Q-Learning with Fine-Grained Gap-Dependent Regret

2025-10-08

AI Summary

Existing model-free reinforcement learning algorithms suffer from overly coarse gap-dependent regret bounds that fail to capture the intrinsic structure of suboptimality gaps. Method: This paper establishes the first fine-grained, gap-dependent regret analysis framework for model-free RL in tabular MDPs. Contributions: (1) It derives the first tight, fine-grained upper bound for UCB-Hoeffding; (2) it introduces a novel analytical paradigm that explicitly separates optimal and suboptimal state-action pairs; (3) it corrects the design flaw in AMB and provides the first rigorous fine-grained regret guarantee for non-UCB-type algorithms (e.g., corrected AMB variants); (4) it proposes a simplified ULCB-Hoeffding algorithm, integrating Hoeffding confidence intervals, martingale difference analysis, and optimized Q-value updates. Theoretically, ULCB-Hoeffding achieves a tighter regret bound; empirically, it outperforms AMB, while the corrected AMB significantly improves convergence speed and stability.

Technology Category

• Machine Learning: Online Learning & Bandits
• Search and Optimization: Learning to Search
• Reasoning under Uncertainty: Sequential Decision Making

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
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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 $Q$-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.

Problem

• Establishing fine-grained gap-dependent regret bounds for model-free RL
• Developing novel analytical framework for UCB-based algorithms
• Addressing algorithm design issues in non-UCB-based reinforcement learning

Innovation

• Novel analytical framework separating optimal/suboptimal pairs
• ULCB-Hoeffding algorithm with simplified structure
• Refined AMB version fixing truncation and martingale issues