accuracy over marginal coverage methods.

Efficient optimization algorithms for the proposed loss function.

Successful application to synthetic classification and regression tasks.

Abstract

We consider supervised learning problems in which set predictions provide explicit uncertainty estimates. Using Choquet integrals (a.k.a. Lov{\'a}sz extensions), we propose a convex loss function for nondecreasing subset-valued functions obtained as level sets of a real-valued function. This loss function allows optimal trade-offs between conditional probabilistic coverage and the ''size'' of the set, measured by a non-decreasing submodular function. We also propose several extensions that mimic loss functions and criteria for binary classification with asymmetric losses, and show how to naturally obtain sets with optimized conditional coverage. We derive efficient optimization algorithms, either based on stochastic gradient descent or reweighted least-squares formulations, and illustrate our findings with a series of experiments on synthetic datasets for classification and regression tasks, showing improvements over approaches that aim for marginal coverage.