Optimal structure learning and conditional independence testing

TL;DR

This paper reveals a fundamental link between optimal structure learning and conditional independence testing, showing that the minimax rates for both are interconnected, and introduces a modified PC algorithm for optimal structure learning.

Contribution

It establishes a general reduction between structure learning and independence testing, deriving optimal rates for various models and proposing a modified PC algorithm for optimal learning.

Findings

Minimax optimal rates for structure learning are determined by those for conditional independence testing.

A general reduction between structure learning and testing is established for poly-forests.

A modified PC algorithm achieves optimal structure learning in these settings.

Abstract

We establish a fundamental connection between optimal structure learning and optimal conditional independence testing by showing that the minimax optimal rate for structure learning problems is determined by the minimax rate for conditional independence testing in these problems. This is accomplished by establishing a general reduction between these two problems in the case of poly-forests, and demonstrated by deriving optimal rates for several examples, including Bernoulli, Gaussian and nonparametric models. Furthermore, we show that the optimal algorithm in these settings is a suitable modification of the PC algorithm. This theoretical finding provides a unified framework for analyzing the statistical complexity of structure learning through the lens of minimax testing.