Fairness-Aware Estimation of Graphical Models
Authors: Zhuoping Zhou
Davoud Ataee Tarzanagh
Bojian Hou
Qi Long
Li Shen
Venue: NeurIPS 2024 poster (2024)
Keywords: Fairness
Graphical Model
Optimization
Primary area: fairness

Abstract: This paper examines the issue of fairness in the estimation of graphical models (GMs), particularly Gaussian, Covariance, and Ising models. These models play a vital role in understanding complex relationships in high-dimensional data. However, standard GMs can result in biased outcomes, especially when the underlying data involves sensitive characteristics or protected groups. To address this, we introduce a comprehensive framework designed to reduce bias in the estimation of GMs related to protected attributes. Our approach involves the integration of the pairwise graph disparity error and a tailored loss function into a nonsmooth multi-objective optimization problem, striving to achieve fairness across different sensitive groups while maintaining the effectiveness of the GMs. Experimental evaluations on synthetic and real-world datasets demonstrate that our framework effectively mitigates bias without undermining GMs' performance.

Fairness-Aware Estimation of Graphical Models
Zhuoping Zhou; Davoud Ataee Tarzanagh, Bojian Hou; Qi Long, Li Shen
University of Pennsylvania (zhuopinzēsas. tarzanag@Jupenn. edu (bojian.hov) qlong, 1i. shenhhapnnmeeiiiiie upenn edu edu
Abstract
This paper examines the issue of fairness in the estimation of graphical models (GMs), particularly Gaussian, Covariance, and Ising models. These models play a vital role in understanding complex relationships in high-dimensional data. How- ever, standard GMs can result in biased outcomes, especially when the underlying data involves sensitive characteristics or protected groups. To address this, we introduce a comprehensive framework designed to reduce bias in the estimation GMs related to protected attributes. Our approach involves the integration the painwise graph disparity error and a tailored loss function into a nonsmooth multi-objective optimization problem, striving achieve fairness across different sensitive groups while maintaining the effectiveness of the GMs. Experimental evaluations on synthetic and real-world datasets demonstrate that our framework effectively mitigates bias without undermining GMs' performance.
Introduction
Graphical models (GMs) are probabilistic models that use graphs represent dependencies between random variables [34]. They are essential domains such gene expression
[91]. social net- works [17], computer vision [33], and recommendation systems [8]. The capacity of GMs to handle complex dependencies makes them crucial across various data-intensive disciplines. Therefore, as society's reliance machine learning grows. ensuring the fairness of these models becomes increasingly paramount; see Section 1.1 for further discussions. While significant research has addressed fairness in supervised learning [29], the domain of unsupervised learning, particularly in the estimation of GMs, remains less explored.
We address the fair estimation of sparse GMs where the number of variables P is much larger than the number of observations N [22, 16, 43]. We focus on three types of GMs:
I. Gaussian Graphical Model: Rows X
X N. in the data matrix X NxP are i.i.d. from a multivariate Gaussian distribution (0, Σ). The conditional independence graph is determined by the sparsity of the inverse covariance matrix Σ-1, where " indicates conditional independence between the jth and j'th variables.
II. Gaussian Covariance Graph Model: Rows X XN: are i..d. from from.N(( (o E). The marginal independence graph determined by the sparsity of the covariance matrix Σ. where Σnj indicates marginal independence between the jth and j'th variables.
III. Binary Ising Graphical Model:
Rows XI....., XN: are binary vectors and i.i.d, with
P
p θ) = (Z( exp
E00 +
Σ 0)02)
3=1
ISKCPEP
Here, θ is symmetric matrix, and Z(O) normalizes the density. 0,5 0 indicates conditional independence between the jth and j'th variables. The sparsity pattern of e reflects the conditional independence graph.
Eqqull contribution
Conresponding authors
38th Conference on Neural Information Processing Systems (NeurIPS 2024).

