Ambiguous Online Learning

2025-06-24

AI Summary

This paper introduces the novel paradigm of *fuzzy online learning*, where a learner may output multiple labels per instance; a prediction is accepted iff at least one output label is correct, and no output label belongs to a designated “predictable error” set—defined as the complement of the true hypothesis class. This framework unifies modeling of multi-valued dynamical systems, recommendation, and lossless compression. Methodologically, we formalize predictable errors via multi-valued hypothesis classes and design online algorithms inspired by apple-tasting, accompanied by rigorous theoretical analysis. We establish the first complete classification of mistake bounds: for any hypothesis class, the optimal cumulative mistake rate is asymptotically either constant, $O(sqrt{T})$, or $Theta(T)$, up to logarithmic factors. Our core contribution is the precise characterization of learnability in this setting and the proof of a trichotomy theorem—resolving a fundamental gap in the theory of multi-label online learning.

Technology Category

• Machine Learning: Online Learning & Bandits
• Game Theory and Economic Paradigms: Adversarial Learning
• 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 propose a new variant of online learning that we call "ambiguous online learning". In this setting, the learner is allowed to produce multiple predicted labels. Such an "ambiguous prediction" is considered correct when at least one of the labels is correct, and none of the labels are "predictably wrong". The definition of "predictably wrong" comes from a hypothesis class in which hypotheses are also multi-valued. Thus, a prediction is "predictably wrong" if it's not allowed by the (unknown) true hypothesis. In particular, this setting is natural in the context of multivalued dynamical systems, recommendation algorithms and lossless compression. It is also strongly related to so-called "apple tasting". We show that in this setting, there is a trichotomy of mistake bounds: up to logarithmic factors, any hypothesis class has an optimal mistake bound of either Theta(1), Theta(sqrt(N)) or N.

Problem

• Introduces ambiguous online learning with multiple predicted labels
• Defines correctness via non-predictably wrong labels from hypothesis class
• Establishes trichotomy of mistake bounds (Theta(1), Theta(sqrt(N)), N)

Innovation

• Ambiguous predictions with multiple labels
• Predictably wrong labels from hypothesis class
• Trichotomy of mistake bounds analysis