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do we also want to consider the subset of invalid actions for the $\max\limits_{a}Q(s_{t+1},a)$ No. Doing so would go against the theory behind the Bellman equation from which the update derives. The value of $r_{t+1} + \gamma \max\limits_{a'}Q(s_{t+1},a')$ needs to match to a realisable trajectory, otherwise the eventual expected values may be estimates ...

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