Paper:
Learning Causal Graphs with Small Interventions
We consider the problem of learning causal networks with interventions, when each intervention is limited in size under Pearl's Structural Equation Model with independent errors (SEM-IE). The objective is to minimize the number of experiments to discover the causal directions of all the edges in a causal graph. Previous work has focused on the use of separating systems for complete graphs for this task. We prove that any deterministic adaptive algorithm needs to be a separating system in order to learn complete graphs in the worst case. In addition, we present a novel separating system construction, whose size is close to optimal and is arguably simpler than previous work in combinatorics. We also develop a novel information theoretic lower bound on the number of interventions that applies in full generality, including for randomized adaptive learning algorithms. For general chordal graphs, we derive worst case lower bounds on the number of interventions. Building on observations about induced trees, we give a new deterministic adaptive algorithm to learn directions on any chordal skeleton completely. In the worst case, our achievable scheme is an ↵-approximation algorithm where ↵ is the independence number of the graph. We also show that there exist graph classes for which the sufficient number of experiments is close to the lower bound. In the other extreme, there are graph classes for which the required number of experiments is multiplicatively ↵ away from our lower bound. In simulations, our algorithm almost always performs very close to the lower bound, while the approach based on separating systems for complete graphs is significantly worse for random chordal graphs.

Review:
This paper addresses the problem of learning causal networks with interventions, when each intervention is limited to size k. The paper is generally well-written and addresses a relevant question, as it is generally not feasible to learn the causal structure from observational data alone. Moreover, in some cases, it may also not be possible to perform arbitrarily large interventions, but it is possible to perform interventions over smaller subsets of the variables. The authors prove a number of results around the number of interventions required to learn complete and chordal graphs, and, while I was not able to check all the proofs in detail, the results are as expected (and appear to be correct).The results on chordal graphs are applicable to general causal structures in the sense that application of conditional independence learning and Meek rules results in a chain graph with chordal chain components.In addition, the authors implement a novel adaptive deterministic algorithm and test it on randomly generated chordal graphs.The simulations show that the algorithm performs close to the lower bound for worst case chordal graphs using adaptive or non-adaptive algorithms.Additionally, it significantly outperforms the naive separating systems approach for complete graphs. Overall, this is a strong paper that should be accepted. I list below the specific contributions. - The paper gives a novel separating system construction, where the size is larger than that of Wegener (1979) but has a simpler description. This construction is later used for the results on adaptive and non-adaptive algorithms for a complete graph.- It had previously been shown that any non-adaptive algorithm needs a (n, k) separating system to learn a complete graph. The authors show that the same is true for any adaptive deterministic algorithm.- The authors show that to learn a complete causal DAG on n variables using k-size interventions, n/2k interventions are necessary. This bound holds for any algorithm.- The authors give a straightforward extension of a result by Hu et al. (2014), which showed that randomized adaptive algorithms need only log log n interventions with high probability for the unbounded case for complete graphs. The authors show that O(n/k log log k) interventions suffice with high probability when the interventions are bounded by k in size.- The authors extend the lower bound for adaptive algorithms for general chordal graphs and show that the number of experiments from a ((G), k) separating system is needed where (G) is the chromatic number of the skeleton graph.- The authors give two extreme examples; one for which the number of interventions is close to the lower bound is sufficient and the other for which the number of interventions needed is close to the upper bound.- The authors give a novel adaptive deterministic algorithm that uses the idea of separating systems together with adaptability to Meek rules. They test their algorithm on randomly generated chordal graphs and observe that it performs close to the (, k) separating system and significantly outperforms the (n, k) separating system (for complete graphs).** Additional Comments: - The "Meek rules" described in Section 2.2 were introduced by Verma and Pearl (1992) and Meek (1995) proved certain properties of them. You should cite Verma92.- Top of pg. 5, replace "also provided a tighter upper bound in [15]" with "also provided a tighter upper bound than the one in [15] -On pg. 8, the "Information Theoretic LB" is described as /2k.Where is this bound given? References: Verma, Thomas, and Pearl, Judea. "An algorithm for deciding if a set of observed independencies has a causal explanation." Proceedings of the Eighth international conference on uncertainty in artificial intelligence. Morgan Kaufmann Publishers Inc., 1992.