Give step-by-step solution with explanation and final answer:Q2 (10 points) Derive the relationship between V7 (s) and Q™(s, a). Describe the procedure to compute
Q7(s, a) if the state-value function

Question

Give step-by-step solution with explanation and final answer:Q2 (10 points) Derive the relationship between V7 (s) and Q™(s, a). Describe the procedure to compute
Q7(s, a) if the state-value function V7 (s') for all subsequent states s' is known. Conversely, describe
the procedure to compute V7 (s) if the state-action value function Q7 (s, a) for all actions a is known.

VIPSolutions · Accepted Answer

## Step-by-Step Solution

### 1. **Relationship between $ V^\pi(s) $ and $ Q^\pi(s, a) $**

**Definitions:**
- **State-value function:**  
  $ V^\pi(s) = \mathbb{E}_\pi \left[ \sum_{t=0}^{\infty} \gamma^t r_t \mid s_0 = s \right] $
- **State-action value function:**  
  $ Q^\pi(s, a) = \mathbb{E}_\pi \left[ \sum_{t=0}^{\infty} \gamma^t r_t \mid s_0 = s, a_0 = a \right] $

#### **Relationship:**
$$
V^\pi(s) = \mathbb{E}_{a \sim \pi(\cdot|s)} [Q^\pi(s, a)]
$$
or more explicitly:
$$
V^\pi(s) = \sum_{a} \pi(a|s) Q^\pi(s, a)
$$

$$
Q^\pi(s, a) = \mathbb{E}_{s'} [r(s, a) + \gamma V^\pi(s')]
$$
or more explicitly:
$$
Q^\pi(s, a) = \sum_{s'} P(s'|s, a) \left[ r(s, a, s') + \gamma V^\pi(s') \right]
$$

---

### 2. **Procedure to Compute $ Q^\pi(s, a) $ given $ V^\pi(s') $ for all $ s' $:**

For each $ (s, a) $:

1. **Compute expected immediate reward and next state value:**
   $$
   Q^\pi(s, a) = \sum_{s'} P(s'|s, a) \left[ r(s, a, s') + \gamma V^\pi(s') \right]
   $$
2. **Steps:**
   - For each possible next state $ s' $:
     - Get transition probability $ P(s'|s, a) $
     - Get reward $ r(s, a, s') $
     - Get $ V^\pi(s') $
   - Multiply and sum over all $ s' $

---

### 3. **Procedure to Compute $ V^\pi(s) $ given $ Q^\pi(s, a) $ for all $ a $:**

For each state $ s $:

1. **Compute expected value under policy $ \pi $:**
   $$
   V^\pi(s) = \sum_{a} \pi(a|s) Q^\pi(s, a)
   $$
2. **Steps:**
   - For each possible action $ a $:
     - Get policy probability $ \pi(a|s) $
     - Get $ Q^\pi(s, a) $
   - Multiply and sum over all $ a $

---

## **Summary Table**

| Given | To Compute | Formula |
|-------|------------|---------|
| $ V^\pi(s') $ | $ Q^\pi(s, a) $ | $ Q^\pi(s, a) = \sum_{s'} P(s'|s, a) [ r(s, a, s') + \gamma V^\pi(s') ] $ |
| $ Q^\pi(s, a) $ | $ V^\pi(s) $ | $ V^\pi(s) = \sum_{a} \pi(a|s) Q^\pi(s, a) $ |

---

## **Final Answers**

- The relationship:  
  $$
  V^\pi(s) = \sum_{a} \pi(a|s) Q^\pi(s, a)
  $$
  $$
  Q^\pi(s, a) = \sum_{s'} P(s'|s, a) [ r(s, a, s') + \gamma V^\pi(s') ]
  $$

- **To compute $ Q^\pi(s, a) $ from $ V^\pi(s') $:** Use the second formula above.
- **To compute $ V^\pi(s) $ from $ Q^\pi(s, a) $:** Use the first formula above.

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