Having established the methods for learning environment dynamics ( $P_\theta(s'|s, a)$ ) and reward functions ( $R_\phi(s, a, s')$ ), a natural next step is to leverage these learned models for decision-making. While simple trajectory sampling can provide some benefit, integrating the learned model with a powerful planning algorithm like Monte Carlo Tree Search (MCTS) offers a more structured approach to lookahead planning, potentially yielding significantly stronger policies.

MCTS excels at exploring large search spaces by intelligently balancing exploration and exploitation during simulated trajectories. In its traditional applications (like game playing), MCTS relies on a perfect simulator or generative model of the environment. In model-based RL, we replace this perfect simulator with our learned, approximate model. This allows the agent to "think ahead" using its internal understanding of the world, even if that understanding isn't perfect.

The MCTS Cycle with a Learned Model

The core MCTS process remains the same, consisting of four steps repeated many times to build a search tree rooted at the current state $s_t$ : Selection, Expansion, Simulation, and Backpropagation. The integration points for the learned model are primarily within the Expansion and Simulation steps.

Selection: Starting from the root node (representing the current state $s_t$ ), recursively select child nodes based on a tree policy until a leaf node is reached. A common tree policy is UCT (Upper Confidence bounds applied to Trees), which balances exploiting nodes with high estimated values $Q(s, a)$ and exploring nodes with fewer visits $N(s, a)$ . The selection process itself doesn't directly use the learned dynamics model, but relies on values updated via simulations that do use the model.
Expansion: When the selection process reaches a leaf node $L$ representing state $s_L$ , if the state is non-terminal and the node hasn't been fully expanded (i.e., not all actions have been tried), choose an untried action $a$ . Use the learned transition model $P_\theta(s'|s_L, a)$ to sample a possible next state $s'$ . Create a new child node representing $s'$ . This is a critical point where the learned model dictates the structure of the search tree being built. If $P_\theta$ is deterministic, there's only one $s'$ ; if it's probabilistic, we sample one outcome.
Simulation (Rollout): From the newly expanded node (state $s'$ ), perform a simulation or rollout until a terminal state or a predefined maximum depth is reached. This simulation proceeds by repeatedly:
- Choosing an action $a_{sim}$ according to a default rollout policy $\pi_{rollout}$ (often simple, like random action selection).
- Sampling the next state $s'_{sim}$ using the learned transition model: $s'_{sim} \sim P_\theta(\cdot | s_{sim}, a_{sim})$ .
- Obtaining the reward $r_{sim}$ using the learned reward model: $r_{sim} = R_\phi(s_{sim}, a_{sim}, s'_{sim})$ or sampled if the reward model is probabilistic. The cumulative discounted reward $G$ obtained during this rollout is the result of the simulation. This step heavily relies on both the learned transition and reward models to generate hypothetical future trajectories.
Backpropagation: Update the statistics (visit count $N$ and value estimate $Q$ ) of all nodes along the path from the newly expanded node back up to the root node using the result $G$ of the simulation. For a node representing state $s$ where action $a$ was taken leading down the path, update $N(s,a)$ and $Q(s,a)$ . Typically, $Q(s,a)$ is updated towards the average return observed after taking action $a$ from state $s$ :
$N(s,a) \leftarrow N(s,a) + 1$ $Q(s,a) \leftarrow Q(s,a) + \frac{G - Q(s,a)}{N(s,a)}$

After performing many iterations of these four steps (a computational budget often measured in number of simulations or time), the agent chooses an action to take in the real environment. This is typically the action corresponding to the most visited child node of the root, as visit counts often serve as a robust indicator of the best action found by the search.

The MCTS cycle leveraging a learned environment model. The learned transition model $P_\theta$ determines state transitions during Expansion and Simulation. The learned reward model $R_\phi$ provides rewards during Simulation.

Sophistications: Learning Policy and Value Networks (AlphaZero Style)

A significant advancement in integrating MCTS with learned components came from systems like AlphaGo and its successor AlphaZero. Instead of relying solely on random rollouts for the Simulation step and simple UCT for Selection/Expansion, these systems use neural networks trained alongside the MCTS planning:

Policy Network $\pi_\psi(a|s)$ : This network predicts promising actions from a given state $s$ . Its output can be used to bias the MCTS selection towards more promising branches and potentially guide the expansion step, focusing the search effort more effectively than vanilla UCT.
Value Network $V_\omega(s)$ : This network estimates the expected future return from state $s$ . Its output can be used to truncate the Simulation step early, replacing the noisy Monte Carlo rollout result $G$ with the learned value estimate $V_\omega(s')$ , potentially reducing variance and computational cost.

These networks are typically trained using data generated from the MCTS searches themselves. The visit counts from MCTS serve as training targets for the policy network (encouraging it to predict actions that MCTS explored more), and the final outcome of the MCTS search (often improved value estimates derived from the search results) serves as a target for the value network. This creates a powerful feedback loop where better planning leads to better networks, which in turn lead to even better planning. While implementing such a system is complex, it demonstrates the potential depth of integration between learned models/functions and search algorithms.

Advantages and Considerations

Integrating MCTS with learned models offers several potential advantages:

Improved Performance: MCTS performs lookahead search, which can find better solutions than purely reactive policies, especially in tasks requiring planning.
Directed Exploration (via Planning): The search process itself can be seen as a form of directed exploration within the learned model's state space.
Potential Sample Efficiency Gains: If the learned model is reasonably accurate, generating simulated experiences via MCTS can be much faster and cheaper than interacting with the real environment, potentially reducing the need for real-world samples (though model learning itself requires samples).

However, this approach is not without challenges:

Model Accuracy is Paramount: The performance of MCTS planning is highly sensitive to the accuracy of the learned models $P_\theta$ and $R_\phi$ . Errors in the model can accumulate during rollouts and mislead the search, leading to poor action choices. This is often referred to as the problem of model bias or compounding model error.
Computational Cost: Running thousands of MCTS simulations at each time step can be computationally expensive, potentially limiting applicability in real-time scenarios. The complexity scales with the desired search depth and breadth.
Balancing Model Learning and Planning: There's a trade-off between spending resources on improving the model versus spending resources on planning with the current model.

Integrating MCTS with learned models represents a sophisticated approach within model-based RL. It allows agents to combine the benefits of learning from experience with the deliberate planning capabilities offered by tree search, providing a powerful tool for tackling complex sequential decision-making problems, provided the challenges of model accuracy and computational cost can be managed effectively.