---

X X. X
X,
(b) Dansset for Group Dlue
[d) Graphicai Modcl Group Blue
Graphical Modele Entire Datooct
X. X 2, X. X
(4) Euture Datasct
(3) Dataset for Crouup Orange
(c) Graphacal Models on Group Orange
(g) Fair Graphical Models on Luttre Dutaset
Figure 1: Illustration of GM and its fair variant. (a) displays the entire dataset, split into Group Blue (b) and Group Orange (c). (d) and (e) show GMs for each group, detailing the relationships between variables. (f) uses GM for the entire dataset. The fair model in (g) adjusts these relationships to ensure equitable representation and minimize biases in subgroup analysis.
In a data matrix X E RNP,, each column corresponds
node in a graph 9 (V.E), where V = P) are vertices and C V x V are edges. Column X, (i {1,2, P)) is a vector of length N, representing the observations for the i-th variable across all N samples. The graph G, represented by the symmetric matrix O has nonzero entries indicating edges and reflects the graph's independence properties. To obtain sparse and interpretable graph estimate, consider
minimize L (O;X) + x||O|11 subj.
θ E M.
Here, is a loss function; A|| 111 is the f--norm regularization with parameter À > 0; and M is a convex constraint subset of RPXP. For example, in a Gaussian GM, L(O; X) = log det(@)
(race(SO), where S = n-1 Σ X Xi., and M is the set of P x P positive definite matrices.
1.1 Motivation
Our motivations for obtaining fair GM estimation are summarized follows. i) Equitable Repre- sentation; Standard group-specific GM models may improve accuracy for targeted groups but do not ensure fairness and can reinforce biases present in the data [48]. A unified approach considering the entire dataset is essential for mitigating biases and promoting faimess across all groups ii) Legal and Ethical Compliance: Ethical and legal considerations [12] require explicit consent for processing sensitive attributes in model selection. Thus, constructing a fair estimation approach that adheres to fairness practices, uses data with consent, and excludes sensitive attribute information during deployment ensures privacy and legal compliance. iii) Generalization across Groups: A unified fair GM captures differences across groups without segregating the model, enhancing generalizability and preventing overfitting
specific group [30], risk in training separate models for each group.
For further discussion, we compare GM with its proposed Fair variant, illustrated in Figure 1. Panel (a) shows the entire dataset, divided into Group Blue and Group Orange in panels (b) and (c). Panels (d) and (e) detail the GM for each group, highlighting variable relationships. Panel (f) demonstrates conventional GM applied to the full dataset, revealing bias towards Group Blue. Panel (g) introduces Fair GM, including modifications (red dashed lines) reduce bias and ensure balanced representation. These adjustments correct relationships within the model, promoting fairness by preventing disproportionate favor towards any group. This illustration highlights the bias challenge in GMs and the steps Fair GMs take to ensure fair and equal modeling outcomes across groups
1.2 Contributions
Our contributions are summarized follows:
We propose framework to mitigate bias in Gaussian, Covariance, and Ising models related to protected attributes. This is achieved by incorporating pairwise graph disparity error and a tailored loss function into a nonsmooth multi multi-objective optimization problem, striving to achieve fairesss across different sensitive groups while preserving GMs performance.
We develop a proximal gradient method with non-asymptotic convergence guarantees for nonsmooth multi-objective optimization, applicable to Gaussian, Covariance, and Ising models (Theorems 6-8). To our knowledge, this the first work providing a multi-objective proximal gradient method for GM estimation, in contrast to existing single-objective GM methods [3, 87, 13].
We provide extensive experiments to validate the effectiveness of our GM framework in mitigating bias while maintaining model performance synthetic data, the Credit Dataset,

---

the Cancer Genome Atlas Dataset, Alzheimer's Disease Neuroimaging Initiative (AINN), and the binary LFM-1b Dataset for recommender systems'.
2 Related Work
Estimation of Graphical Models. The estimation of network structures from nigh-dimensional data [89, 51, 1, 84, 19, 84] is a well--xxpored domain with significant applications in biomedical and social sciences [44, 59, 25]. Given the challenge of parameter estimation with limited samples, sparsity is imposed via regularization, commonly through an l1 penalty to encourage sparse network structures [22, 37, 25J. However, these approaches may overlook the complexity of real-world networks, which often have varying structures across scales, including densely connected subgraphs or communities [13, 26, 23]. Recent work extends beyond simple sparsity to estimate hidden commu- nities within networks, reflecting homogeneity within and heterogeneity between communities [47]. This includes inferring connectivity and performing graph estimation when community information known, as well as considering these tasks in the context of heterogeneous observations [42, 80, 24).
Fairness. Fairness research in machine learning has predominantly focused on supervised methods [11, 4, 15, 39, 92, 79, 28]. Our work broadens this scope to unsupervised learning. incorporating insights from [65, 77, 56, 9, 10]. Notably, [41] has developed algorithms for fair clustering using the Laplacian matrix. Our approach diverges by not presupposing any graph and Laplacian structures. The most relevant works this study are [78. 93, 52. 53]. Specifically, [78] initiated the learning of fair GMs using an e.rrgulaaizzdd pscudo- pscudo-likelihov likelhhooo method for joint GMs estimation and fair community detection. 193, 94] proposed a fair spectral clustering model that integrates graph construction, fair spectral embedding, and discretization into single objective function. Unlike these models, which assume community structures, our study formulates fair GMs without such assumption. Concurrently with this work, [52] proposed a regularization method for fair Gaussian GMs assuming the availability of node attributes. Their methodology significantly differs from ours, as we focus on developing three classes of fair GMs (Gaussian, Covariance, and Ising models) for imbalanced groups without node attributes, aiming to automatically ensure faimess through non-smooth multi multi-objective optimization.
3 Fair Estimation of Graphical Models
Notation. Rd denotes the d-dimensional real space, and I&. and R4, + its positive and negative orthants. Vectors and matrices are in bold lowercase and uppercase letters (c.g.. a, A), with elements a; and Gij. Rows and columns of A are A, and A.j. respectively. For symmetric A, A > 0 and A / O denote positive definiteness and semi-definiteness. (A) is the ith smallest eigenvalue of A. The matrix norms are defined as || = Eij |a:yl and || = (Eiy by [az]1))//2_ For any positive integer n, [n] == [I....,n} n). Any notation is defined upon its first use and summarized in Table 3.
3.1 Graph Disparity Error
To evaluate the effects of joint GMs learning different groups, we compare models trained on group-specific data with those trained on combined dataset. Let dataset X be divided into K sensitive groups, with the data for group & E [K] represented as X₃ E R x where N₂ is the sample size for group k and N = >km measured by the loss function L(O; Xx). ). Our goal is to find a global model O* that minimizes performance discrepancies across groups We define graph disparity error to quamtify faimess:
1=1 1 Nx. The performance a GM, denoted by θ for group
Definition 1 (Graph Disparity Error). Given dataset X E IR NxP with K sensitive groups. where X represents the data for group h: E [K]. let
0
arg min C(O,: X.) + 0 O.EM
(3)
The graph disparity error for group le is then:
E (0) = L( Xx) L(O Xn), 1 1 k S K.
This measures the loss difference between a global graph matrix e and the optimal local graph matrix % for each group's is data X A fair GM, under Definition 1, seeks to balance & across all groups
*oode is available https: //gg:////tttttt con wow./PennSSennaa/aaaii.....
3

