Bilinear Classes: A Structural Framework for Provable Generalization in RL (2103.10897v3)

Abstract: This work introduces Bilinear Classes, a new structural framework, which permit generalization in reinforcement learning in a wide variety of settings through the use of function approximation. The framework incorporates nearly all existing models in which a polynomial sample complexity is achievable, and, notably, also includes new models, such as the Linear $Q^/V^$ model in which both the optimal $Q$-function and the optimal $V$-function are linear in some known feature space. Our main result provides an RL algorithm which has polynomial sample complexity for Bilinear Classes; notably, this sample complexity is stated in terms of a reduction to the generalization error of an underlying supervised learning sub-problem. These bounds nearly match the best known sample complexity bounds for existing models. Furthermore, this framework also extends to the infinite dimensional (RKHS) setting: for the the Linear $Q^/V^$ model, linear MDPs, and linear mixture MDPs, we provide sample complexities that have no explicit dependence on the explicit feature dimension (which could be infinite), but instead depends only on information theoretic quantities.

• The paper introduces Bilinear Classes, a novel framework that extends RL models to address unseen settings with polynomial sample efficiency.
• It presents an algorithmic reduction from reinforcement learning to supervised learning, linking sample complexity to intrinsic information-theoretic quantities.
• The framework unifies traditional and new model classes, enabling sample-efficient learning even in infinite-dimensional and non-parametric environments.

Bilinear Classes: A Structural Framework for Provable Generalization in RL

This paper presents a novel framework called Bilinear Classes for addressing generalization in reinforcement learning (RL) using function approximation. This framework is relevant for a wide array of RL settings, including new models that were previously not encompassed by other frameworks. The key contribution of this work is showing that Bilinear Classes can permit efficient reinforcement learning by providing polynomial sample complexity.

Overview of Bilinear Classes

Bilinear Classes introduce a structured framework where the generalization capability of RL models is explored through function approximation. The framework is applicable to almost all existing models that achieve polynomial sample complexity and introduces new models like the Linear $Q^*/V^*$ model, where both the optimal $Q$-function and $V$-function are linear in a known feature space.

The framework provides an algorithmic approach to RL, based on a reduction to a supervised learning sub-problem. The sample complexity bound is linked to the generalization error of this sub-problem, and is nearly comparable to the best-known bounds for existing models. This aspect of Bilinear Classes is particularly noteworthy since it ties RL performance to well-understood supervised learning paradigms.

Sample Complexity and Information-Theoretic Measures

A significant contribution of Bilinear Classes is extending to infinite-dimensional (RKHS) settings, offering a reduction in sample complexity that does not explicitly depend on feature dimension. Instead, the complexity depends on intrinsic information-theoretical quantities. This is crucial because it allows the framework to adapt to non-parametric settings, thereby addressing shortcomings of existing parametric approaches, which may not gracefully handle model misspecification or approximation errors.

Examples and Models Captured

The Bilinear Classes framework incorporates many previously studied models, including Linear BeLLMan Complete Models and Linear Mixture Models, while also encompassing new categories like Linear $Q^\star/V^\star$ and models with Low Occupancy Complexity. The latter models benefit from unique properties of Bilinear Classes, enabling sample-efficient learning in scenarios where it was previously deemed impossible.

Implications and Future Directions

The introduction of Bilinear Classes as a framework demonstrates a unifying approach to RL that potentially closes gaps between distinct model types. By aligning RL generalization with information-theoretic principles traditionally associated with supervised learning, Bilinear Classes could pave the way for more robust RL applications. This is particularly relevant for RL's deployment in complex, real-world scenarios where traditional sample inefficiencies are untenable.

The paper suggests that future research could focus on exploring alternate RL models within the Bilinear Classes paradigm. Additionally, there may be room to enhance algorithmic strategies leveraging this framework, further closing the gap towards realizing universally efficient RL methodologies. As RL continues to interface with intricate environments and high-dimensional data, frameworks like Bilinear Classes could be instrumental in shaping the next generation of RL tools and techniques.