# Policy iteration

## Policy iteration

Last chapter we discuss actor-critic algorithms, but what if we just use critic (value), without an actor(policy)? Surely we can do this, by extracting a policy from a value function.

Firstly, $$A^\pi(s\_t,a\_t)$$ measures how much is $$a\_t$$ than the average action according to $$\pi$$, and $$\arg \max\_{a\_t}A^\pi(s\_t,a\_t)$$ means the best action from $$s\_t$$ if we then follow $$\pi$$, which can be viewed as a substitute for policy $$\pi(s\_t|a\_t)$$. Notice that the "policy" obtained by max trick is at least as good as any normal policy, regardless of what $$\pi(a\_t|s\_a)$$ is.

So forget policies, let's just use max trick.

$$
\pi'(a\_t|s\_t)=
\begin{cases}
1&\text{if }a\_t=\arg \max\_{a\_t}A^\pi(s\_t,a\_t)\\
0 &\text{otherwise}
\end{cases}
$$

![Policy Iteration](https://4133958719-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-LigLKy0c06y4iTEtrkI%2F-Lo5SZswCzdKfaAyVDZK%2F-Lo5S_qJuuY_pQatTL5h%2F1567770415812.png?generation=1567771484909498\&alt=media)

At a high level, we got the policy iteration algorithms:

> Policy Iteration:
>
> repeat until converge
>
> \====1: evaluate $$A^\pi(s\_t,a\_t)$$
>
> \====2: set $$\pi\leftarrow \pi'$$

As before $$A^\pi(s,a)=r(s,a)+\gamma \mathbb{E}\[V^\pi(s')]-V^\pi(s)$$. So now the key problem is to evaluate $$V^\pi(s)$$.

## Dynamic programming

First of all, some basic hypothesis: assume we know the dynamic $$p(s'|s,a)$$, and $$s$$ and $$a$$ are both discrete (and small). For example:

![Dynamic Programming](https://4133958719-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-LigLKy0c06y4iTEtrkI%2F-Lo5SZswCzdKfaAyVDZK%2F-Lo5S_qLcMbyB1bGh2BF%2F1567770828100.png?generation=1567771484967557\&alt=media)

16 states, 4 actions per state. Can store full $$V^\pi(s)$$ in a table. $$\mathcal{T}$$ is a $$16\times16\times4$$ tensor.

We can use bootstrapped update:

$$
V^\pi(s)\leftarrow \mathbb{E}*{a\sim\pi(a|s)}\left\[r(s,a)+\gamma\mathbb{E}*{s'\sim p(s'|s,a)}\[V^\pi(s's)]\right]
$$

Since we use deterministic policy $$\pi(s)=a$$, the update equation can be even simplified:

$$
V^\pi(s)\leftarrow r(s,\pi(s))+\gamma\mathbb{E}\_{s'\sim p(s'|s,\pi(s))}\[V^\pi(s's)]
$$

![Simplified policy iteration](https://4133958719-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-LigLKy0c06y4iTEtrkI%2F-Lo5SZswCzdKfaAyVDZK%2F-Lo5S_qNHmYa2g4tubzx%2F1567771344789.png?generation=1567771484981000\&alt=media)

> Simplified Policy Iteration:
>
> repeat until converge:
>
> \==== 1: evaluate $$V^\pi(s)$$
>
> \==== 2: set $$\pi\leftarrow \pi'$$


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