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Fair GLASSO: Estimating Fair Graphical Models with Unbiased Statistical Behavior

Madeline Navarro, Samuel Rey, Andrei Buciulea, Antonio G. Marques, Santiago Segarra

TL;DR

This work tackles fair estimation of Gaussian graphical models from biased data by formulating two convex fairness penalties that quantify demographic parity-like balance in the conditional dependencies across nodal groups. It introduces Fair GLASSO, a convex, regularized graphical-lasso objective with a sparsity penalty and a fairness term, solved efficiently via a proximal-gradient (FISTA) algorithm with guaranteed convergence. Theoretical results characterize the tradeoff between fairness and accuracy, providing finite-sample error bounds that depend on the true graph bias, and showing conditions under which accuracy can be preserved. Empirical results on synthetic and real-world networks demonstrate that Fair GLASSO reduces bias while maintaining or improving estimation accuracy compared with standard GLASSO and related baselines, validating its practical impact for fair graph learning in high-stakes domains.

Abstract

We propose estimating Gaussian graphical models (GGMs) that are fair with respect to sensitive nodal attributes. Many real-world models exhibit unfair discriminatory behavior due to biases in data. Such discrimination is known to be exacerbated when data is equipped with pairwise relationships encoded in a graph. Additionally, the effect of biased data on graphical models is largely underexplored. We thus introduce fairness for graphical models in the form of two bias metrics to promote balance in statistical similarities across nodal groups with different sensitive attributes. Leveraging these metrics, we present Fair GLASSO, a regularized graphical lasso approach to obtain sparse Gaussian precision matrices with unbiased statistical dependencies across groups. We also propose an efficient proximal gradient algorithm to obtain the estimates. Theoretically, we express the tradeoff between fair and accurate estimated precision matrices. Critically, this includes demonstrating when accuracy can be preserved in the presence of a fairness regularizer. On top of this, we study the complexity of Fair GLASSO and demonstrate that our algorithm enjoys a fast convergence rate. Our empirical validation includes synthetic and real-world simulations that illustrate the value and effectiveness of our proposed optimization problem and iterative algorithm.

Fair GLASSO: Estimating Fair Graphical Models with Unbiased Statistical Behavior

TL;DR

This work tackles fair estimation of Gaussian graphical models from biased data by formulating two convex fairness penalties that quantify demographic parity-like balance in the conditional dependencies across nodal groups. It introduces Fair GLASSO, a convex, regularized graphical-lasso objective with a sparsity penalty and a fairness term, solved efficiently via a proximal-gradient (FISTA) algorithm with guaranteed convergence. Theoretical results characterize the tradeoff between fairness and accuracy, providing finite-sample error bounds that depend on the true graph bias, and showing conditions under which accuracy can be preserved. Empirical results on synthetic and real-world networks demonstrate that Fair GLASSO reduces bias while maintaining or improving estimation accuracy compared with standard GLASSO and related baselines, validating its practical impact for fair graph learning in high-stakes domains.

Abstract

We propose estimating Gaussian graphical models (GGMs) that are fair with respect to sensitive nodal attributes. Many real-world models exhibit unfair discriminatory behavior due to biases in data. Such discrimination is known to be exacerbated when data is equipped with pairwise relationships encoded in a graph. Additionally, the effect of biased data on graphical models is largely underexplored. We thus introduce fairness for graphical models in the form of two bias metrics to promote balance in statistical similarities across nodal groups with different sensitive attributes. Leveraging these metrics, we present Fair GLASSO, a regularized graphical lasso approach to obtain sparse Gaussian precision matrices with unbiased statistical dependencies across groups. We also propose an efficient proximal gradient algorithm to obtain the estimates. Theoretically, we express the tradeoff between fair and accurate estimated precision matrices. Critically, this includes demonstrating when accuracy can be preserved in the presence of a fairness regularizer. On top of this, we study the complexity of Fair GLASSO and demonstrate that our algorithm enjoys a fast convergence rate. Our empirical validation includes synthetic and real-world simulations that illustrate the value and effectiveness of our proposed optimization problem and iterative algorithm.
Paper Structure (26 sections, 66 equations, 4 figures, 3 tables, 1 algorithm)

This paper contains 26 sections, 66 equations, 4 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Notation
  3. Fair Gaussian Graphical Models
  4. Bias Metric for Fair Graphical Models
  5. Fair GLASSO
  6. Graphical Lasso for Fair GGMs
  7. Fair GLASSO Theoretical Analysis
  8. Algorithmic Implementation
  9. Experiments
  10. Estimating Fair Graphs with Biased Data
  11. Performance as Graph Size Increases
  12. Social Network with Synthetic Signals
  13. Fair GGMs for Real-World Data
  14. Conclusion
  15. Related Work
  16. ...and 11 more sections

Figures (4)

  • Figure 1: Three real-world networks with node groups denoted by color. Within-group edges are in blue and across-group edges in red, while edge widths correspond to edge weight magnitudes. For each network, we present ("M") the modularity of the graphs with respect to group membership masrour2020bursting, ("W") the ratio of positive to negative estimated partial correlations for within-group edges, and ("A") an analogous ratio for across-group edges. Networks in (a) and (c) show high group-wise modularity, while (b) and (c) show significant preferences for positive correlations in the same group.
  • Figure 2: Estimation performance in terms of error and bias. (a) Bias and error for estimating a fair graph as data becomes more biased. (b) Bias and error as graph size $p$ grows for ER graphs. (c) Bias and error for a biased real-world network zachary1977information as the number of observations $n$ grows.
  • Figure 3: Estimated Karate club network via graphical lasso with and without penalties $H$ and $H_\mathrm{node}$. Node colors denote group membership, while edge thickness denotes edge weight magnitude and edge color its sign, with blue (red) as positive (negative) correlation. (a) Estimation via GL. (b) Estimation via FGL. (c) Estimation via NFGL.
  • Figure : Fair GLASSO from Gaussian observations.