Monte Carlo methods in reinforcement learning are a way to estimate value functions and derive optimal policies from them. They are based on averaging sample returns. Since MC methods require actual returns for learning, they can only be applied to episodic tasks.
Initialization:
Policy Execution:
Episode Processing:
Policy Improvement (for MC Control):
Repeat:
MC methods are fundamentally about learning from episodes of experience.
Sampling: Instead of knowing the complete dynamics of the environment (like in Dynamic Programming methods), MC methods rely on episodes sampled from the environment to gather information.
Averaging Returns: The core idea behind MC is straightforward: to find the value of a state, simply average the total returns observed after visiting that state over many episodes.
Epsilon-Greedy Exploration: To ensure adequate exploration of all state-action pairs, an ε-greedy policy (or other exploration strategies) might be employed, especially in MC control where the goal is to find the optimal policy.
No Bootstrapping: Unlike Temporal Difference methods, MC does not bootstrap. This means it waits until the final outcome (end of the episode) to calculate the value of a state based on actual returns rather than estimating it based on other estimated values.
Imagine teaching an agent to play a simple card game like Blackjack.
Over time, after many games, the value estimates will converge to the true expected returns, and the policy (if being optimized) will approach the optimal strategy for playing Blackjack.
Key equations for the Monte Carlo (MC) methods and explain each:
For a given policy , the objective is to estimate the state-value function .
Key Equation:
Here, the return is the total discounted reward from time onwards:
Explanation: The value of a state is updated by averaging over all the returns that have been observed after visiting that state across different episodes.
For a given policy , the objective is to estimate the action-value function .
Key Equation:
With the return defined as before.
Explanation: Similar to state values, but here the value is associated with taking a specific action in a specific state . The update is an average of all returns observed after taking action in state across different episodes.
The goal is to improve the policy over time to converge to the optimal policy .
Key Equations:
Policy Improvement:
Where chooses the action that maximizes for state .
Explanation: After estimating values using MC prediction, the policy is improved by choosing the action that maximizes expected return in each state. Exploration is ensured using strategies like ε-greedy, with decreasing over time to ensure the "Greedy in the Limit" property.
This is a variant of MC Control that uses a constant step size for updates.
Key Equation:
Explanation: Instead of averaging all returns for a state-action pair, this method takes a weighted average of the old value and the new return. The weight is given by , the learning rate. This approach can be more suitable for non-stationary environments where recent returns might be more relevant than older ones.
In all these methods, the core idea is to use actual returns observed from episodes to update value estimates, either for evaluation (MC Prediction) or for improving the policy (MC Control).
Get the latest updates, exclusive content and special offers delivered directly to your mailbox. Subscribe now!