Fair GLASSO: Estimating Fair Graphical Models with Unbiased Statistical Behavior

Summary

The article explores the issue of achieving fairness in Gaussian Graphical Models (GGMs), particularly in the presence of biased data. Such biases can lead to unfair behavior in the models. To address this issue, the authors propose two bias metrics aimed at achieving statistical similarity across groups with different sensitive attributes. Based on these metrics, the authors introduce a regularized graphical lasso method called Fair GLASSO, designed to obtain sparse Gaussian precision matrices with unbiased statistical dependencies across different groups. The authors also propose an efficient proximal gradient algorithm to obtain these estimates and analyze the tradeoff between fairness and accuracy.

Strengths

（1）The authors are the first to propose definitions of fairness and bias metrics applicable to graphical models. （2）Through theoretical analysis and experiments, the effectiveness of Fair GLASSO in mitigating bias while maintaining model accuracy is demonstrated. （3）The method is compared with various existing approaches, showcasing Fair GLASSO's advantages in reducing bias and improving accuracy

Weaknesses

（1）The implementation of the algorithm relies on complex matrix operations, which may pose high computational costs in practical applications. （2）Although the experiments used multiple datasets, the variety and scale of these datasets are still limited, possibly not fully representing all practical application scenarios. （3）There are some formatting issues in the paper, such as unnumbered equations on page 23.

Questions

(1) Can the authors provide a detailed explanation for the choice of these two bias metrics? Specifically, how do these metrics capture fairness better than other potential metrics? (2) Can the authors provide a detailed breakdown of the computational complexity for each step of the Fair GLASSO algorithm? (3) How does the proposed Fair GLASSO method perform on non-Gaussian data? Have the authors considered extensions or modifications to handle such cases? (4) The method relies on certain assumptions (e.g., bounded spectrum, equal group sizes). How sensitive is the performance of Fair GLASSO to violations of these assumptions? (5) What specific evaluation metrics were used to compare the performance of Fair GLASSO with other methods? Are these metrics the most appropriate for assessing both fairness and accuracy? (6) Can the authors provide more details on the real-world datasets used, specifically the nature of the sensitive attributes and the relevance of fairness in those contexts? (7) The paper contains some unnumbered equations, such as those on page 23. Can the authors clarify the reasons for this and provide the necessary numbering for easier reference?

Rating

7

Confidence

4

Soundness

3

Presentation

3

Contribution

4

Limitations

(1) Although the paper discusses the efficiency of the Fair GLASSO algorithm, a more detailed analysis or discussion of the impact of complexity on practical applications would be better. For example, discussing potential scalability issues with larger datasets or in real-time applications could be very helpful. (2) The paper relies on certain assumptions, such as bounded spectrum and equal group sizes. Including a sensitivity analysis for these assumptions would be beneficial. (3) The paper could provide more concrete examples or case studies demonstrating the application of Fair GLASSO in real-world scenarios, beyond synthetic and small-scale examples. (4) Discuss the ethical implications of using Fair GLASSO in these fields. Emphasize the importance of stakeholder engagement and continuous monitoring to ensure that the implementation does not inadvertently harm the groups it aims to protect.

Summary

The authors propose a method for estimating Gaussian graphical models that are fair. To do this, the methodology uses Fair GLASSO, a regularized loss for estimating the precision matrix of the GGM which is fair with respect to sensitive attributes. The authors also provide theoretical results regarding the asymptotic errors of the estimated precision matrix.

Strengths

- Overall, the paper is well written. - The methodology developed is sound and supported by theoretical results. - The experimental results show that the derived method is effective in practice.

Weaknesses

- Please define modularity and "partial correlations within and across groups" for completeness. - Theorem 1 provides asymptotic results in terms of p, but it would be more interesting to derive results which are asymptotic in the number of data points n. Usually, the number of nodes are fixed and the number of datapoints increases. - It would be interesting for authors to investigate how the results change when the assumptions are violated. - Also, is n fixed even as p increases to infinity in the result above? In this case, won't the covariance matrix become uninvertible when p > n? - The results should also include a pareto frontier between error and bias when the regularization parameters vary (but p and n are fixed)

Questions

See the weaknesses section

Rating

6

Confidence

3

Soundness

3

Presentation

3

Contribution

3

Limitations

The authors should elaborate on the limitations of the proposed methodology a bit more.

