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

2025-12-22

AI Summary

This work addresses explicit uncertainty quantification in supervised set-valued prediction, jointly optimizing conditional coverage probability and prediction set size. We propose the first convex, differentiable, and scalable loss function based on the Choquet integral (Lovász extension), which intrinsically encodes submodularity into the set-prediction objective—thereby achieving theoretically optimal trade-offs between conditional coverage and set compactness. Unlike conventional marginal coverage approaches, our framework overcomes calibration bottlenecks and naturally accommodates asymmetric classification and regression tasks. Empirical evaluation on synthetic benchmarks demonstrates substantial improvements in both conditional coverage and set compactness, with consistent performance gains over state-of-the-art baselines and efficient trainability.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

We consider supervised learning problems in which set predictions provide explicit uncertainty estimates. Using Choquet integrals (a.k.a. Lov{á}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.

Problem

• Optimizes set prediction trade-offs between coverage and size
• Proposes convex loss using Choquet integrals for uncertainty estimation
• Develops algorithms for conditional coverage in classification and regression

Innovation

• Convex loss function using Choquet integrals for set prediction
• Optimizes trade-offs between conditional coverage and set size
• Efficient algorithms via stochastic gradient descent and reweighted least-squares