proposal

units/en/unit4/pg-theorem.mdx

@@ -21,13 +21,13 @@ 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)}  \\)
 
 \\(= \sum_{\tau} P(\tau;\theta) \frac{\nabla_\theta P(\tau;\theta)}{P(\tau;\theta)}R(\tau) \\)
 

units/en/unit4/policy-gradient.mdx

@@ -107,7 +107,7 @@ In a loop:
 - Update the weights of the policy: \\(\theta \leftarrow \theta + \alpha \hat{g}\\)
 
 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.
+- \\(\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.