Off-Policy TD Control

# Expected Sarsa

Expected Sarsa is like Q-learning but instead of taking the maximum over next state-action pairs, we use the expected value, taking into account how likely each action is under the current policy.

$$Q\left(S_{t}, A_{t}\right) \leftarrow Q\left(S_{t}, A_{t}\right) + \alpha\left[R_{t+1} + \gamma\sum_{a}\pi\left(a\mid{S_{t+1}}\right)Q\left(S_{t+1}, a\right) - Q\left(S_{t}, A_{t}\right)\right]$$

Except for this change to the update rule, the algorithm otherwise follows the scheme of Q-learning. It is more computationally expensive than Sarsa but it eliminates the variance due to the random selection of $A_{t+1}$.

Source: Sutton and Barto, Reinforcement Learning, 2nd Edition

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