Universal Lower Bound for Learning Causal DAGs with Atomic Interventions

Abstract

A well-studied challenge that arises in the structure learning problem of causal directed acyclic graphs (DAG) is that using observational data, one can only learn the graph up to a "Markov equivalence class" (MEC). The remaining undirected edges have to be oriented using interventions, which can be very expensive to perform in applications. Thus, the problem of minimizing the number of interventions needed to fully orient the MEC has received a lot of recent attention, and is also the focus of this work. We prove two main results. The rst is a new universal lower bound on the number of atomic interventions that any algorithm (whether active or passive) would need to perform in order to orient a given MEC. Our second result shows that this bound is, in fact, within a factor of two of the size of the smallest set of atomic interventions that can orient the MEC. Our lower bound is provably better than previously known lower bounds. The proof of our lower bound is based on the new notion of clique-block shared-parents (CBSP) orderings, which are topological orderings of DAGs without v-structures and satisfy certain special properties. Further, using simulations on synthetic graphs and by giving examples of special graph families, we show that our bound is often signi cantly better.

Causal Structure Learning: a Combinatorial Perspective

“…Recent work [160] establishes novel submodularity properties for greedy objectives in this settings, allowing for efficient optimization over the choice of interventions in each batch. Taken together, these recent advances suggest several future directions, including (1) characterizing the number of multi-target interventions needed by an oracle in the adaptive case [124], (2) approximation guarantees for experimental design, compared to either oracles or optimal strategies, and (3) experimental design in settings with latent confounding [94,2], selection bias, and cycles.…”

Causal Structure Learning: A Combinatorial Perspective

“…Recent work [161] establishes novel submodularity properties for greedy objectives in this settings, allowing for efficient optimization over the choice of interventions in each batch. Taken together, these recent advances suggest several future directions, including (1) characterizing the number of multi-target interventions needed by an oracle in the adaptive case [125], (2) approximation guarantees for experimental design, compared to either oracles or optimal strategies, and (3) experimental design in settings with latent confounding [2,94], selection bias, and cycles.…”