I'm writing my thesis on a topic in machine learning. In some parts I defined a summation over a set. I sent my thesis draft to my professor and then I met him. He briefly mentioned that "we don't sum over sets" and said that I need to be careful and read my thesis again. The meeting was just 1 minute and I couldn't ask him which parts or lines had the issue, so I couldn't really find where the problem is.
I believe that the professor meant the training datasets. But I might be wrong.
My question: is it correct mathematically to sum over sets?!
Edit:
One way I define such a context is like this: given a set $D=\{s_1, s_2, \dots, s_n\}$ which represents a training dataset that contains training samples $s_1, \dots, s_n$, then a summation over $D$ is $\sum_{s_i \in D} s_i$
Best Answer
You have to be careful summing over sets because the order in which you sum might make a difference. For example if a series converges but not absolutely, then you can rearrange the terms to converge to any real number. So when you write:
$\sum_{n=1}^\infty a_n$ you have a clear order. But if $D$ is a countably infinite set with no clear ordering, then the expression $\sum_{n\in D} a_n$ may not be well defined. If, however, you know all $a_n\geq 0$ then it would make sense.