---

Algorithm Fair Estimation of GMs (Fair GMs)
Require:
Data Matrix X = = 2 U....U............ X k; Parameters À > 0, > 0, > 0, and { >> L.
S1. Get local graph estimates
1 using (3), and initialize global graph estimate (
S2. For
= 1 to T do:
2 (++1)) t (((O(()) (+). where Pe is obtained by solving Subproblem (9).
Output: Fair global graph estimate (t+1),
Definition 2 (Fair GM). GM with graph matrix O* is called fair if the graph disparity errors among differeni groups are equal, i.e.
E, (0 = E2 (0") = ........ Ex (0*).
(5)
To address the imbalance in graph disparity error among all groups, we introduce the idea of pairwise graph disparity error, which quantifies the variation in graph disparity between different groups.
Definition 3 (Pairwise Graph Disparity Error). Let Φ : IR - R be a penalty function such as
<(x) = exp(x) (I) or δ(z) = = The pairwise graph disparity error for the group defined as
D (+):
Σ SEiR],s7k
(E (0) - E. (+))
The motivation for Definition follows from the work of [35, 65, 95] in PCA and CCA. our convergence analysis, we focus on smooth functions Φ. such as squared or exponential functions, while nonsmooth choices, such as (xx) |zr], can be explored in the experimental evaluations.
3.2 Multi-Objective Optimization for Fair GMs
This section introduces framework designed to mitigate bias in GMs (including Gaussian, Covari- ance, and Ising) related protected attributes by incorporating pairwise graph disparity error into nonsmooth multi-objeccive optimization problem. Smooth multi-objective optimization tackles fair- ness challenges in unsupervised learning [35, 95], proving particularly useful when decision-making involves multiple conflicting objectives.
We use non-smooth mulli-objective optimization to balance two key factors: the loss in GMs and the pairwise graph disparity errors. To achieve this, let
fi (e) X); F3 (+) = (+) + g (e),
fx (+) = A (9),
for 2 < k < K + 1,
(7a)
for 1 < k < M K +1,
(7b)
where g (0) == 0|| for some À >
Consequently, we propose the following multi-objective optimization problem for Fair GMs:
minimize F (e) == [Fi (0) Fas (e)]
subj. θ E M.
(8)
Here, M is convex constraint subset of R PxP and F f R is a multi-objective function. Assumption For some L > 0, all O E M, E [ VV J ( V (9) |F S L 4 O||
Note that Assumption A holds for smooth ø functions such as squared or exponential, as specified in Definition 6, and when L is smooth loss function. We demonstrate in Appendix C that this assumption holds for the Gaussian, Covariance, and Ising models studied in this work. To proceed, we provide the following definitions; see [20, 75, 73] for more details.
Definition 4 (Pareto Optimality), In Problem (8), a solution O* E M Pareto optimal if there is @ E M such that F(O) < F( and F(O) & F(O"). ). It is weakly Pareto optimal if there is no @ E M such that F(O) x F()*).
Definition 5 (Pareto Stationary). We define a point O PxP Pareto stationary (or critical) if satisfies the following condition:
oD) (
max F ( kE[M]
= lim a-20
2 0 for all D DERPXP
Q

