Suppose that an environment has six states, encoded as integers 0 - 5. An agent is attempting to learn the value function Vpi(s) for a policy pi by applying Monte Carlo prediction. After a few iterations of Monte Carlo prediction, the agent's estimate for Vpi(s) is as indicated by the dictionary shown below.

 

V_estimate = {0:10.4, 1:16.7, 2:12.8, 3:24.2, 4:8.2, 5:16.7}

 

The agent generates a new episode starting in state 0. During the episode, the agent performs a sequence of four actions. The sequence of states visited as a result of taking these actions, and the rewards earned by the agent are provided by the lists below.

 

states_visited = [0, 4, 3, 1, 5]

rewards_earned = [3, 2, -5, 6]


Using a discount rate of 0.9 and a learning rate of 0.1, determine the agent's new estimate for Vpi(0)after performing an update based on the episode.

To update the agent's estimate of \( V_{\pi}(0) \), we will follow the Monte Carlo prediction approach, which involves calculating the total return \( G_t \) from the episode, discounted by the given discount rate (0.9 in this case), and updating the value function accordingly using the learning rate (0.1).

Here are the steps:

### Step 1: Compute the total return \( G_t \)
The return \( G_t \) for each time step is calculated using the formula:

\[
G_t = R_{t+1} + \gamma R_{t+2} + \gamma^2 R_{t+3} + \ldots
\]

where:
- \( R_{t+1}, R_{t+2}, \ldots \) are the rewards earned after state \( t \),
- \( \gamma \) is the discount rate (0.9 in this case).

We'll compute \( G_0 \), which corresponds to the total return starting from state 0.

Given:
- \( \text{rewards\_earned} = [3, 2, -5, 6] \)
- Discount rate \( \gamma = 0.9 \)

The return starting from state 0 is:

\[
G_0 = 3 + 0.9(2) + 0.9^2(-5) + 0.9^3(6)
\]

### Step 2: Update the value estimate \( V_{\pi}(0) \)
The update rule for Monte Carlo prediction is:

\[
V_{\pi}(s) \leftarrow V_{\pi}(s) + \alpha (G_t - V_{\pi}(s))
\]

where:
- \( \alpha \) is the learning rate (0.1 in this case),
- \( G_t \) is the total return starting from state \( s \),
- \( V_{\pi}(s) \) is the current estimate of the value of state \( s \).

We'll update the value estimate for \( s = 0 \), and the current estimate \( V_{\pi}(0) \) is 10.4.

Let's compute the total return and update the value estimate.

I'll perform the calculation.The total return \( G_0 \) starting from state 0 is 5.124. Using the Monte Carlo update rule, the new estimate for \( V_{\pi}(0) \) becomes approximately 9.87.

### Summary:
- Total return \( G_0 = 5.124 \)
- Updated \( V_{\pi}(0) = 9.87 \)

Would you like more details or have any questions?

Here are five related questions to deepen your understanding:
1. How would the estimate change with a different discount rate (e.g., 0.95)?
2. How does the learning rate affect the convergence speed in Monte Carlo methods?
3. What is the difference between first-visit and every-visit Monte Carlo prediction?
4. How would the agent's value estimates change if rewards were negative throughout?
5. What other methods can be used to estimate \( V_{\pi}(s) \), besides Monte Carlo?

**Tip:** In Monte Carlo methods, larger learning rates may cause faster but more unstable updates, while smaller rates yield smoother convergence.

Suppose that an environment has six states, encoded as integers 0 - 5. An agent is attempting to learn the value function Vpi(s) for a policy pi by applying Monte Carlo prediction. After a few iterations of Monte Carlo prediction, the agent's estimate for Vpi(s) is as indicated by the dictionary shown below.

V_estimate = {0:10.4, 1:16.7, 2:12.8, 3:24.2, 4:8.2, 5:16.7}

The agent generates a new episode starting in state 0. During the episode, the agent performs a sequence of four actions. The sequence of states visited as a result of taking these actions, and the rewards earned by the agent are provided by the lists below.

states_visited = [0, 4, 3, 1, 5]
rewards_earned = [3, 2, -5, 6]

Using a discount rate of 0.9 and a learning rate of 0.1, determine the agent's new estimate for Vpi(0) after performing an update based on the episode.

Learn how to update the value function Vpi(s) using Monte Carlo prediction in reinforcement learning. This problem includes calculating discounted returns with a discount rate of 0.9 and applying a learning rate of 0.1. Suitable for students with knowledge of reinforcement learning at the undergraduate level.

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