Partial Feedback Online Learning

TL;DR

This paper introduces a new partial-feedback online learning framework, developing theoretical tools like the PFLdim and PMSdim to characterize learnability and regret bounds.

Contribution

It extends classical online learning theory by defining collection version spaces and new dimensions for partial feedback settings, resolving open questions.

Findings

Characterized learnability using PFLdim and PMSdim.

Identified conditions where deterministic and randomized learnability coincide.

Showed minimax regret can be linear even with small hypothesis spaces beyond set realizability.

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.