Math Problem Statement

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.

Solution

To update the agent's estimate of Vπ(0)V_{\pi}(0), we will follow the Monte Carlo prediction approach, which involves calculating the total return GtG_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 GtG_t

The return GtG_t for each time step is calculated using the formula:

Gt=Rt+1+γRt+2+γ2Rt+3+G_t = R_{t+1} + \gamma R_{t+2} + \gamma^2 R_{t+3} + \ldots

where:

  • Rt+1,Rt+2,R_{t+1}, R_{t+2}, \ldots are the rewards earned after state tt,
  • γ\gamma is the discount rate (0.9 in this case).

We'll compute G0G_0, which corresponds to the total return starting from state 0.

Given:

  • rewards_earned=[3,2,5,6]\text{rewards\_earned} = [3, 2, -5, 6]
  • Discount rate γ=0.9\gamma = 0.9

The return starting from state 0 is:

G0=3+0.9(2)+0.92(5)+0.93(6)G_0 = 3 + 0.9(2) + 0.9^2(-5) + 0.9^3(6)

Step 2: Update the value estimate Vπ(0)V_{\pi}(0)

The update rule for Monte Carlo prediction is:

Vπ(s)Vπ(s)+α(GtVπ(s))V_{\pi}(s) \leftarrow V_{\pi}(s) + \alpha (G_t - V_{\pi}(s))

where:

  • α\alpha is the learning rate (0.1 in this case),
  • GtG_t is the total return starting from state ss,
  • Vπ(s)V_{\pi}(s) is the current estimate of the value of state ss.

We'll update the value estimate for s=0s = 0, and the current estimate Vπ(0)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 G0G_0 starting from state 0 is 5.124. Using the Monte Carlo update rule, the new estimate for Vπ(0)V_{\pi}(0) becomes approximately 9.87.

Summary:

  • Total return G0=5.124G_0 = 5.124
  • Updated Vπ(0)=9.87V_{\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 Vπ(s)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.

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Math Problem Analysis

Mathematical Concepts

Monte Carlo Methods
Reinforcement Learning
Value Function Estimation
Discounted Return
Learning Rate

Formulas

G_t = R_{t+1} + \gamma R_{t+2} + \gamma^2 R_{t+3} + \dots
V_{\pi}(s) \leftarrow V_{\pi}(s) + \alpha (G_t - V_{\pi}(s))

Theorems

Monte Carlo Prediction Update Rule

Suitable Grade Level

Undergraduate