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I'm trying to make sense of the "The Bandit Gradient Algorithm as Stochastic Gradient Ascent" proof in Sutton and Barto's intro to RL textbook. I'm stuck on the line

$E[(q_*(A_t)-B_t)\frac{\partial\pi_t(A_t)}{\partial H_t(a)}/\pi_t(A_t)] = E[(R_t-\bar{R}_t)\frac{\partial\pi_t(A_t)}{\partial H_t(a)}/\pi_t(A_t)]$$

They say they can replace $q_*(A_t)$ with $R_t$ because $E[R_t|A_t]=q_*(A_t)$, but I can't figure out how to show this rigorously.

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It's the definition of the optimal $Q$-function: $$ Q* = \mathbb{E}[R_t|S_t,A_t] $$

(the state is omitted as in Bandits there is only 1 state)

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