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Causal discovery for linear causal model with correlated noise: an Adversarial Learning Approach

Mujin Zhou, Junzhe Zhang

TL;DR

This work tackles causal discovery in linear systems with unmeasured confounding by learning a binary causal structure M = (S_B,S_Σ) that encodes both direct and confounding edges. It reframes model selection as minimizing Bayesian free energy, which is equivalent to minimizing the KL divergence between the true data distribution and the model distribution, and solves this via the f-GAN framework with a differentiable relaxation of discrete graph structures using Gumbel-Softmax. The method, fGAN-CD, unifies Bayesian model selection with adversarial learning to recover general ADMGs without restrictive bow-free assumptions, demonstrated on synthetic data with dense confounding where baselines fail. The approach yields state-of-the-art structural recovery and accurate independence constraints, highlighting its potential for causal discovery in singular models and real-world confounded settings.

Abstract

Causal discovery from data with unmeasured confounding factors is a challenging problem. This paper proposes an approach based on the f-GAN framework, learning the binary causal structure independent of specific weight values. We reformulate the structure learning problem as minimizing Bayesian free energy and prove that this problem is equivalent to minimizing the f-divergence between the true data distribution and the model-generated distribution. Using the f-GAN framework, we transform this objective into a min-max adversarial optimization problem. We implement the gradient search in the discrete graph space using Gumbel-Softmax relaxation.

Causal discovery for linear causal model with correlated noise: an Adversarial Learning Approach

TL;DR

This work tackles causal discovery in linear systems with unmeasured confounding by learning a binary causal structure M = (S_B,S_Σ) that encodes both direct and confounding edges. It reframes model selection as minimizing Bayesian free energy, which is equivalent to minimizing the KL divergence between the true data distribution and the model distribution, and solves this via the f-GAN framework with a differentiable relaxation of discrete graph structures using Gumbel-Softmax. The method, fGAN-CD, unifies Bayesian model selection with adversarial learning to recover general ADMGs without restrictive bow-free assumptions, demonstrated on synthetic data with dense confounding where baselines fail. The approach yields state-of-the-art structural recovery and accurate independence constraints, highlighting its potential for causal discovery in singular models and real-world confounded settings.

Abstract

Causal discovery from data with unmeasured confounding factors is a challenging problem. This paper proposes an approach based on the f-GAN framework, learning the binary causal structure independent of specific weight values. We reformulate the structure learning problem as minimizing Bayesian free energy and prove that this problem is equivalent to minimizing the f-divergence between the true data distribution and the model-generated distribution. Using the f-GAN framework, we transform this objective into a min-max adversarial optimization problem. We implement the gradient search in the discrete graph space using Gumbel-Softmax relaxation.
Paper Structure (20 sections, 20 equations, 2 figures, 1 table, 1 algorithm)

This paper contains 20 sections, 20 equations, 2 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Linear SEM with unmeasured confounding
  4. Variational Divergence Minimization (f-GAN)
  5. Definition of f-divergence
  6. Variational Lower Bound
  7. Generative Adversarial Training Objective:
  8. Method
  9. Bayesian Free Energy and Model Selection
  10. Optimization via f-GAN Framework
  11. Deriving the Standard GAN Objective Function from f-GAN:
  12. Generative Process
  13. Differentiable Relaxation and Constraints
  14. Gumbel-Softmax Relaxation:
  15. Algorithm Summary
  16. ...and 5 more sections

Figures (2)

  • Figure 1: Comparison of learned structures. Red dashed lines represent bidirected edges (confounders). It can be seen that fGAN-CD has a greater advantage in learning the arrow directions.
  • Figure 2: Comparison of Learned Structures (Top Row) and their corresponding PAGs (Bottom Row). In the ADMGs, red dashed lines indicate unmeasured confounding.