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Interventional Fairness on Partially Known Causal Graphs: A Constrained Optimization Approach

Aoqi Zuo, Yiqing Li, Susan Wei, Mingming Gong

TL;DR

The proposed approach involves modeling fair prediction using a Partially Directed Acyclic Graph (PDAG), specifically, a class of causal DAGs that can be learned from observational data combined with domain knowledge that is used to measure causal fairness.

Abstract

Fair machine learning aims to prevent discrimination against individuals or sub-populations based on sensitive attributes such as gender and race. In recent years, causal inference methods have been increasingly used in fair machine learning to measure unfairness by causal effects. However, current methods assume that the true causal graph is given, which is often not true in real-world applications. To address this limitation, this paper proposes a framework for achieving causal fairness based on the notion of interventions when the true causal graph is partially known. The proposed approach involves modeling fair prediction using a Partially Directed Acyclic Graph (PDAG), specifically, a class of causal DAGs that can be learned from observational data combined with domain knowledge. The PDAG is used to measure causal fairness, and a constrained optimization problem is formulated to balance between fairness and accuracy. Results on both simulated and real-world datasets demonstrate the effectiveness of this method.

Interventional Fairness on Partially Known Causal Graphs: A Constrained Optimization Approach

TL;DR

The proposed approach involves modeling fair prediction using a Partially Directed Acyclic Graph (PDAG), specifically, a class of causal DAGs that can be learned from observational data combined with domain knowledge that is used to measure causal fairness.

Abstract

Fair machine learning aims to prevent discrimination against individuals or sub-populations based on sensitive attributes such as gender and race. In recent years, causal inference methods have been increasingly used in fair machine learning to measure unfairness by causal effects. However, current methods assume that the true causal graph is given, which is often not true in real-world applications. To address this limitation, this paper proposes a framework for achieving causal fairness based on the notion of interventions when the true causal graph is partially known. The proposed approach involves modeling fair prediction using a Partially Directed Acyclic Graph (PDAG), specifically, a class of causal DAGs that can be learned from observational data combined with domain knowledge. The PDAG is used to measure causal fairness, and a constrained optimization problem is formulated to balance between fairness and accuracy. Results on both simulated and real-world datasets demonstrate the effectiveness of this method.
Paper Structure (50 sections, 19 theorems, 14 equations, 20 figures, 3 algorithms)

This paper contains 50 sections, 19 theorems, 14 equations, 20 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Structural causal model and causal graph
  4. Causal inference and interventional fairness
  5. Problem formulation
  6. Fairness under MPDAGs as a graphical problem
  7. $\epsilon$-Approximate interventional fairness
  8. Interventional Fairness under MPDAGs
  9. Modeling and verification
  10. Identification of fairness criteria
  11. Solving the optimisation problem
  12. Applicability to other causal fairness notions
  13. Experiment
  14. Synthetic Data
  15. Real Data
  16. ...and 35 more sections

Key Result

Lemma 3.1

Let $\mathcal{G}$ be the causal graph of the given model $(U,V,F)$. Then $\hat{Y}$ will be interventionally fair if it is a function of the admissible set $\mathbf{X}_{ad}$ and non-descendants of $\mathbf{A}$.

Figures (20)

  • Figure 1: (a) is an underlying causal DAG $\mathcal{D}$ with three variables $X_{[1]}$, $X_{[2]}$ and $X_{[3]}$ in $\mathbf{X}$; (b) is a causal DAG $\mathcal{D^*}$ under modeling on $\hat{Y}$; (c) is an example MPDAG $\mathcal{G}$ such that $\mathcal{D} \in [\mathcal{G}]$; (d) is an example MPDAG $\mathcal{G^*}$ such that $\mathcal{D^*} \in [\mathcal{G^*}]$.
  • Figure 2: Accuracy fairness trade-off.
  • Figure 3: Density plots of the predicted $Y_{A\leftarrow a}$ and $Y_{A\leftarrow a'}$ in synthetic data.
  • Figure 4: Accuracy fairness trade-off.
  • Figure 5: Density plot of the predicted $Y_{A\leftarrow a}$ and $Y_{A\leftarrow a'}$ in Student data.
  • ...and 15 more figures

Theorems & Definitions (38)

  • Definition 2.1: Interventional fairness
  • Lemma 3.1
  • Definition 3.1: $\epsilon$-Approximate Interventional Fairness
  • Definition 4.1: Augmented-$\mathcal{G}$ with $\hat{Y}$
  • Theorem 4.1
  • Proposition 4.1
  • Lemma A.1
  • Lemma A.2
  • Corollary A.1: Bucket Decomposition
  • Lemma A.3
  • ...and 28 more