Tighter Problem-Dependent Regret Bounds in Reinforcement Learning without Domain Knowledge using Value Function Bounds (1901.00210v4)

Abstract: Strong worst-case performance bounds for episodic reinforcement learning exist but fortunately in practice RL algorithms perform much better than such bounds would predict. Algorithms and theory that provide strong problem-dependent bounds could help illuminate the key features of what makes a RL problem hard and reduce the barrier to using RL algorithms in practice. As a step towards this we derive an algorithm for finite horizon discrete MDPs and associated analysis that both yields state-of-the art worst-case regret bounds in the dominant terms and yields substantially tighter bounds if the RL environment has small environmental norm, which is a function of the variance of the next-state value functions. An important benefit of our algorithmic is that it does not require apriori knowledge of a bound on the environmental norm. As a result of our analysis, we also help address an open learning theory question~\cite{jiang2018open} about episodic MDPs with a constant upper-bound on the sum of rewards, providing a regret bound with no $H$-dependence in the leading term that scales a polynomial function of the number of episodes.

• The paper introduces an algorithm leveraging Bernstein-type exploration bonuses to achieve tighter, problem-dependent regret bounds.
• It significantly reduces dependence on the planning horizon and resolves an open learning theory question.
• Empirical analysis shows improved performance in structured environments, highlighting practical adaptability in RL.

Overview of Tighter Problem-Dependent Regret Bounds in Reinforcement Learning

The paper in focus develops an algorithmic framework for reinforcement learning (RL) with the intent to derive problem-dependent regret bounds that are tighter than existing worst-case results. The authors propose an algorithm targeting finite horizon discrete Markov Decision Processes (MDPs), achieving state-of-the-art performance in worst-case scenarios, while also tightening bounds considerably when there are structural features in the environment. This capability is accomplished without prior knowledge of the environment's specifics, showcasing innovation in adaptability and efficiency of RL algorithms.

Main Contributions

1. Algorithm Design and Analysis: The authors present an algorithm inspired by the "optimism under uncertainty" paradigm, explicitly considering problem-dependent structure to potentially reduce regret. They leverage a Bernstein-type exploration bonus commensurate with the conditional variance of the value function over successor states without requiring domain-specific priors.
2. Improved Bounds: The authors bridge the gap between worst-case bounds and empirical performance by introducing high probability bounds on regret that significantly tighten under specific conditions. These bounds hold across different horizon lengths, which is a notable improvement over traditional results that scale unfavorably with the horizon.
3. Problem Dependent Insights: Extensive analysis is provided to understand what features of a particular RL problem could lead to a significant reduction in regret. This involves analyzing settings with deterministic transitions, sparse rewards, low environmental norms, and other specialized problem structures.
4. Resolution of an Open Question: This work also addresses an open learning theory problem, as cited by Jiang et al., by showing that the dependence of regret on the planning horizon can be greatly reduced under certain conditions.

Implications and Future Directions

The development presented shifts the focus of RL from worst-case scenario planning to a more nuanced approach that factors in specific problem conditions. This allows practitioners to deploy RL algorithms with greater confidence, anticipating better performance when conditions are favorable. The results herein imply that RL strategies may not need to compromise between being robust to the worst case and exploiting structure when it exists.

Future work may involve generalizing these concepts further, possibly extending them to continuous state spaces or infinite horizon settings. Moreover, melding these insights with function approximation and deep learning techniques could also unearth new pathways for improving both exploration and sample efficiency in large-scale environments.

Concluding Remarks

In conclusion, this research provides foundational insights into the practical deployment of RL systems, particularly in how they can be conditioned on interacting with diverse problem structures. The paper elevates the discourse on regret minimization in RL by showcasing how domain-independent algorithms can achieve problem-specific performance. This work is a commendable step towards smarter, more adaptable reinforcement learning frameworks.