3
$\begingroup$

The task (exercise 3.13 in the RL book by Sutton and Barto) is to express $q_\pi(s,a)$ as a function of $p(s',r|s,a)$ and $v_\pi(s)$.

$q_\pi(s,a)$ is the action-value function, that states how good it is to be at some state $s$ in the Markov Decision Process (MDP), if at that state, we choose an action $a$, and after that action, the policy $\pi(s,a)$ determines future actions.

Say that we are at some state $s$, and we choose an action $a$. The probability of landing at some other state $s'$ is determined by $p(s',r|s,a)$. Each new state $s'$ then has a state-value function that determines how good is it to be at $s'$ if all future actions are given by the policy $\pi(s',a)$, therefore:

$$q_\pi(s,a) = \sum_{s' \in S} p(s',r|s,a) v_\pi(s')$$

Is this correct?

$\endgroup$
5
$\begingroup$

Not quite. You are missing the reward at time step $t+1$.

The definition you are looking for is (leaving out the $\pi$ subscripts for ease of notation)

$$q(s,a) = \mathbb{E}[R_{t+1} + \gamma v(s') | S_t=s,A_t=a] = \sum_{r,s'}(r + v(s'))p(s',r|s,a)\;.$$

Because $q(s,a)$ relates to expected returns at time $t$, and returns are defined as $G_t = \sum_{b = 0}^\infty \gamma ^b R_{t+b+1}$, thus $R_{t+1}$ is also a random variable at time $t$ that we need to take expectation with respect to, not just the state that we transition into.

| improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.