# Value functions and Q-functions

## Definition: Q-function

$$
Q^\pi (s\_t,a\_t)=\sum\_{t'=t}^T E\_{\pi\_\theta}\[r(s\_{t'},a\_{t'})|s\_t,a\_t]
$$

Total reward from taking $$a\_t$$ in $$s\_t$$

## Definition: Value function

Total reward from $$s\_t$$

$$
V^\pi(s\_t)=\sum\_{t'=t}^T E\_{\pi\_\theta}\[r(s\_{t'},a\_{t'})|s\_t]
$$

and can rewrite with Q-function. (The relation between Q-function and Value function)

$$
V^\pi(s\_t)=E\_{a\_t\sim \pi\_\theta (a\_t|s\_t)}\[Q^\pi (s\_t,a\_t)]
$$

Besides, the RL objective can rewrite with value function

$$
J(\theta)=E\_{s\_1\sim p(s\_1)}\[V^\pi(s\_1)]
$$

## Using Q-functions and Value function

### Idea 1

$$Q^\pi (s\_t,a\_t) \Rightarrow \text{improve policy }\pi$$

set $$\pi'(a|s)=1$$ if $$a=\arg\max\_a Q^\pi (s,a)$$. This new policy $$\pi'$$ is at least as good as $$\pi$$ (and probably better) and it doesn't matter what $$\pi$$ is.

### Idea 2

Compute gradient to increase probability of good actions $$a$$

if $$Q^\pi(s,a)> V^\pi (s)$$, the $$a$$ is better than average, recall that $$V^\pi(s)=E\[Q^\pi(s,a)]$$ under $$\pi(a|s)$$

So we can modify $$\pi(a|s)$$ to increase probability of $$a$$, like compute gradient.


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