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6

The weights do sum to one. Note that in the second line where we have $$\frac{\epsilon}{|\mathcal{A}(s)|} \sum_a q_{\pi}(s,a) + (1-\epsilon)\max_aq_{\pi}(s,a) \; ,$$ the sum is over the whole action space, including the greedy action, so the sum of the weights will be $\frac{\epsilon}{|\mathcal{A}(s)|} \times |\mathcal{A}(s)| + (1-\epsilon) = 1$.

3

There is a difference between accurate value function estimates, and optimal value functions. An optimal value function is more specifically the value function of an optimal policy. Value functions are always specific to some policy, which is why you will often see the subscript $\pi$ in e.g. $v_{\pi}(s)$ when there is a defined policy. The policy evaluation ...

2

Where the author mentions the policy evaluation being stopped after one state, they are referring to the part of the algorithm that evaluates the policy -- the pseudocode you have listed is the pseudocode for Value Iteration, which consists of iterating between policy evaluation and policy improvement. In normal policy evaluation, you would apply the update $... 2 A policy can be stochastic or deterministic. A deterministic policy is a function of the form$\pi_{\text{deterministic}}: S \rightarrow A$, that is, a function from the set of states to the set of actions. A stochastic policy is a map of the form$\pi_{\text{stochastic}} : S \rightarrow P(A)$, where$P(A)$is a set of probability distributions ($P(A) = \{ ...

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