Integer Programming for Learning Directed Acyclic Graphs from Non-identifiable Gaussian Models

TL;DR

This paper introduces a mixed-integer programming approach for learning directed acyclic graphs from Gaussian data that handles heteroscedastic noise and provides optimality guarantees, outperforming existing methods.

Contribution

It develops a novel computational framework that overcomes limitations of current algorithms by incorporating heteroscedastic noise and offering asymptotic optimality and consistency guarantees.

Findings

Outperforms state-of-the-art algorithms in numerical experiments.

Robust to heteroscedastic noise, unlike some competing methods.

Provides an early stopping criterion for asymptotic optimality.

Abstract

We study the problem of learning directed acyclic graphs from continuous observational data, generated according to a linear Gaussian structural equation model. State-of-the-art structure learning methods for this setting have at least one of the following shortcomings: i) they cannot provide optimality guarantees and can suffer from learning sub-optimal models; ii) they rely on the stringent assumption that the noise is homoscedastic, and hence the underlying model is fully identifiable. We overcome these shortcomings and develop a computationally efficient mixed-integer programming framework for learning medium-sized problems that accounts for arbitrary heteroscedastic noise. We present an early stopping criterion under which we can terminate the branch-and-bound procedure to achieve an asymptotically optimal solution and establish the consistency of this approximate solution. In addition, we show via numerical experiments that our method outperforms state-of-the-art algorithms and is robust to noise heteroscedasticity, whereas the performance of some competing methods deteriorates under strong violations of the identifiability assumption. The software implementation of our method is available as the Python package \emph{micodag}.