Learning Near Optimal Policies with Low Inherent Bellman Error (2003.00153v3)

Abstract: We study the exploration problem with approximate linear action-value functions in episodic reinforcement learning under the notion of low inherent BeLLMan error, a condition normally employed to show convergence of approximate value iteration. First we relate this condition to other common frameworks and show that it is strictly more general than the low rank (or linear) MDP assumption of prior work. Second we provide an algorithm with a high probability regret bound $\widetilde O(\sum_{t=1}^H d_t \sqrt{K} + \sum_{t=1}^H \sqrt{d_t} \IBE K)$ where $H$ is the horizon, $K$ is the number of episodes, $\IBE$ is the value if the inherent BeLLMan error and $d_t$ is the feature dimension at timestep $t$. In addition, we show that the result is unimprovable beyond constants and logs by showing a matching lower bound. This has two important consequences: 1) it shows that exploration is possible using only \emph{batch assumptions} with an algorithm that achieves the optimal statistical rate for the setting we consider, which is more general than prior work on low-rank MDPs 2) the lack of closedness (measured by the inherent BeLLMan error) is only amplified by $\sqrt{d_t}$ despite working in the online setting. Finally, the algorithm reduces to the celebrated \textsc{LinUCB} when $H=1$ but with a different choice of the exploration parameter that allows handling misspecified contextual linear bandits. While computational tractability questions remain open for the MDP setting, this enriches the class of MDPs with a linear representation for the action-value function where statistically efficient reinforcement learning is possible.

• The paper introduces a novel RL algorithm that utilizes the low inherent Bellman error framework to achieve optimal statistical rates in cumulative regret.
• The paper generalizes the common low-rank MDP assumption, enabling broader applications in RL through the use of approximate linear action-value functions.
• The rigorous regret analysis, supported by matching lower bounds, confirms the statistical tightness and potential efficacy of the proposed method in exploring complex environments.

Learning Near Optimal Policies with Low Inherent BeLLMan Error

This paper addresses the exploration problem in reinforcement learning (RL) using approximate linear action-value functions within the framework of low inherent BeLLMan error (IBE). This condition, the authors argue, is more general than the low-rank MDP assumption common in prior work, and is leveraged to develop an algorithm demonstrating high-probability regret bounds. The implications of these bounds are further substantiated by a matching lower bound analysis, suggesting the results are statistically tight.

Summary of Contributions

The paper makes several critical contributions to the field of reinforcement learning:

1. Generalization of Low-Rank Assumptions: The notion of low inherent BeLLMan error (IBE) is shown to be a broader framework compared to low-rank MDPs. The authors make a compelling case that the IBE condition is more versatile, accommodating a wider range of scenarios in which RL can be statistically efficient.
2. Algorithm Development: A novel reinforcement learning algorithm is proposed that achieves an optimal statistical rate in terms of cumulative regret. The algorithm's regret bound is given by $$\widetilde O(\sum_{t=1}^H d_t \sqrt{K} + \sum_{t=1}^H \sqrt{d_t} K)$$, where $H$ represents the horizon, $K$ the number of episodes, and $d_t$ the feature dimension at each timestep.
3. Regret Analysis: The regret bound is shown to be unimprovable beyond constants and logarithmic factors by a complementary lower bound proof. Such comprehensive treatment underscores the theoretical solidity of the paper's claims.
4. Algorithm Performance: The proposed method is capable of effective exploration in settings where batch assumptions hold, and the online setting's inherent BeLLMan errors are only amplified by $\sqrt{d_t}$. The algorithm can handle misspecified contextual linear bandits, reducing to LinUCB with modified exploration parameters for such cases.
5. Future Work and Computational Tractability: While the statistical efficiency is addressed, the authors note an open challenge regarding the computational efficiency in the MDP setting, inviting further research to bridge this gap.

Implications and Speculations on Future AI

The implications of this work are both theoretical and practical. Theoretically, it extends our understanding of when and how exploration strategies can be optimized in environments modeled by MDPs with linear architectures. On a practical front, the ideas presented could influence the development of more sophisticated learning algorithms capable of functioning efficiently in complex environments often characterized by misspecified models.

Going forward, computational tractability remains an open area for development. The reinforcement learning community may see advancements in algorithm design that address these computational challenges, potentially through novel computational frameworks or algorithmic improvements that maintain the statistical strengths noted here.

This work sits at the intersection of function approximation and exploration-exploitation strategies, and may inspire further studies exploring the generalization capabilities of RL systems under various assumptions about environmental dynamics. Such research could prove pivotal in designing AI systems that learn effectively in real-world, imperfectly specified environments.