We study the representation complexity of model-based and model-free reinforcement learning (RL) in the context of circuit complexity. We prove theoretically that there exists a broad class of MDPs such that their underlying transition and reward functions can be represented by constant depth circuits with polynomial size, while the optimal $Q$-function suffers an exponential circuit complexity in constant-depth circuits. By drawing attention to the approximation errors and building connections to complexity theory, our theory provides unique insights into why model-based algorithms usually enjoy better sample complexity than model-free algorithms from a novel representation complexity perspective: in some cases, the ground-truth rule (model) of the environment is simple to represent, while other quantities, such as $Q$-function, appear complex. We empirically corroborate our theory by comparing the approximation error of the transition kernel, reward function, and optimal $Q$-function in various Mujoco environments, which demonstrates that the approximation errors of the transition kernel and reward function are consistently lower than those of the optimal $Q$-function. To the best of our knowledge, this work is the first to study the circuit complexity of RL, which also provides a rigorous framework for future research.

通过实证研究，我们证明了在电路复杂度的背景下，基于模型和无模型的强化学习在表示复杂度方面的差异，揭示了模型为基础的算法通常比无模型的算法在样本复杂度上表现更好的原因，并为未来的研究提供了一个严格的框架。

关于基于模型和无模型强化学习的表示复杂性