Each observation in the data consists of a binary outcome variable $Y$, a set of labelled measurements $(M_1,L_1), (M_2,L_2)...$ (the set size can vary between observations) where each label is one of the $k$ labels from $\lambda = \{l_1,l_2 .. l_k\}$. Assume there exists a hidden function $h: \lambda \rightarrow \{c_1,c_2,c_3\}$ such that the following relation holds: $$ logit(Prob[Y=1])=a_1\sum\limits_{h(L_i)=c_1} M_i + a_2\sum\limits_{h(L_i)=c_2} M_i + a_3\sum\limits_{h(L_i)=c_3} M_i $$
The problem is to learn the function $h$ and the coefficients $a_i$ from the data when $k$ is of moderate size (large enough to restrict us from studying each $l_i$ individually.). It would we helpful if someone can point to a framework within which such problems are studied.
Thanks in advance.
