  Reinforcement Learning (RL) encompasses diverse paradigms, including
model-based RL, policy-based RL, and value-based RL, each tailored to
approximate the model, optimal policy, and optimal value function,
respectively. This work investigates the potential hierarchy of representation
complexity -- the complexity of functions to be represented -- among these RL
paradigms. We first demonstrate that, for a broad class of Markov decision
processes (MDPs), the model can be represented by constant-depth circuits with
polynomial size or Multi-Layer Perceptrons (MLPs) with constant layers and
polynomial hidden dimension. However, the representation of the optimal policy
and optimal value proves to be $\mathsf{NP}$-complete and unattainable by
constant-layer MLPs with polynomial size. This demonstrates a significant
representation complexity gap between model-based RL and model-free RL, which
includes policy-based RL and value-based RL. To further explore the
representation complexity hierarchy between policy-based RL and value-based RL,
we introduce another general class of MDPs where both the model and optimal
policy can be represented by constant-depth circuits with polynomial size or
constant-layer MLPs with polynomial size. In contrast, representing the optimal
value is $\mathsf{P}$-complete and intractable via a constant-layer MLP with
polynomial hidden dimension. This accentuates the intricate representation
complexity associated with value-based RL compared to policy-based RL. In
summary, we unveil a potential representation complexity hierarchy within RL --
representing the model emerges as the easiest task, followed by the optimal
policy, while representing the optimal value function presents the most
intricate challenge.