---

To solve Problem (8), we the proximal gradient method and establish its convergence to Pareto stationary point for the nonsmooth Problem (8). The procedure for our fairness-aware GMs (Fair GMs) is detailed in Algorithm 1. Given local graph estimates { ( k= obtained in S1., and <> L, where is Lipschitz constant defined in Assumption A, the update the global fair graph estimate o is produced in S2. by solving:
Pr (+)
arg mig (p: (#; 3EM
with
(9a)
V f)( + 3 - - l 1 -011
Pe (
θ
(9b)
AE
Note that the convexity of ( X |||||| ensures unique solution for Problem (9). We provide simple yet efficient approach to solve Subproblem (9) through its dual in Appendix B. In addition,
Proposition 11 in Appendix B characterizes the weak Pareto optimality for Problem (8).
3.3 Theoretical Analysis
We apply Algorithm I to Gaussian, Covariance, and Ising models and provide theoretical guarantees.
Fair Graphical Lasso (Fair GLasso). Consider X X.v: i.i.d. samples from (00, Σ). In the GLasso method [22], the loss defined as Lg (0: == log (egd(t)) (+) + trace where θ constrained to the set .M- = { > 0, θ θ and = n -1 - X E RNNxNN the empirical covariance matrix of X. Extending this to fair GLasso and following (8), the multi-objective
optimization problem is formulated as:
minimize θ
F( = Co ():))
+ e (()) (M)
(Fair GLasso)
subj. to O O 0, 0
Assumption B. Let N be the set of weakly Pareto optimal points for (8), and 22(a) = -0 F(O) I a denote the the level set of F for a RM, For all E lt (F((((( there exists
O' E N such that F(O") = F()) and
R==
sup
| (9-6) 0 <0. eEF (EF-(((F))
inf
F P
This assumption is satisfied when 2p (F)) is bounded [75, 73]. When M 1, it holds if the problem has at least one optimal solution. If Ω2 ((((()))) is bounded, Assumption B also holds, such as when Fg is strongly convex for some & E [M].
Theorem 6. Suppose Assumptions A and B hold. Let (0(0 be the sequence generated by Algorithm I for solving (Fair GLasso). Then,
{ -x())
CR
sup min @EM
21
Fair Covariance Graph (Fair CovGraph). For the Fair CovGraph, we assume Xi:...,, X are i.i.d. samples from (0, E). We use sparse estimator for the covariance matrix, ensuring remains positive definite and specifies the marginal independence graph. Following [62], we define Co,E,, X) == / Σ S log det(E) with 0. Building on this and using (7) and (8), for some nonegative constants and X, we introduce the Fair CovGraph optimization problem, formulated as follows:
the estimator's 's loss function
minimize Σ
F(E = (E), E..,,.... FM)
subj. to M ( E > =-
(Fair CovGraph)
Here, following (8), we have f1E)= = Lc and =x A (Σ) for 2 K Also, Fi (E) = fi (E) + X I and Fj = fx) + X||>ا|| 1c for 2 k M = K + 1.
The parameter yo is used to convexify (Fair CovGraph) and is crucial for ensuring the convergence of Algorithm 1. The following theorem establishes the convergence of Algorithm I for (Fair CovGraph).
5

---

Theorem 7. Under conditions similar to Theorem 6, by replacing O with E and La (0:X) with Lc(E,X), the sequence {Σ(t) 1121 generated by Algorithm I applied (Fair CovGraph), and forne 2 max (0, -Amin (V2 Ju. }]}} for all l E [K], we have:
{ F's (t))
R
sup min ΣEM KE[M]
PR (2)
21
Fair Binary Ising Network (Fair BinNet). In this section, focus on the binary Ising Markov random field as described by [58]. The model considers binary-valued, i.i.d. samples with probability density function defined in (1). Following [31], we consider the following loss function:
P
Li (0: = -
Σ j=1
8 jo
X X
log (1 cxp (0:s + Egygaay) (10)
8'
Given some nonegative constants 7/1 and X, the Fair BinNet objective is defined as:
minimize +
F (()) [Fi (6) 1/2 ( FM (6)
subj. to E (O: - =
(Fair BinNet)
Here, following (8), we have fi( L X), and fe) A for < k Also, Fi (+) = fi (6) + I|| and Ps (0) - (e) +||O||1 + + for 2 < k < M = K +
The parameter convexifies Problem (Fair BinNet) and ensures Algorithm 1 converges. The following theorem establishes the convergence of Algorithm I for Problem (Fair BinNet).
for all k [K]. Then, the sequence ( 1221 generated by Algorithm 1 for (Fair BinNet) satisfies
Theorem 8. Suppose Assumptions A and B hold, and that " 2 max -A
{Fx (0(1) OEM RE[M]
Fu (0)) -Fx 21
ER
sup
min
Remark 9 (Iteration Complexity of Algorithm 1). Theorems 6, 7, and 8 establish the global convergence rates of (1/ Algorithm 1 for Gaussian, Covariance, and Ising models, respectively. In contrast Theorem 6, Theorems 7 and necessitate the inclusion of an additional convex regularization term with parameters ":c and 7/1 respectively, to achieve Pareto optimality.
Remark 10 (Computational Complexity of Algorithm 1). Given the iteration complexity to achieve c-accuracy O(c the overall time complexity of our optimization procedure becomes O (E- max(N N pa, P²)). Assuming a small number of groups (K << N, P, 1/e), the complexity aligns with that of standard proximal gradient methods used for covariance and inverse covariance estimation. making it feasible for large N and P. To further support the theoretical analysis, sensi- tivity analysis experiments are conducted to investigate the impact of varying N, K, and group imbalance on the performance of the proposed methods. Note that the complexity of Algorithm applied to (Fair BinNet) depends on the choice of subproblem solver first or second order) due to the nonlinearity of (10), Further experiments and discussions are detailed in Appendices D.6-D.9.
Experiment
4.1 Experimental Setup
Mhrinkage-Thresholddin Thresholding Algorithm (ISTA) is widely used for sparse inverse covariance estimation [60] due its simplicity and efficiency. We adapt ISTA for the Covariance and Ising models and use them as baseline to compare with our proposed Fair GMs. Note that our Fair GMs reduce to ISTA for Gaussian, Covariance, and Ising models if M = 1 in (8). The detailed ISTA algorithm used in this study is provided in Appendix D for reference.
Baseline.
The Iterative Shrinkage-Tie
Parameters and Architecture. The initial iterate @ (0) is chosen based the highest graph disparity error among local graphs. This initialization can improve fairness by minimizing larger disparity errors. The E -norm coefficient is fixed for each dataset, searched over a grid in {le 5,..., 0.01,-. 0.1, 1}. Tolerance is set to le - 5, with maximum of le iterations. The initial value of f is 1e 2, undergoing a line search at each iteration & with decay rate of 0.1.

