Partial Feedback Online Learning

2026-01-29

AI Summary

This work investigates online learning with partial feedback, where only a single correct label is observed per round despite the existence of multiple valid labels. Under the set realizability assumption, the authors introduce two new complexity measures—the Partial-Feedback Littlestone dimension (PFLdim) and the Partial-Feedback Measure Shattering dimension (PMSdim)—which precisely characterize the minimax regret for deterministic and randomized learners, respectively, and reveal their indistinguishability under broad conditions. Tight regret bounds are established via set-valued version spaces, auxiliary dimension constructions, and combinatorial information-theoretic tools such as Helly numbers and nested structures. Furthermore, the study demonstrates that the problem may become information-theoretically unsolvable in the non-realizable setting, thereby underscoring the necessity of the proposed dimensions and resolving a key open question in set-valued online learning.

Technology Category

• Machine Learning: Online Learning & Bandits
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Learning to Search

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 a new learning protocol, termed partial-feedback online learning, where each instance admits a set of acceptable labels, but the learner observes only one acceptable label per round. We highlight that, while classical version space is widely used for online learnability, it does not directly extend to this setting. We address this obstacle by introducing a collection version space, which maintains sets of hypotheses rather than individual hypotheses. Using this tool, we obtain a tight characterization of learnability in the set-realizable regime. In particular, we define the Partial-Feedback Littlestone dimension (PFLdim) and the Partial-Feedback Measure Shattering dimension (PMSdim), and show that they tightly characterize the minimax regret for deterministic and randomized learners, respectively. We further identify a nested inclusion condition under which deterministic and randomized learnability coincide, resolving an open question of Raman et al. (2024b). Finally, given a hypothesis space H, we show that beyond set realizability, the minimax regret can be linear even when |H|=2, highlighting a barrier beyond set realizability.

Problem

• partial feedback
• online learning
• minimax regret
• set realizability
• learnability

Innovation

• Partial-Feedback Littlestone dimension
• Minimax regret
• Collection version space
• Set-realizable online learning
• Measure shattering dimension