[Paper Review] Efficient Intervention Design for Causal Discovery with Latents

Raghavendra Addanki, Shiva Prasad Kasiviswanathan+2 more|arXiv (Cornell University)|May 24, 2020

Bayesian Modeling and Causal Inference31 references4 citations

TL;DR

This paper presents efficient algorithms for causal discovery in the presence of latent variables, minimizing intervention costs under two models: linear cost (where intervention cost depends on variables targeted) and identity cost (where all interventions have equal cost). It introduces p-colliders—specific colliders arising from conditioning in the causal graph—and provides a flow-based method to detect them, achieving near-optimal intervention counts with theoretical guarantees, including a 2-approximation for linear cost and improved bounds using p-collider structure.

ABSTRACT

We consider recovering a causal graph in presence of latent variables, where we seek to minimize the cost of interventions used in the recovery process. We consider two intervention cost models: (1) a linear cost model where the cost of an intervention on a subset of variables has a linear form, and (2) an identity cost model where the cost of an intervention is the same, regardless of what variables it is on, i.e., the goal is just to minimize the number of interventions. Under the linear cost model, we give an algorithm to identify the ancestral relations of the underlying causal graph, achieving within a $2$-factor of the optimal intervention cost. This approximation factor can be improved to $1+ε$ for any $ε> 0$ under some mild restrictions. Under the identity cost model, we bound the number of interventions needed to recover the entire causal graph, including the latent variables, using a parameterization of the causal graph through a special type of colliders. In particular, we introduce the notion of $p$-colliders, that are colliders between pair of nodes arising from a specific type of conditioning in the causal graph, and provide an upper bound on the number of interventions as a function of the maximum number of $p$-colliders between any two nodes in the causal graph.

Motivation & Objective

• To minimize intervention cost in causal discovery when latent variables are present, a common but challenging scenario in real-world applications.
• To address the gap in existing work by handling general intervention cost models (linear and identity) without assuming causal sufficiency (i.e., no unobserved confounders).
• To develop a method that identifies ancestral relations and the full causal graph efficiently, even when latent variables affect the structure.
• To introduce the novel concept of p-colliders—colliders formed via specific conditioning in the causal graph—and use them to bound intervention requirements.
• To provide theoretical guarantees on intervention cost and number of interventions, improving upon prior work in sparse and realistic random graph models.

Proposed method

• Proposes a strongly separating set system to model intervention sets that recover ancestral relations, ensuring each pair of nodes is separated in at least one intervention.
• Uses a greedy algorithm to construct intervention sets that minimize total cost by prioritizing high-cost variables in as few sets as possible, achieving a 2-approximation under the linear cost model.
• Introduces p-colliders as a structural property: a node is a p-collider between i and j if there exist two vertex-disjoint paths from i and j to it, with both paths having incoming edges.
• Employs a flow network transformation to detect p-colliders: splits nodes into in/out components, assigns unit capacities, and uses max flow from a source T_ij to v_k to test for two disjoint paths.
• For the identity cost model, bounds the number of interventions as O(nτ log n + n log n), where τ is the maximum number of p-colliders between any two observable nodes.
• Applies the p-collider framework to derive improved bounds over prior methods, especially when τ ≪ d²/n, where d is the maximum undirected degree.

Experimental results

Research questions

• RQ1 Can we design intervention sets that recover ancestral relations in a causal graph with latent variables at near-optimal cost under a general linear cost model?
• RQ2 How can we efficiently detect p-colliders—specific types of colliders arising from conditioning—using graph-theoretic flow techniques?
• RQ3 What is the minimum number of interventions required to recover the full causal graph, including latent variables, under the identity cost model?
• RQ4 How does the p-collider structure improve intervention efficiency compared to existing methods in sparse random graph models?
• RQ5 Can we achieve better bounds than prior work when τ (maximum p-colliders per pair) is small relative to the graph’s degree?

Key findings

• The proposed algorithm achieves a 2-approximation for intervention cost in the linear cost model, matching the best-known bound in the absence of latents.
• Under mild assumptions, the algorithm can be modified to achieve a (1+ε)-approximation for any ε>0, improving on the 2-approximation.
• For the identity cost model, the number of interventions needed to recover the full causal graph is bounded by O(nτ log n + n log n), where τ is the maximum number of p-colliders between any two observable nodes.
• In random graph models—bipartite, power-law, and Erdős–Rényi—τ is significantly smaller than d²/n, making the new bound superior to prior work when τ ≪ d²/n.
• The method reduces intervention count from O(d log²n) to O(τ log n) for recovering only the observable graph G, offering substantial gains in sparse graphs.
• The p-collider detection algorithm correctly identifies p-colliders via a max-flow formulation on a transformed graph with vertex-splitting and unit capacities.