The Role of Inherent Bellman Error in Offline Reinforcement Learning with Linear Function Approximation

The Role of Inherent Bellman Error in Offline Reinforcement Learning with Linear Function Approximation

This paper investigates the offline reinforcement learning (RL) problem using linear function approximation, an area gaining significant traction due to the increasing complexity of RL environments. The authors focus on minimizing the inherent Bellman error, introducing a novel algorithm that provides guarantees based on single-policy coverage.

Key Contributions

1. Formalizing Inherent Bellman Error: The paper introduces the concept of inherent Bellman error, denoted as (\epsilon_{be}). Inherent Bellman error measures the maximum deviation in value-function approximation under the Bellman backup operator. This parameter is critical as it encapsulates the fundamental challenge of value iteration success in offline RL, where the dataset might not fully cover the state-action space.

2. Algorithm Development: The authors propose a computationally efficient algorithm that leverages single-policy coverage to output policies with satisfactory performance. The algorithm ensures that the suboptimality scales with (\sqrt{\epsilon_{be}}), offering a robust performance even in cases where the Bellman error is non-zero. This contrasts with existing methods that typically exhibit linear degradation in performance with the increase of the Bellman error.

3. Performance Guarantees: Theoretical guarantees show that under the assumption of linear Bellman completeness ((\epsilon{be} = 0)), the algorithm achieves the first known guarantee under single-policy coverage. Moreover, the authors establish that no approach can outperform the (\sqrt{\epsilon{be}}) scaling, presenting a lower bound that confirms this result.

Algorithmic Approach

The proposed algorithm is built around an actor-critic framework. The Actor generates policies using a no-regret learning method optimized for single-policy coverage. At every iteration, the Critic constructs pessimistic value function estimates based on the given dataset, ensuring that the value of the policy generated by the Actor does not overestimate the possible rewards due to incomplete state-action coverage.

The key insight here is the utilization of perturbed linear policies, where the policy is chosen by adding random perturbations to the linear value function at each step. This strategy allows the algorithm to explore the feature space more broadly, preventing the overly optimistic value estimations that can occur when certain state-action pairs are underexplored in the dataset.

Theoretical Implications

The paper's theoretical analysis reveals several vital implications for offline RL:

Suboptimality Bound:

The suboptimality of the algorithm, that is, the difference between the value of the output policy and the optimal value, scales with (\sqrt{\epsilon_{be}}). This result is backed by rigorous proofs showing that such scaling is unavoidable, thus providing a definitive answer to the performance limits imposed by inherent Bellman error.

Single-Policy Coverage:

The algorithm's reliance on single-policy coverage underlines its practicality. Instead of requiring comprehensive exploration of the entire state-action space, the algorithm can perform well even when the available data is gathered from executing reasonably good policies.

Comparison with Bellman Restricted Closedness:

The results contrast with settings where Bellman restricted closedness (completeness) is assumed. Bellman restricted closedness requires stronger assumptions, typically leading to linear dependence on the approximation error. The authors' approach demonstrates that by targeting perturbed linear policies under single-policy coverage, one can achieve efficient learning in more relaxed conditions.

Practical Implications and Future Directions

In practice, the findings suggest that offline RL can be significantly robust when the learning algorithm properly manages the inherent Bellman error. Practically, this means that even in environments where data is imperfect and gathered from suboptimal policies, efficient learning is feasible.

Future developments in this domain might explore extending the theoretical guarantees to broader classes of function approximations, such as neural networks. Additionally, expanding the perturbation methods to adapt dynamically based on observed data distributions could further enhance performance, particularly in highly stochastic environments.

Conclusion

Overall, this paper makes significant strides in understanding the offline RL landscape under linear function approximation. By focusing on the inherent Bellman error and utilizing a novel algorithmic approach, the authors provide valuable insights and a robust methodological framework that could influence both theoretical research and practical applications in reinforcement learning.