Regret-Oracle Complexity Tradeoffs in Agnostic Online Learning

2026-05-07

AI Summary

This work addresses the intractability of standard optimal algorithms in agnostic online learning and the prohibitively high oracle query complexity of existing oracle-efficient methods. The authors propose a dynamic, adaptive reduction mechanism based on a weakly consistent oracle, which reduces complexity by pruning unrealizable label sequences online. Innovatively leveraging the VC dimension to dynamically control the number of active paths, the method lowers the oracle query complexity from doubly exponential to polynomial, specifically $O(T^{d_{\mathrm{VC}}+1})$, while significantly reducing memory overhead and preserving a near-optimal expected regret bound. The study further establishes a theoretical trade-off between regret and query complexity, providing a lower bound of $\Omega(T/Q)$ on regret under a constrained query budget $Q$.

Abstract

Agnostic online learning is classically solved via a reduction to the realizable setting, utilizing Littlestone's Standard Optimal Algorithm (SOA) as a base learner. However, the SOA is computationally intractable to execute even for a single round. To overcome this barrier, recent work in oracle-efficient online learning replaces the SOA with a realizable base learner that accesses the concept class exclusively through an offline empirical risk minimization (ERM) oracle. While such agnostic learners achieve near-optimal expected regret, they suffer from a doubly-exponential oracle complexity of $O\big(T^{2^{O(d_\mathrm{LD})}}\big)$, where $d_\mathrm{LD}$ is the Littlestone dimension and $T$ is the number of rounds. In this work, we significantly improve this oracle complexity while relying on an even weaker primitive: a weak-consistency oracle, which merely decides whether a given labeled dataset is realizable. At the core of our approach is an adaptive and dynamic agnostic-to-realizable reduction that actively prunes non-realizable label sequences on the fly. By using the VC dimension ($d_\mathrm{VC}$) to bound the number of dynamically maintained active paths, our algorithm reduces the total query complexity down to $O(T^{d_\mathrm{VC}+1})$ while perfectly preserving near-optimal expected regret. Crucially, this dynamic pruning also yields a memory reduction over the standard reduction. Furthermore, we formally quantify the regret--oracle complexity tradeoff, providing upper bounds that smoothly interpolate between restricted query budgets and attainable expected regret. We complement these with lower bounds proving that any learner restricted to $Q = o(\sqrt{T})$ queries must suffer an expected regret of $Ω(T/Q)$.

Problem

• agnostic online learning
• oracle complexity
• regret
• realizable reduction
• computational efficiency

Innovation

• agnostic online learning
• oracle complexity
• weak-consistency oracle
• dynamic pruning
• regret-complexity tradeoff