Optimal Mistake Bounds for Transductive Online Learning

2025-12-14

AI Summary

This paper resolves a 30-year-old open problem: precisely characterizing the gap in learnability between transductive online learning (which leverages unlabeled data) and standard online learning. Building on the Littlestone dimension (d), we establish for the first time the optimal mistake bound for transductive online learning as (Theta(sqrt{d})), revealing a quadratic improvement over the (Theta(d)) bound of standard online learning. Methodologically, we integrate combinatorial game-theoretic analysis, information-theoretic lower-bound constructions, and algorithmic upper-bound design—raising the prior best lower bound from (Omega(log d)) to (Omega(sqrt{d})) and providing a matching (O(sqrt{d})) upper bound. This tight characterization conclusively settles the problem and furnishes a rigorous theoretical foundation for leveraging unlabeled data to enhance online learning performance.

Technology Category

• Machine Learning: Online Learning & Bandits
• Search and Optimization: Learning to Search
• Game Theory and Economic Paradigms: Adversarial Learning

Application Category

• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• 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

Abstract

We resolve a 30-year-old open problem concerning the power of unlabeled data in online learning by tightly quantifying the gap between transductive and standard online learning. In the standard setting, the optimal mistake bound is characterized by the Littlestone dimension $d$ of the concept class $H$ (Littlestone 1987). We prove that in the transductive setting, the mistake bound is at least $Ω(sqrt{d})$. This constitutes an exponential improvement over previous lower bounds of $Ω(loglog d)$, $Ω(sqrt{log d})$, and $Ω(log d)$, due respectively to Ben-David, Kushilevitz, and Mansour (1995, 1997) and Hanneke, Moran, and Shafer (2023). We also show that this lower bound is tight: for every $d$, there exists a class of Littlestone dimension $d$ with transductive mistake bound $O(sqrt{d})$. Our upper bound also improves upon the best known upper bound of $(2/3)d$ from Ben-David, Kushilevitz, and Mansour (1997). These results establish a quadratic gap between transductive and standard online learning, thereby highlighting the benefit of advance access to the unlabeled instance sequence. This contrasts with the PAC setting, where transductive and standard learning exhibit similar sample complexities.

Problem

• Resolves the gap between transductive and standard online learning
• Proves a lower bound of Ω(√d) for transductive mistake bounds
• Establishes a quadratic gap highlighting unlabeled data benefit

Innovation

• Proves transductive mistake bound is at least Ω(√d)
• Shows tight upper bound O(√d) for Littlestone dimension d
• Establishes quadratic gap between transductive and standard online learning