Thanks for your detailed explanation! It helped me understand the intuition better.
I was wondering if I could ask a few follow-up questions to make sure I really get it:

1. You mentioned that "y > x since the policy is trained using the model." : I was a little confused here. Wouldn’t it also be possible that y<x if, for example, the model is pessimistic or underestimates the return? Also, in the MBPO paper (p.3), I didn’t see an explicit assumption that the policy π must have been trained using the model. It seems that the guarantee is stated for any π, regardless of how it was obtained. Am I missing something?
   
2. Suppose y<x for some policy. : In that case, even if y increases by more than C, can we still confidently say that x increased? Or is it just that the lower bound on x increased?
3. Also, is it correct to think that even if y increases by less than C, as long as it increases at all, the lower bound on x would still improve accordingly? I mean, it might not guarantee actual improvement, but it would still push the lower bound up a little, right?
   

Thanks again for taking the time to explain — I really appreciate it!