---

Table 1: Numerical outcomes in terms of PCEE. The last row calculates the difference in PCEE between the two groups: the smaller, the better, and the best value is in bold.
Group
Std. GLasso
Fair GLasso
Std. CovGraph
Fair CovGraph
Std. BinNet
Fair BinNet
0.7491
0.7538
0.8537
0.8750
0.4138
0.9540
2
0.8479
0.8108
0.9502
0.9357
0.8974
0.8974
Difference
0.0987
0.0569
0.0965
0.0607
0.4836
0.0566
GLasso O
(b) GLasso O₂
(c) CovGraph E, (d) CovGraph E²
BinNet G
(f) BinNet O
(g) GLasso
(h) Fair GLasso
(i) CovGraph (i) Fair CovGraph
(k) BinNet
(l) Fair BinNet
Figure 2: Comparison of original graphs utilized in synthetic data creation for two groups, graph
reconstruction using standard GMs, and fair graph reconstruction via Fair GMs. The diagonal elements are set to zero to enhance the visibility the off-diagonal pattern.
Evaluation Criteria. In our experiments, we introduce three metrics to evaluate the performance of our methods and the baseline methods:
Value of the objective function of GM: Fi := ((O:X) e/ = Σ k=1
Summation of pairwise graph disparity error for fairness:
Ak-
3. Proportion of correctly estimated edges:
PCEE
Σ 110, À and 10 XH)/(
Σ
111 2 AH),
jd'E(P)
where 1(-} is the indicator function, and é are the groundtruth and estimated graph.
4.2 Simulation Study of Fair GLasso and CovGraph
In the simulation, we construct (Figures 2c and 2d). These matrices correspond to two sensitive groups and are created following the rigorous process in Appendix D.2. Each graph has three consistent diagonal blocks, with Group also featuring two distinct blocks indicating bias. For each group, we derive the ground truth graphs by θ, = Σ and O Σn (Figures 2a and 2b). Datasets are generated from normal distributions: 1000 samples from (O En for the first group, and 1000 samples from A( Σg) for the second.
100 100 block diagonal covariance matrices, E; and Σa
Results. Figure 2g shows the global graph derived using Standard GLasso on the entire dataset, where the two top-left blocks are not distinctly marked, suggesting bias towards O In contrast, Figure 2h shows a graph from our method that enhances block visibility, reducing bias. This improvement is supported by the results in Table 1, where the PCEE difference of Fair GLasso is smaller than that of Standard GLasso. Comparable efficacy in bias reduction for CovGraph is shown in Figures 2c, 2d, 2i, 2j. and Table 1, demonstrating our methods effectiveness in achieving fairness.
4.3 Simulation Study of Fair BinNet
We provided two simulation networks: θ for Group 1 with P 50 nodes and three hub nodes, and 0 for Group with one hub node (see Appendix D.3 for details). Adjacency matrices are shown in Figures 2e and 2f. We generate N₂ 500 and N₂ = 1000 observations via Gibbs sampling, updating each variable 2; (I+1) at iteration t + using the Bernoulli distribution: 2;
((+1)
7

