Definition of Submodular

In this paper, we focus on the case where the functions \(f_i\) are monotone and submodular set functions and each task \(T_i\) amounts to maximizing \(f_i\) under the \(k\)-cardinality constraint. That is, the task \(T_i\) is to select a subset \(S \subseteq V\) of size \(k\) such that the value of \(f_i(S)\) is maximized. Submodularity of \(f_i\) means that for any \(A, B \subseteq V\) we have \(f_i(A) + f_i(B) \geq f_i(A \cup B) + f_i(A \cap B)\). Furthermore, fi is called monotone if for any \(A \subseteq B\) we have \(f_i(A) \leq f_i(B)\).

Submodular Meta-Learning