Explanation on some steps of AdaBoost.R2

Question

I am trying to understand AdaBoost.R2 in order to implement it and apply it to a regression problem. In this circumstances I need to understand it perfectly, however there's some step i don't really get.

The paper is available here, and Adaboost.R2 is presented in section 3:
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.31.314&rep=rep1&type=pdf

• In step 4, $\operatorname{sup}|.|$ is used; I've never seen that notation, what does it mean exactly?
• In step 7, "** means exponentation", in that case that would mean $w_i\beta *\operatorname{exp}([1-L_i])$, right?

Answer 1

Q1. $\operatorname{sup}$ is the supremum, or least upper bound. You may think of it as the maximum, but relaxed in case the set of values doesn't actually have a maximum. (And the pipes are just absolute values.) https://en.wikipedia.org/wiki/Infimum_and_supremum

Q2. No, they mean $w_i \cdot \beta^{1-L_i}$. See also https://stats.stackexchange.com/a/299266/232706.