On the Complexity of Identification in Linear Structural Causal Models

TL;DR

This paper introduces a new polynomial-space algorithm for generic identification in linear causal models, significantly improving computational efficiency over previous methods, and proves that certain identification problems are computationally hard.

Contribution

It presents a sound, complete polynomial-space algorithm for generic identification and establishes the computational hardness of specific identification problems.

Findings

New polynomial-space algorithm for generic identification

Exponential time complexity of the algorithm in practice

Identification problem is coNP-hard in general

Abstract

Learning the unknown causal parameters of a linear structural causal model is a fundamental task in causal analysis. The task, known as the problem of identification, asks to estimate the parameters of the model from a combination of assumptions on the graphical structure of the model and observational data, represented as a non-causal covariance matrix. In this paper, we give a new sound and complete algorithm for generic identification which runs in polynomial space. By standard simulation results, this algorithm has exponential running time which vastly improves the state-of-the-art double exponential time method using a Gr\"obner basis approach. The paper also presents evidence that parameter identification is computationally hard in general. In particular, we prove, that the task asking whether, for a given feasible correlation matrix, there are exactly one or two or more parameter sets explaining the observed matrix, is hard for $\forall R$, the co-class of the existential theory of the reals. In particular, this problem is $coNP$-hard. To our best knowledge, this is the first hardness result for some notion of identifiability.