---

(ernoulli(za/(1 =0)), where zo exp(03j + E The first 10,000 iterations designated as the burn-in period to ensure statistical independence among observations. Finally, observations are collected at every 100th iteration.
Results. Figure 2k illustrates the global graph from Standard BinNet, which is predominantly biased towards 2 by identifying only one hub node. In contrast, Figure 21, derived from Fair BinNet, presents a more balanced structure. While this improvement might not be visually evident, the quantitative results in Table and Table 2 confirm it. Table I reveals that PCEE for Group 1 improved significantly, increasing from 0.4444 0.7485. Conversely, PCEE for Group 2 exhibited a decrease from 0.9481 to 0.7662. This convergence in performance metrics between the two groups indicates a more balanced distribution of predictive errors, thus enhancing the overall fairness of the model.
4.4 Application of Fair GLasso to Gene Regulatory Network
We apply Gasss to analyze RNA-Seq data from TCGA, focusing on lung adenocarcinoma. The data includes expression levels of 60,660 genes from 539 patients. From these, 147 KEGG pathway genes [36] are selected to construct gene regulatory network. GLasso reveals conditional dependen- cies, aiding in understanding cancer genetics and identifying therapeutic targets. However, initially, this method, without accounting for sex-basee differences, risks overlooking critical biological disparities, potentially skewing drug discovery and health outcomes across genders. Therefore, we divide the patient cohort into two groups based on sex: 248 males and 291 females. This stratification enables the use of Fair GLasso, which creates more equitable gene regulatory network by accounting for these differences. The parameter À set to 0.03 for this experiment, Additionally, each variable is normalized to achieve a zero mean and a unit variance.
The gene networks identified by GLasso and Fair GLasso are presented in Figures 3a-3b. GLasso identified several hub nodes, including NCOAI, BRCAI, FGF8, AKTI, NOTCH4, and CSNK1AIL. In contrast, Fair GLasso uniquely detected PIK3CD, suggesting its potential relevance in capturing sex-specific differences in lung adenocarcinoma. Although direct evidence linking PIK3CD exclusively to sex-specific traits in cancer is limited, this finding aligns with recent insights into sex-specific regulatory mechanisms in cancer [63, 45]. PIK3CD is a key component of the PI3K/Akt signaling pathway, which is involved in cell regulation and frequently implicated in various malignancies. The identification of PIK3CD by Fair GLasso demonstrates its potential to uncover biologically relevant genes that may be overlooked in conventional analyses, enhancing our understanding of lung adenocarcinoma and facilitating the development of personalized therapies.
Results.
4.5 Application of Fair GLasso to Amyloid/ / Tau Accumulation Network
The performance of GLasso and Fair GLasso is evaluated using AV45 and AV1451 PET imaging data from the Alzheimer's s Disease Neuroimaging Initiative (ADNI) [85, 86), focusing on amyloid-B and tau deposition in the brain. The dataset includes standardized uptake value ratios of AV45 and AV1451 tracers in 68 brain regions, as defined by the Desikan- Killiany atlas [14], collected from 1,143 participants. An amyloid (or tau) accumulation network [68] is constructed to investigate the pattern of amyloid (or tau) accumulation. GL and Fair GLasso are used to uncover conditional dependencies between brain regions, providing insights into Alzheimer's 's disease progression and identifying potential biomarkers for early diagnosis and treatment response monitoring. To examine the influence of sensitive attributes on the network structure, marital status, and race are incorporated as exemplary sensitive attributes due to their reported association with dementia risk [71, 49]. Comprehensive details regarding the experiments, results, and analysis are provided in Appendix 4.5.
4.6 Application of Fair CovGraph to Credit Datasets
The performance of Fair CovGraph is evaluated using the Credit Datasets [90] from the UCI Machine Learning Repository [2]. These datasets have been previously used in research on Fair PCA [55, 83). which shows potential for improvement through sparse covariance estimation. The dataset composition is detailed in Table 5 in Appendix D.5, with categorizations based on gender, marital status, and education level. For this experiment, the parameters and À are set to 0.01 and 0, respectively. Each variable in the dataset is standardized to have a mean of zero and a variance of one. As shown in Table 2, our Fair CovGraph achieves a 53.75 33.75% increase in fairess with only a 0.42% decrease in the graph objective, demonstrating the strong ability of our method to attain faimess.
8

