Learning with Multiple Correct Answers -- A Trichotomy of Regret Bounds under Different Feedback Models

2026-02-10

AI Summary

This work addresses the online learning setting where a single instance may admit multiple correct labels, yet not all outputs are acceptable. It introduces, for the first time, three distinct feedback models tailored to this multi-correct-answer scenario, systematically characterizing the optimal mistake bound under realizable conditions and establishing a trichotomy of regret bounds in the agnostic case. By proposing a novel combinatorial dimension to capture learning complexity, and integrating techniques from online learning theory with reductions to batch learning, the paper derives tight mistake and regret bounds. Furthermore, it leverages this combinatorial dimension to obtain an upper bound on the sample complexity of batch learning in this setting.

Technology Category

• Machine Learning: Online Learning & Bandits
• Multiagent Systems: Multiagent Learning
• Game Theory and Economic Paradigms: Adversarial Learning

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Social networks and social learning
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

We study an online learning problem with multiple correct answers, where each instance admits a set of valid labels, and in each round the learner must output a valid label for the queried example. This setting is motivated by language generation tasks, in which a prompt may admit many acceptable completions, but not every completion is acceptable. We study this problem under three feedback models. For each model, we characterize the optimal mistake bound in the realizable setting using an appropriate combinatorial dimension. We then establish a trichotomy of regret bounds across the three models in the agnostic setting. Our results also imply sample complexity bounds for the batch setup that depend on the respective combinatorial dimensions.

Problem

• online learning
• multiple correct answers
• feedback models
• regret bounds
• valid labels

Innovation

• multiple correct answers
• online learning
• regret trichotomy
• combinatorial dimension
• feedback models