Learning Large DAGs is Harder than you Think: Many Losses are Minimal for the Wrong DAG

Structure learning is a crucial task in science, especially in fields such as medicine and biology, where the wrong identification of (in)dependencies among random variables can have significant implications. The primary objective of structure learning is to learn a Directed Acyclic Graph (DAG) that represents the underlying probability distribution of the data. Many prominent DAG learners rely on least square losses or log-likelihood losses for optimization. It is well-known from regression models that least square losses are heavily influenced by the scale of the variables. Recently it has been demonstrated that the scale of data also affects performance of structure learning algorithms, though with a strong focus on linear 2-node systems and simulated data. Moving beyond these results, we provide conditions under which square-based losses are minimal for wrong DAGs in $d$-dimensional cases. Furthermore, we also show that scale can impair performance of structure learners if relations among variables are non-linear for both square based and log-likelihood based losses. We confirm our theoretical findings through extensive experiments on synthetic and real-world data.