# What does the term $|\mathcal{A}(s)|$ mean in the $\epsilon$-greedy policy?

I've been looking online for a while for a source that explains these computations but I can't find anywhere what does the $$|A(s)|$$ mean. I guess $$A$$ is the action set but I'm not sure about that notation:

$$\frac{\varepsilon}{|\mathcal{A}(s)|} \sum_{a} Q^{\pi}(s, a)+(1-\varepsilon) \max _{a} Q^{\pi}(s, a)$$

Here is the source of the formula.

I also want to clarify that I understand the idea behind the $$\epsilon$$-greedy approach and the motivation behind the on-policy methods. I just had a problem understanding this notation (and also some other minor things). The author there omitted some stuff, so I feel like there was a continuity jump, which is why I didn't get the notation, etc. I'd be more than glad if I can be pointed towards a better source where this is detailed.

• From the pseudo code, it is pretty clear that $A(s)$ refers to the set of all possible actions, since in step c) the algorithm iterates through all actions ($a$) (taken from that set). That it is about the actions becomes apparent from the use of $a$. Jul 14 '20 at 20:34
• Yes I realized that I was talking more about the notation $|A(s)|$ but I get it now. Thanks. Jul 14 '20 at 20:44

This expression: $$|\mathcal{A}(s)|$$ means

• $$|\quad|$$ the size of

• $$\mathcal{A}(s)$$ the set of actions in state $$s$$

or more simply the number of actions allowed in the state.

This makes sense in the given formula because $$\frac{\epsilon}{|\mathcal{A}(s)|}$$ is then the probability of taking each exploratory action in an $$\epsilon$$-greedy policy. The overall expression is the expected return when following that policy, summing expected results from the exploratory and greedy action.

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– nbro
Jan 10 at 15:56