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I am working through Sutton and Barto's RL book. So far in the text, when backup diagrams are drawn, the reward and next state are iterated together (i.e. the equations always have $\sum_{s',r}$), because the text uses the four-place function $p(s',r|s,a)$. Starting from a solid circle (state-action pair), each edge has a reward labeled along the edge and the next state labeled on the open circle. (See page 59 for an example diagram, or see Figure 3.4 here.)

However, exercise 3.29 asks to rewrite the Bellman equations in terms of $p(s'|s,a)$ and $r(s,a)$. This means that the reward is an expected value (i.e. we don't want to iterate over rewards like $\sum_r \cdots (r + \cdots)$), whereas the next states should be iterated (i.e. we want something like $\sum_{s'} p(s'|s,a) (\cdots)$).

I think writing the Bellman equations themselves isn't too difficult; my current guess is that they look like this: $$v_\pi(s) = \sum_a \pi(a|s) \left(r(s,a) + \gamma \sum_{s'} p(s'|s,a) v_\pi(s')\right)$$

$$q_\pi(s,a) = r(s,a) + \gamma \sum_{s'} p(s'|s,a) \sum_{a'} \pi(a'|s') q_\pi(s',a')$$

My problem instead is that I want to be able to draw the backup diagrams corresponding to these equations. Given the "vocabulary" for backup diagrams given in the book (e.g. solid circle = state-action pair, open circle = state, rewards along the edge, probabilities below nodes, maxing over denoted by an arc), I don't know how to represent the fact that the reward and next state are treated differently. Two ideas that don't seem to work:

  • If I draw a bunch of edges after the solid circle, that looks like I'm iterating over rewards.
  • If I come up with a special kind of edge that represents an expected reward, then it looks like only a single next state is being considered.
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