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1

Recall that the definition of a value function is $$v_\pi(s) = \mathbb{E}\left[G_t | S_t = s\right]\;.$$ That is, the expected future returns given from state $s$ at time $t$ when we follow our policy $\pi$ -- i.e. our trajectory is generated according to $\pi$. Using Monte Carlo methods we typically will estimate our value function by looking at the ...

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We estimate a value using sampling on whole episodes, and we take this values to construct the target policy. The crucial bit that you are missing is that there is no single value of $V(s)$ (or $Q(s,a)$) of a state (or a state action pair). These value functions are always defined with respect to some policy $\pi(a|s)$ and is given the notation of $V^{\pi}(... 4 Famous example is AlphaZero. It doesn't do unrolls, but consults the value network for leaf node evaluation. The paper has the details on how the update is performed afterwards: The leaf$s'$position is expanded and evaluated only once by the network to gene-rate both prior probabilities and evaluation,$(P(s′ , \cdot),V(s ′ )) = f_\theta(s′ )$. Each edge$...

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There's are some solutions to calculating q-values; find the exact values: Brute-force the action sequence to find max (not pratical) Do recursion on Bellman equation to get max (the same like action sequence brute-force, not pratical) Estimate the q-values: Based on different problems to solve, apply some classic algorithms or human logics to estimate; ...

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