Merge pull request #488 from PierreCounathe/pierrecounathe/unit-4-pro…

…positions

Unit 4 Proposal Updates

units/en/unit4/pg-theorem.mdx

@@ -21,17 +21,16 @@ So we have:
 
 We can rewrite the gradient of the sum as the sum of the gradient:
 
-\\( =  \sum_{\tau} \nabla_\theta P(\tau;\theta)R(\tau) \\)
+\\( =  \sum_{\tau} \nabla_\theta (P(\tau;\theta)R(\tau)) = \sum_{\tau} \nabla_\theta P(\tau;\theta)R(\tau) \\) as \\(R(\tau)\\) is not dependent on \\(\theta\\)
 
 We then multiply every term in the sum by \\(\frac{P(\tau;\theta)}{P(\tau;\theta)}\\)(which is possible since it's = 1)
 
 \\( = \sum_{\tau} \frac{P(\tau;\theta)}{P(\tau;\theta)}\nabla_\theta P(\tau;\theta)R(\tau) \\)
 
-We can simplify further this since 
-
-\\( \frac{P(\tau;\theta)}{P(\tau;\theta)}\nabla_\theta P(\tau;\theta) =  P(\tau;\theta)\frac{\nabla_\theta P(\tau;\theta)}{P(\tau;\theta)} \\)
 
+We can simplify further this since \\( \frac{P(\tau;\theta)}{P(\tau;\theta)}\nabla_\theta P(\tau;\theta)\\). 
 
+Thus we can rewrite the sum as \\( =  P(\tau;\theta)\frac{\nabla_\theta P(\tau;\theta)}{P(\tau;\theta)}  \\)
 
 \\( P(\tau;\theta)\frac{\nabla_\theta P(\tau;\theta)}{P(\tau;\theta)}= \sum_{\tau} P(\tau;\theta) \frac{\nabla_\theta P(\tau;\theta)}{P(\tau;\theta)}R(\tau) \\)
 

units/en/unit4/policy-gradient.mdx

@@ -109,8 +109,8 @@ In a loop:
 
 We can interpret this update as follows:
 
-- \\(\nabla_\theta log \pi_\theta(a_t|s_t)\\) is the direction of **steepest increase of the (log) probability** of selecting action at from state st.
-This tells us **how we should change the weights of policy** if we want to increase/decrease the log probability of selecting action  \\(a_t\\) at state  \\(s_t\\).
+- \\(\nabla_\theta log \pi_\theta(a_t|s_t)\\) is the direction of **steepest increase of the (log) probability** of selecting action \\(a_t\\) from state \\(s_t\\).
+This tells us **how we should change the weights of policy** if we want to increase/decrease the log probability of selecting action \\(a_t\\) at state \\(s_t\\).
 
 - \\(R(\tau)\\): is the scoring function:
   - If the return is high, it will **push up the probabilities** of the (state, action) combinations.