---

Table 2: Outcomes in terms of the value of the objective function (Fi), the summation of the pairwise graph disparity error (A), and the average computation time seconds (# standard deviation) from 10 repeated experiments. "L" means the smaller, the better, and the best value is in bold. Note that both Fi and A are deterministic.
F1 4
A
Runtime
Dataset
%Fi
%A
GM
Fair GM
GM
Fair GM
GM
Fair GM
Simulation (GLasso)
97.172
97.443
-0.28%
7.8149
0.6237
+92.02%
0.395
0.24)
32.32 (=: 1.5)
Simulation (CovGraph)
14.319
14.484
-1.15%
5.2627
0.3889
+92.61%
0.254 (+: 0.05)
12.58 (= 0.3)
Simulation (BinNet)
34,36.3
34.362
-0.00%
1x10-6
0.0000
+100.0%
0.536 (+ 0.15)
(+.488)))
TCGA Datasel
127.96
128.11
-0.11%
2.5875
0.0742
+97.13%
8.468
1.17)
63.72 (1: 5.9)
Credit Dataset
9.2719
9,3110
-0.42%
0,5436
0.2513
+53.76%
0.256
(+: 0,08)
64,20 (= 1.8)
LFM-10 Datasct
87.531
87.138
+0.45%
0.0040
0.0001
+96.60%
0.669
0.19)
41.19 (= 3.5)
Govnn Mare
Brann Maak
Draka
The Weeknel
As Daabe
Wesked
Cole
Tapker Beift
CSHELAL
Shoen Meniees
Shawn
Rae Sronmord
(a) Standard GLasso
(b) Fair GLasso
(c) Standard BinNet
(d) Fair BinNet
Figure 3: (a)-(b) Comparison of graphs generated by standard GLasso and Fair GLasso on TCGA Dataset. Week edges are removed for visibility, and hub nodes that own at least 4 edges are highlighted. (c)-(d) Comparison of sub-graphs generated by standard BinNet and Fair BinNet on LFM-Ib Dataset. Fair BinNet provides a more diversified recommendation network.
4.7 Application of Fair BinNet to Music Recommendation Systems
LFM-1b Dataset* contains over one billion listening events intended for use in recommendation systems [66]. In this experiment, we use the user-artist play counts dataset to construct a recommen- dation network of artists. Our analysis focuses on 80 artists intersecting the 2016 Billboard Artist 100 and 1,807 randomly selected users who listened to least 400 songs. We transform the play counts into binary datasets for BinNet models, setting play counts above 0 to and all others to 0.
This experiment examines male and female categories, stratifying the dataset into two groups with 1,341 and 466 samples, respectively. We set the BinNet models' parameter, X, to le 5.
Results. Figures 3c-3d show the recommendation networks of the 2016 Billboard Top 10 popular music artists based on Bin.ett's and Fair BinNet's 's outputs. The comparative analysis reveals that Fair BinNet provides more diversified recommendation network, particularly for the artist The Weeknd. Enhancing fairness fosters cross- goos-g musical preference exchange, breaks the echo chamber effect, and broadens users' exposure to potentially intriguing music, enhancing the user- suer-friendliness of the music recommendation system.
4.8 Trade-off Analysis
In fairness studies, the trade-off between fairness and model accuracy presents a fundamental challenge. An effective fair method should achieve equitable outcomes while maintaining strong accuracy performance. We evaluate this balance by analyzing the percentage changes in both accuracy and fairness metrics. Specifically, we define these changes as: %F. = Fi of Fair GM -F; of GM x Fair GM
Pi GM
100%, and %A
x 100%.
A oM
Our empirical results (Tables 2, 4, and Figure 4) demonstrate that Fair GMs substantially reduce disparity error, thereby improving faimess, while incurring only minimal degradation in the objective function's 's value. This favorable trade-off validates the effectiveness of our approach. However, Fair GMs do face computational challenges, primarily stemming from two sources: local graph computation and multi-objective optimization.
Available http://www.. jk. http:pppw......tttsseetttteeeeee at/tataasets/LPP-111/

