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Q-Learning with Fine-Grained Gap-Dependent Regret

ICLR 2026 Conference SubmissionAnonymous Authors
OpenReview Score: 6.5Download Report PDF
Reinforcement LearningQ-LearningRegretSuboptimality Gap.
Abstract:

We study fine-grained gap-dependent regret bounds for model-free reinforcement learning in episodic tabular Markov Decision Processes. Existing model-free algorithms achieve minimax worst-case regret, but their gap-dependent bounds remain coarse and fail to fully capture the structure of suboptimality gaps. We address this limitation by establishing fine-grained gap-dependent regret bounds for both UCB-based and non-UCB-based algorithms. In the UCB-based setting, we develop a novel analytical framework that explicitly separates the analysis of optimal and suboptimal state-action pairs, yielding the first fine-grained regret upper bound for UCB-Hoeffding (Jin et al., 2018). To highlight the generality of this framework, we introduce ULCB-Hoeffding, a new UCB-based algorithm inspired by AMB (Xu et al., 2021) but with a simplified structure, which enjoys fine-grained regret guarantees and empirically outperforms AMB. In the non-UCB-based setting, we revisit the only known algorithm AMB, and identify two key issues in its algorithm design and analysis: improper truncation in the QQ-updates and violation of the martingale difference condition in its concentration argument. We propose a refined version of AMB that addresses these issues, establishing the first rigorous fine-grained gap-dependent regret for a non-UCB-based method, with experiments demonstrating improved performance over AMB.

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This report is AI-GENERATED using Large Language Models and WisPaper (A scholar search engine). It analyzes academic papers' tasks and contributions against retrieved prior work. While this system identifies POTENTIAL overlaps and novel directions, ITS COVERAGE IS NOT EXHAUSTIVE AND JUDGMENTS ARE APPROXIMATE. These results are intended to assist human reviewers and SHOULD NOT be relied upon as a definitive verdict on novelty.
NOTE that some papers exist in multiple, slightly different versions (e.g., with different titles or URLs). The system may retrieve several versions of the same underlying work. The current automated pipeline does not reliably align or distinguish these cases, so human reviewers will need to disambiguate them manually.
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Overview

Overall Novelty Assessment

The paper establishes fine-grained gap-dependent regret bounds for model-free reinforcement learning in tabular episodic MDPs. It resides in the 'UCB-Based Optimistic Algorithms' leaf, which contains four papers total including the original work. This leaf sits within the broader 'Tabular Episodic MDP Algorithms with Gap-Dependent Bounds' branch, indicating a moderately populated research direction. The paper's focus on separating analysis of optimal versus suboptimal state-action pairs and introducing ULCB-Hoeffding represents an effort to refine existing UCB-based approaches in a well-established but still-active subfield.

The taxonomy reveals neighboring leaves addressing bootstrap methods and non-UCB optimistic approaches, both pursuing gap-dependent bounds through alternative mechanisms. The broader branch structure shows parallel efforts in function approximation, risk-sensitive objectives, and non-stationary environments. The paper's emphasis on fine-grained analysis connects it to theoretical foundations exploring instance-dependent complexity, while its UCB-based methodology distinguishes it from posterior sampling and bootstrap-based alternatives. The taxonomy's scope and exclude notes clarify that this work targets standard stationary episodic settings without function approximation or multi-agent constraints.

Among thirty candidates examined, the first contribution (novel analytical framework for UCB-based algorithms) showed no clear refutation across ten candidates, suggesting potential novelty in the separation technique for optimal and suboptimal pairs. The second contribution (ULCB-Hoeffding algorithm) similarly encountered no refuting candidates among ten examined, indicating the specific algorithmic design may be new. The third contribution (refined AMB algorithm) found one refutable candidate among ten examined, suggesting some overlap with prior AMB-related work. These statistics reflect a limited semantic search scope rather than exhaustive coverage of the field.

The analysis suggests moderate novelty within a focused research direction, with the UCB-based framework and ULCB-Hoeffding appearing more distinctive than the AMB refinement based on the limited search. The taxonomy positioning in a four-paper leaf indicates neither a highly crowded nor entirely sparse area. Readers should note that these assessments derive from top-thirty semantic matches and may not capture all relevant prior work, particularly in adjacent leaves or recent preprints outside the search scope.

