The Sample Complexity of Online Reinforcement Learning: A Multi-Model Perspective

Abstract

We study the sample complexity of online reinforcement learning in the general non-episodic setting of nonlinear dynamical systems with continuous state and action spaces. Our analysis accommodates a large class of dynamical systems ranging from a finite set of nonlinear candidate models to models with bounded and Lipschitz continuous dynamics, to systems that are parametrized by a compact and real-valued set of parameters. In the most general setting, our algorithm achieves a policy regret of $\mathcal{O}(N \epsilon^2 + d_\mathrm{u}\mathrm{ln}(m(\epsilon))/\epsilon^2)$, where $N$ is the time horizon, $\epsilon$ is a user-specified discretization width, $d_\mathrm{u}$ the input dimension, and $m(\epsilon)$ measures the complexity of the function class under consideration via its packing number. In the special case where the dynamics are parametrized by a compact and real-valued set of parameters (such as neural networks, transformers, etc.), we prove a policy regret of $\mathcal{O}(\sqrt{d_\mathrm{u}N p})$, where $p$ denotes the number of parameters, recovering earlier sample-complexity results that were derived for *linear* *time-invariant* dynamical systems. While this article focuses on characterizing sample complexity, the proposed algorithms are likely to be useful in practice, due to their simplicity, their ability to incorporate prior knowledge, and their benign transient behaviors.