Reinforcement Learning with General Value Function Approximation: Provably Efficient Approach via Bounded Eluder Dimension (2005.10804v3)

Abstract: Value function approximation has demonstrated phenomenal empirical success in reinforcement learning (RL). Nevertheless, despite a handful of recent progress on developing theory for RL with linear function approximation, the understanding of general function approximation schemes largely remains missing. In this paper, we establish a provably efficient RL algorithm with general value function approximation. We show that if the value functions admit an approximation with a function class $\mathcal{F}$, our algorithm achieves a regret bound of $\widetilde{O}(\mathrm{poly}(dH)\sqrt{T})$ where $d$ is a complexity measure of $\mathcal{F}$ that depends on the eluder dimension [Russo and Van Roy, 2013] and log-covering numbers, $H$ is the planning horizon, and $T$ is the number interactions with the environment. Our theory generalizes recent progress on RL with linear value function approximation and does not make explicit assumptions on the model of the environment. Moreover, our algorithm is model-free and provides a framework to justify the effectiveness of algorithms used in practice.

• The paper presents a novel RL algorithm that achieves provable efficiency by deriving regret bounds with general value function approximation.
• It employs a stable upper-confidence bonus through sensitivity and importance sampling to manage exploration without explicit feature extraction.
• The approach extends beyond linear paradigms, robustly addressing both accurate and misspecified models in complex environments.

Overview of "Reinforcement Learning with General Value Function Approximation: Provably Efficient Approach via Bounded Eluder Dimension"

The paper "Reinforcement Learning with General Value Function Approximation: Provably Efficient Approach via Bounded Eluder Dimension" presents a notable advancement in the reinforcement learning domain by addressing the theoretical understanding of general value function approximation. While function approximation has proven empirically successful in reinforcement learning applications, solid theoretical guarantees for algorithms using these approximations have remained limited, especially in the context of general function classes like deep neural networks.

Main Contributions

The authors propose a new reinforcement learning algorithm that achieves provable efficiency with general value function approximation, providing regret bounds that are pivotal for assessing the performance of such algorithms. Specifically, they demonstrate that their algorithm achieves a regret bound of $O(poly(dH)VT)$, where $d$ is a measure of complexity derived from eluder dimension and log-covering numbers, $H$ represents the planning horizon, and $T$ denotes the number of interactions with the environment. This approach generalizes prior progress focused on linear function approximation and does not require explicit assumptions regarding the environment model, offering a robust and model-free solution.

Algorithmic Insights

At the core of the proposed approach is a Q-learning algorithm that operates efficiently with any specified value function class, sidestepping the need for explicit feature extractors or assumptions related to the transition model. The formulation uses a stable upper-confidence bonus function, designed via a sensitivity sampling mechanism, to manage exploratory actions efficiently. Furthermore, the algorithm incorporates importance sampling to reduce computational complexity and stabilize the bonus function, which plays a significant role in guaranteeing a concentrated estimation of the Q-function.

Theoretical Implications

The paper rigorously derives regret bounds under both precise and misspecified modeling conditions. Under model misspecification, the algorithm still performs effectively by accounting for potential inaccuracies in the model through refined modifications of the core algorithmic components. The algorithm demonstrates an ability to adapt to settings where conventional assumptions do not hold, showcasing its general applicability.

Comparison to Prior Work

While the algorithm provides slightly worse regret bounds compared to some specialized algorithms designed for linear cases, it stands out by addressing the broader and more complex setting of general function classes. This paper’s approach leverages the eluder dimension to bound the complexity, aligning with recent research in reinforcement learning and bandit problems. The researchers meticulously handle the dependencies between collected samples and estimations—an area often neglected or oversimplified in theoretical analyses of RL algorithms.

Practical and Future Directions

Practically, the algorithm offers insights into why contemporary RL methods employing neural network approximations perform well despite their theoretical limitations. By exploring the structure intrinsic to general function classes, this research potentially unlocks new avenues for developing RL methods in complex environments, such as those with continuous state spaces.

Future work could include extending these theoretical results to policy-based reinforcement learning methods or exploring even broader classes of function approximation. The algorithm offers a solid foundation for leveraging the eluder dimension in reinforcement learning, holding promise for further exploration in optimization and representation learning.

Overall, the paper represents a substantial advancement toward understanding and implementing reinforcement learning algorithms with general value function approximation, bridging important gaps between empirical successes and theoretical guarantees.