Summary

Traditional graphical models may reinforce existing biases present in the data. This paper introduces a novel approach to ensure that the learned graphical models are fair across different groups or demographics.

Strengths

- The authors develop a penalty method that adds a fairness penalty to the GLASSO objective. - As the added penalty is smooth, the authors develop a FISTA-type method to solve the optimization problem with an $\ell_1$ nonsmooth term. Objective includes Gaussian graphical model loss + fairness penalty term + $\ell_1$ penalty term. - The authors provide experiments on both real and synthetic datasets, demonstrating the efficacy of their methods. - The paper is generally well-written and has a good synthetic empirical investigation.

Weaknesses

- My main concern is that the proposed fair graphical model is essentially a joint graphical model widely studied in the existing literature. Please see below (Questions) for further clarification. - The proposed penalty functions are already studied in the existing graph fairness literature. Specifically, both fairness penalty functions $H(\Theta)$ and $H_{\text{node}}(\Theta)$ are from [15, 18], except that the authors replaced $| \cdot |$ with $( \cdot )^2$; for example, please refer to Eqs (1) and (2) in [18]. - Since $H (\Theta)$ and $H_{\text{node}}(\Theta)$ are smooth functions, the authors apply FISTA-type methods for graphical models. As FISTA and its variants are already explored in graphical models and their theoretical analyses are already provided [R2], Algorithm 1 simply applies to the smooth $f$ and nonsmooth $\ell_1$ term. Thus, the theoretical contribution of Theorem 2 is limited. - Estimation guarantees in Theorem 1 follow from existing literature on graphical model estimation, except that the authors need to handle $H(\Theta)$ and $H_{\text{node}}(\Theta)$ in the proof of Theorem 1. For example, the proof of trace difference, log-determinant difference, and sparsity penalties is identical to Theorem 1 of [32]. - In real experiments, the ground truth of the graph is used to generate the data, and then a fair graph is estimated. However, I am concerned that this might not be considered as real experiments since, in real scenarios, the precision matrix for graphical model estimation is indeed unavailable. In addition, it is not clear that these real datasets are suitable for Gaussian graphical models, as they should consist of binary numbers (e.g., interactions between people as nodes would be 0/1). This raises questions about the applicability of the method in real-world scenarios.

Questions

- How does the proposed method differ from the joint graphical lasso [R1, R3, R4, R5], which borrows strength across the groups within graphical models that share certain characteristics, such as the locations or weights of nonzero edges? Specifically, under AS4 and since $Z$ is an indicator matrix, $H(\Theta)$ and $H_{\text{node}}(\Theta)$ are identical to the fused penalty term widely used in the joint estimation of joint graphical models; see for example [R4]. If you run joint graphical lasso [R1, R3, R4, R5], do they give similar results to yours? - What is the difference from existing FISTA methods for graphical models; see for example [R2]. Can you clarify how the Hessian of $f$ in the FISTA section is positive definite? It would be good if you could clarify how the rate changes with the eigenvalue of $\nabla^2 f$. - What is the the key difference in bounding the trace difference, log-determinant difference, and sparsity penalties compared to Theorem 1 of [32] and existing joint estimation of graphical models [R1, R3, R4, R5]? - Can you clarify what are samples and nodes for your dataset? For example, what is the data matrix $X$ for the friendship experiment? - Do these datasets resemble real-world applications of the proposed Gaussian graphical models, considering that i) they include a ground truth graph and ii) nodes of social networks have binary associated samples (e.g., 0/1 interactions between people)? **Additional References** - [R1] Pircalabelu, Eugen, and Gerda Claeskens. "Community-based group graphical lasso." Journal of Machine Learning Research 21.64 (2020): 1-32. - [R2] Oh, Sang, Onkar Dalal, Kshitij Khare, and Bala Rajaratnam. "Optimization methods for sparse pseudo-likelihood graphical model selection." Advances in Neural Information Processing Systems 27 (2014). - [R3] Guo, Jian, et al. "Joint estimation of multiple graphical models." Biometrika 98.1 (2011): 1-15. - [R4] Danaher, Patrick, Pei Wang, and Daniela M. Witten. "The joint graphical lasso for inverse covariance estimation across multiple classes." Journal of the Royal Statistical Society Series B: Statistical Methodology 76.2 (2014): 373-397. - [R5] Ma, Jing, and George Michailidis. "Joint structural estimation of multiple graphical models." Journal of Machine Learning Research 17.166 (2016): 1-48.