Taxonomy

Core-task Taxonomy Papers
37
3
Claimed Contributions
30
Contribution Candidate Papers Compared
1
Refutable Paper

Research Landscape Overview

Core task: fine-grained gap-dependent regret bounds for model-free reinforcement learning. The field has evolved to address how quickly agents can learn near-optimal policies when the problem structure offers exploitable gaps between action values. The taxonomy reveals several major branches: tabular episodic methods that leverage optimism or posterior sampling to achieve instance-dependent guarantees; function approximation and structured settings that extend these ideas to large or continuous state spaces; specialized extensions covering risk-sensitivity, robustness, and non-stationarity; multi-agent and federated scenarios; and foundational frameworks that unify exploration strategies across diverse problem classes. Works such as Regret-Optimal Q-Learning[4] and Reference-Advantage Decomposition[7] exemplify how UCB-based optimistic algorithms refine gap-dependent analyses in tabular episodic MDPs, while Decision-Estimation Coefficients[6] and Instance-Optimality Theory[11] provide broader theoretical lenses. Meanwhile, branches on posterior sampling (e.g., Posterior Sampling Episodic[27]) and discounted infinite-horizon settings offer alternative algorithmic paradigms, and application-specific algorithms demonstrate how these principles transfer to real-world domains. A particularly active line of research focuses on tightening regret bounds by exploiting finer notions of problem difficulty—moving beyond worst-case horizon or state-action counts to capture suboptimality gaps, variance, or cascaded structure (Cascaded Gaps[17], Sharp Variance Bounds[10]). Another contrasting direction explores robustness and adaptivity: handling adversarial perturbations (Robust Online Decisions[20]), non-stationary environments (Non-Stationary Episodic[29]), or federated learning constraints (Federated Q-Learning[9], Federated Learning Bounds[3]). Fine-Grained Q-Learning[0] sits squarely within the tabular episodic UCB-based cluster, emphasizing refined gap-dependent guarantees for model-free algorithms. It shares methodological kinship with Non-Asymptotic Gap-Dependent[28] and Tail Distribution Regret[25], which also pursue instance-specific bounds, but distinguishes itself by targeting finer-grained characterizations of the learning dynamics. This positioning highlights an ongoing effort to bridge worst-case and problem-dependent perspectives, offering practitioners tighter performance predictions when favorable problem structure is present.

Claimed Contributions

Novel fine-grained analytical framework for UCB-based algorithms

The authors introduce a new analytical framework that separates the analysis of optimal and suboptimal state-action pairs, enabling the first fine-grained gap-dependent regret upper bound for UCB-Hoeffding and extending to ULCB-Hoeffding. This framework analyzes each state-action pair separately and establishes recursive relationships for cumulative weighted visitation counts.

10 retrieved papers
ULCB-Hoeffding algorithm with fine-grained regret guarantees

The authors propose ULCB-Hoeffding, a simplified UCB-based algorithm that removes the problematic multi-step bootstrapping from AMB while retaining the ULCB mechanism. This algorithm achieves fine-grained gap-dependent regret bounds and demonstrates improved empirical performance over AMB.

10 retrieved papers
Refined AMB algorithm with rigorous fine-grained analysis

The authors identify algorithmic and analytical issues in the original AMB algorithm and propose Refined AMB, which removes improper truncations, rigorously proves unbiasedness of multi-step bootstrapping estimators, ensures martingale difference conditions hold, and establishes the first valid fine-grained gap-dependent regret bound for a non-UCB-based method.

10 retrieved papers
Can Refute

Core Task Comparisons

Comparisons with papers in the same taxonomy category

Contribution Analysis

Detailed comparisons for each claimed contribution

Contribution

Novel fine-grained analytical framework for UCB-based algorithms

The authors introduce a new analytical framework that separates the analysis of optimal and suboptimal state-action pairs, enabling the first fine-grained gap-dependent regret upper bound for UCB-Hoeffding and extending to ULCB-Hoeffding. This framework analyzes each state-action pair separately and establishes recursive relationships for cumulative weighted visitation counts.

Contribution

ULCB-Hoeffding algorithm with fine-grained regret guarantees

The authors propose ULCB-Hoeffding, a simplified UCB-based algorithm that removes the problematic multi-step bootstrapping from AMB while retaining the ULCB mechanism. This algorithm achieves fine-grained gap-dependent regret bounds and demonstrates improved empirical performance over AMB.

Contribution

Refined AMB algorithm with rigorous fine-grained analysis

The authors identify algorithmic and analytical issues in the original AMB algorithm and propose Refined AMB, which removes improper truncations, rigorously proves unbiasedness of multi-step bootstrapping estimators, ensures martingale difference conditions hold, and establishes the first valid fine-grained gap-dependent regret bound for a non-UCB-based method.

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