Model-based Reinforcement Learning and the Eluder Dimension (1406.1853v2)

Abstract: We consider the problem of learning to optimize an unknown Markov decision process (MDP). We show that, if the MDP can be parameterized within some known function class, we can obtain regret bounds that scale with the dimensionality, rather than cardinality, of the system. We characterize this dependence explicitly as $\tilde{O}(\sqrt{d_K d_E T})$ where $T$ is time elapsed, $d_K$ is the Kolmogorov dimension and $d_E$ is the \emph{eluder dimension}. These represent the first unified regret bounds for model-based reinforcement learning and provide state of the art guarantees in several important settings. Moreover, we present a simple and computationally efficient algorithm \emph{posterior sampling for reinforcement learning} (PSRL) that satisfies these bounds.

Insights on Model-Based Reinforcement Learning and the Eluder Dimension

In this paper, the authors explore the domain of model-based reinforcement learning (RL) with a focus on optimizing unknown Markov Decision Processes (MDPs). The fundamental contribution of this work lies in establishing regret bounds that are contingent on the dimensionality of the system, specifically through the Kolmogorov and eluder dimensions, rather than its cardinality. As a result, this approach offers a more scalable solution in complex environments where the state and action spaces can be prohibitively large.

Key Findings and Contribution

1. Regret Bounds Based on Dimensionality: The regret bounds provided in the paper are represented as $\tilde{O}(\sqrt{d_K d_E T})$, with $T$ denoting time, $d_K$ representing the Kolmogorov dimension, and $d_E$ the eluder dimension. This characterization marks a significant departure from traditional bounds that scale with the cardinality of state and action spaces.
2. Posterior Sampling for Reinforcement Learning (PSRL): The authors introduce a computationally efficient algorithm, PSRL, which adheres to the derived regret bounds. This algorithm utilizes a Bayesian approach by drawing samples of MDPs from the posterior distribution and employing the optimal policy of the sampled MDP for exploration and exploitation.
3. Unified Analysis for Model-Based RL: By extending the definition of the eluder dimension—a measure initially introduced for bandits—the paper provides a comprehensive analysis applicable to a wide range of RL settings. This forms a cohesive theoretical framework bridging different RL paradigms that handle parameterized environments.

Implications

Theoretical Implications

• Dimensionality vs. Cardinality: The implication of focusing on dimensionality (Kolmogorov and eluder dimensions) instead of cardinality (state and action spaces) suggests a paradigm shift. It hints at broader applicability and efficiency in high-dimensional spaces, akin to trends seen in various AI and machine learning fields.
• Eluder Dimension: By demonstrating the practicality of the eluder dimension in RL, this research potentially paves the way for further studies that leverage this concept in other areas of machine learning, beyond Bayesian and model-based learning frameworks.

Practical Implications

• Computational Resources: The reliance on PSRL provides an avenue for leveraging computational efficiency, offering potential savings in computational resources over exhaustive state-action evaluations typically required in RL.
• Policy Optimization in Complex Environments: The implications for policy optimization in environments with high complexity are significant. Researchers and practitioners might employ these insights for RL applications in robotics, autonomous systems, and adaptive decision-making systems.

Speculative Future Developments

The exploration of regret bounds that depend on dimensionality opens various future research avenues. We might see:

• Enhanced algorithms for real-time AI applications that can operate efficiently in dynamically changing environments without explicit preconditioning on states and actions.
• Exploration into more complex non-linear representations and their tractable analytical frameworks within the boundaries set by this work's findings.
• Further cross-disciplinary adoption of eluder dimension analysis, extending into areas such as natural language processing, where dimensionality significantly impacts learning and inferencing.

Conclusion

The paper fundamentally challenges the traditional approaches to model-based reinforcement learning by reorienting the analytical focus to dimensionality over cardinality. Through the development of PSRL and the leveraging of eluder dimensions, the authors provide both a computationally feasible algorithm and a theoretical lens that could inspire continuing innovation in reinforcement learning and broader AI system design and analysis. While practical implementation may face challenges, particularly in executions involving approximate MDP planning, the theoretical foundation offers a profound stepping stone for subsequent advancements.