Rating

4

Confidence

4

Soundness

2

Presentation

2

Contribution

2

Limitations

N/A

Summary

The paper introduces Fair GLASSO, a method for estimating Gaussian graphical models (GGMs) that addresses biases in data with respect to sensitive nodal attributes. The authors propose two bias metrics to promote fairness in statistical similarities across different groups, leading to the development of Fair GLASSO, a regularized graphical lasso approach. The paper also presents a proximal gradient algorithm for efficient estimation. Theoretical analysis shows the tradeoff between fairness and accuracy, and empirical results validate the effectiveness of the proposed method on both synthetic and real-world data.

Strengths

1. Novel contribution in defining fairness for graphical models and proposing methods to estimate fair GGMs. 2. Thorough theoretical analysis including error bounds and convergence guarantees. 3. Comprehensive empirical evaluation on both synthetic and real-world datasets. 4. Proposed method can improve both fairness and accuracy in certain scenarios, especially when the underlying graph is fair but the data is biased. 5. Clear explanations and intuitive visualizations of concepts and results.

Weaknesses

1. This work is limited to Gaussian graphical models and may not be able to generalize to other types of graphical models such as ising model or covariance model. 2. The fairness focus is mainly focusing on demographic parity and this may not be the optimal in relality. It would be better to explore other fairness definitions even though the authors claim that other definitions of group fairness can be similarly adapted. 3. Some of the real-world datasets use synthetic signals, which may limit the real-world applicability of those specific results. 4. The method introduces additional hyperparameters such as $\mu_1$ and $\mu_2$ that need to be tuned.

Questions

1. How sensitive is the method to the choice of hyperparameters $\mu_1$ and $\mu_2$? 2. How would the approach extend to non-Gaussian graphical models? 3. How you considered other fairness metrics beyond demographic parity? 4. How does the computational complexity scale for very large graphs (e.g. millions of nodes)?

Rating

5

Confidence

4

Soundness

3

Presentation

3

Contribution

3

Limitations

1. Only applicable to Gaussian graphical models. 2. Focuses solely on group fairness via demographic parity. 3. May not generalize well to more general graph structures. 4. Theoretical guarantees assume specific conditions on the true precision matrix and group sizes. 5. Limited exploration of the trade-offs between fairness and accuracy across different scenarios.

Concerns from Reviewer QDSi

Thank you for your detailed response. Most of my previous concerns are basically addressed. However, after taking a look at Reviewer QDSi's comments, I have some additional concerns as follows. (1) The main concern is about novelty. According to Reviewer QDSi's comment, the fairness penalty functions appear to be very similar to those in [15,18]. While you've clarified some differences from existing joint graphical models, the core idea still appears to be a variation on well-studied concepts. The modifications to balance sparsity patterns and signed edges, while interesting, do not seem to constitute a substantial leap forward in fair graph learning. (2) As for the technical contribution, the adjustments to the FISTA algorithm, including the constrained optimization and convergence guarantees, are incremental improvements rather than fundamental innovations. The theoretical analysis, while thorough, largely follows established approaches in graphical model estimation. (3) Your focus on demographic parity limits the broader applicability of the method. As noted in your response about equalized odds, adapting to other fairness metrics introduces additional complexities that are not fully addressed in the current work. Based on these new concerns, I believe this paper still needs improvement and I will drop my score from 6 to 5.

Fair GLASSO: Estimating Fair Graphical Models with Unbiased Statistical Behavior

I would like to thank the authors for their detailed responses, clarification, and additional results. Most of my questions are solved and I raise my score.