On the Number of Conditional Independence Tests in Constraint-based Causal Discovery

TL;DR

This paper introduces a new algorithm for causal discovery that reduces the number of conditional independence tests needed, achieving near-optimal complexity and demonstrating efficiency through simulations and real data.

Contribution

The paper presents an algorithm with improved complexity for constraint-based causal discovery, establishing near-optimal bounds and validating its efficiency empirically.

Findings

The proposed algorithm requires fewer tests than existing methods.

It achieves complexity close to the theoretical lower bound.

Empirical results show improved efficiency on real and synthetic data.

Abstract

Learning causal relations from observational data is a fundamental problem with wide-ranging applications across many fields. Constraint-based methods infer the underlying causal structure by performing conditional independence tests. However, existing algorithms such as the prominent PC algorithm need to perform a large number of independence tests, which in the worst case is exponential in the maximum degree of the causal graph. Despite extensive research, it remains unclear if there exist algorithms with better complexity without additional assumptions. Here, we establish an algorithm that achieves a better complexity of $p^{\mathcal{O}(s)}$ tests, where $p$ is the number of nodes in the graph and $s$ denotes the maximum undirected clique size of the underlying essential graph. Complementing this result, we prove that any constraint-based algorithm must perform at least $2^{\Omega(s)}$ conditional independence tests, establishing that our proposed algorithm achieves exponent-optimality up to a logarithmic factor in terms of the number of conditional independence tests needed. Finally, we validate our theoretical findings through simulations, on semi-synthetic gene-expression data, and real-world data, demonstrating the efficiency of our algorithm compared to existing methods in terms of number of conditional independence tests needed.