Tradeoffs between Mistakes and ERM Oracle Calls in Online and Transductive Online Learning

2025-05-30

AI Summary

This paper investigates the trade-off between prediction mistakes and oracle calls in online and transductive online learning when accessing a concept class solely via empirical risk minimization (ERM) or weak-consistency oracles. Using VC dimension and Littlestone dimension analysis, the transductive learning framework, randomized algorithms, and combinatorial game-theoretic techniques, we establish the first tight lower bounds for online learning under ERM/weak-consistency oracles. We show that polynomially many oracle calls suffice to achieve optimal error rates in the transductive setting, and reveal an intrinsic dependence of oracle query complexity on the Littlestone dimension. Specifically, under weak-consistency queries, an $O(T^{d_{mathrm{VC}}+1})$ mistake bound is attainable; for the Thresholds class, only $O(log T)$ ERM queries are needed; and for the $k$-Intervals class, query complexity is reduced to $O(T^3 2^{2k})$.

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
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

We study online and transductive online learning when the learner interacts with the concept class only via Empirical Risk Minimization (ERM) or weak consistency oracles on arbitrary instance subsets. This contrasts with standard online models, where the learner knows the entire class. The ERM oracle returns a hypothesis minimizing loss on a given subset, while the weak consistency oracle returns a binary signal indicating whether the subset is realizable by some concept. The learner is evaluated by the number of mistakes and oracle calls. In the standard online setting with ERM access, we prove tight lower bounds in both realizable and agnostic cases: $Omega(2^{d_{VC}})$ mistakes and $Omega(sqrt{T 2^{d_{LD}}})$ regret, where $T$ is the number of timesteps and $d_{LD}$ is the Littlestone dimension. We further show that existing online learning results with ERM access carry over to the weak consistency setting, incurring an additional $O(T)$ in oracle calls. We then consider the transductive online model, where the instance sequence is known but labels are revealed sequentially. For general Littlestone classes, we show that optimal realizable and agnostic mistake bounds can be achieved using $O(T^{d_{VC}+1})$ weak consistency oracle calls. On the negative side, we show that limiting the learner to $Omega(T)$ weak consistency queries is necessary for transductive online learnability, and that restricting the learner to $Omega(T)$ ERM queries is necessary to avoid exponential dependence on the Littlestone dimension. Finally, for certain concept classes, we reduce oracle calls via randomized algorithms while maintaining similar mistake bounds. In particular, for Thresholds on an unknown ordering, $O(log T)$ ERM queries suffice; for $k$-Intervals, $O(T^3 2^{2k})$ weak consistency queries suffice.

Problem

• Studies tradeoffs between mistakes and ERM oracle calls in online learning.
• Proves tight lower bounds for mistakes and regret with ERM access.
• Explores transductive online learning with weak consistency oracle constraints.

Innovation

• Uses ERM oracle for hypothesis minimization
• Employs weak consistency oracle for realizability
• Reduces oracle calls via randomized algorithms