# Value and Q-value recursion

There are two forms the expected reward for a given state is encoded:

• v-function: $V^{\pi}(s) = \mathbb{E}_{\pi} \left\{ \sum\limits_{k=0}^{\infty} \gamma^k r_{t+k+1} \lvert s_t = s \right\}$
• q-function: $Q^{\pi}(s,a) = \mathbb{E}_{\pi} \left\{ \sum\limits_{k=0}^{\infty} \gamma^k r_{t+k+1} \lvert s_t = s, a_t = a \right\}$

The v-function is the expected reward given a state whilst the q-function is for a state and action. The recursive aspect of both these two functions can be derived from first principal and it can be shown that the v-function is a function of the q-function.

See RVQ.pdf for the derivation of the recursion and the link between both functional forms.

See RL_Solutions_Chap3.pdf for the effect of sign and constants in the reward function.

We want to find an expression for $\Delta\theta$ which uses an estimator of the expected reward such as the action-value or advantage function.
The key is to able to find an unbiased estimage of the gradient $\Delta\theta$