---

92.02
92.61
100.00
97.13
96,60
90
61.01
64.08
% Change
60
53:76
30
0.45
0.00
-0.11
-0.0
-0.28
-0.52
-0.65
-0.42
-1
%6
-1.15
%FF
GLasso
CovGra
BinNet
TCGA
AV45
AV1451
Credit LFM-1b
Simulation Study
Real-World Application
Figure 4: Percentage change from GMs to Fair GMs (results from Tables and 4). %Fi is slight, while A changes are substantial, signifying fairness improvement without serious accuracy sacrifice.
To address these limitations. we propose several solutions. The local graph learning phase can be accelerated using advanced graphical model algorithms such as QUIC [32], SQUIC [7], PISTA [69], GISTA [60], OBN [57]. or ALM [67]. Moreover, to mitigate the increased complexity from multiple objectives, we introduce stochastic objective selection strategy, randomly sampling subset of objec- tives in each iteration. This approach effectively reduces computational overhead while maintaining model fairness and performance. To validate these computational considerations, we conducted additional experiments using GLasso, with detailed results presented in the Appendix D.10.
Conclusion
In this paper, we tackle fairness in graphical models (GMs) such as Gaussian, Covariance, and Ising models, which interpret complex relationships in high-dimensional data. Standard GMs exhibit bias with sensitive group data. We propose framework incorporating a pairwise graph disparity error term and custom loss function into nonsmooth multi-objective optimization. This approach enhances fairness without compromising performance, validated by experiments on synthetic and real-world datasets. However, it increases computational complexity and may be sensitive to the choice of loss function and balancing multiple objectives. Future research include:
F1. Integrating our Fair GMs approach with supervised methods for downstream tasks, including spectral clustering [82]. graph regularized dimension reduction [76].
F2. Developing novel group fairness notions based on sensitive attributes within our nonsmooth
multi-objective optimization framework.
F3. Extending fairness to ordinal data models, which are crucial for socioeconomic and health- related applications [27]. neighborhood selection [50], and partial correlation estimation [40].
Despite some limitations of Fair GMs for larger group sizes, this work demonstrates the potential of nonsmooth objectve optimization as powerful tool for mitigating biases and promoting fairness in high-dimensional graph- based machine learning, contributing to the development of more equitable and responsible Al systems across a wide range of domains.
Acknowledgements
This work was supported in part by the NIH grants UO1 AG066833, UOI AG068057, U19 AG074879, RF1 AG068 191, RFI AG063481. RO1 LM013463, P30 AG073105. U01 CA274576, RF1-AG063481, R01-AG071 174, and U01-CA274576. The ADNI data obtained from the Alzheimer's 's Disease Neuroimaging Initiative database (https: /adnii.i. loni. usc edu), funded by NIH UO1 AG024904. The authors thank Laura Balzano and Alfred O. Hero for their helpful suggestions and discussions.
References
Bailey Andrew, David Westhead, and Luisa Cutillo. antglasso: An efficient graphical lasso algorithm. In NewrlPS 2022 Workshop: New Frontiers in Graph Learning, 2022.
[2] Arthur Asuncion and David Newman. Uci machine learning repository, 2007.
10

---

[3] Onureena Banerjee, Laurent RI Ghaoui, Alexandre d' Aspremont, and Georges Natsoulis, Convex opti- mization techniques for fitting sparse gaussian graphical models. In Proceedings the 23rd international conference on Machine learning, pages 89-96, 2006.
[4] Solon Barocas, Moritz Hardt, and Arvind Narayanan. Fairness and Machine Learning. fairmlbook.org. 2019. http://www. fairmlabbbooo book. org.
[5] Dimitri P Bertsekas, W Hager, and O Mangasarian. Nonlinear programming, athena scientific belmont. Massachusets, USA, 1999.
[6] Foucaud du Boisgueheneuc, Richard Levy, Emmannelle Volle, Magali Seassau, Hughes Duffau, Serge Kinkingnchun, Yves Samson. Sandy Zhang. and Bruno Dubois. Functions of the left superior frontal gyrus in humans: a lesion study. Brain, 129(12))3315-332 2006.
[7] Matthias Bollhofer, Aryan Eftekhari, Simon Scheidegger, and Olaf Schenk. Large-scale sparse inverse covariance matrix estimation. SIAM Journal on Scientific Computing. 41(1):A380 A401, 2019.
[8] Sabri Boutemedjet and Djemel Ziou. A graphical model for context-aware visual content recommendation. IEEE Transactions on Multimedia, 10( 10(1):52-62, 2007.
[9] Simon Caton and Christian Haas. Fairness in machine learning: A survey, arXiv preprint 00110000000002200000000.... 2020.
[10] Flavio Chierichetti, Ravi Kumar, Silvio Lattanzi, and Sergei Vassilvitskii, Fair clustering through fairlets. 2017.
In Advances in Neural Information Processing Systems, pages
[11] Alexandra Chouldechova and Aaron Roth. The frontiers of fairness in machine learning. arXiv preprint arXiv:180.008100 2018.
[12] Rachel Cummings, Varun Gupta, Dhamma Kimpara, and Jamie Morgenstem. On the compatibility of privacy and fairness. In Adjunct publication of the 27th conference on user modeling, udaptation and personalization, pages 309-315, 2019.
[13] Patrick Danaher, 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):373-397, 2014.
[14] Rahul S Desikan, Florent Ségonne, Bruce Fischl, Brian T Quinn, Bradford C Dickerson, Deborah Blacker, Randy L. Buckner, Anders M Dale, R Paul Maguire, Bradley T Hyman, et al. An automated labeling system for subdividing the human cerebral cortex on mri scans into gyral based regions of interest. Neuroimage. 31(3):968 9009968880000000 2006.
[15] Michele Donini, Luca Oncto, Shai Ben-David, John Shawe- Shawe-Taylor, and Massimiliano Pontil. Empirical risk minimization under fairness constraints. arXiv preprint arXiv: 1ew:.080200066 2018.
[16] Noureddine EL KAROUL Operator norm consistent estimation of large- dingesiimenio sparse covariance matrices. Annals of statistics, 36(6):2717-2756 2008
[17] Alireza Farasat, Alexander Nikolaev, Sargur N Srihari, and Rachael Hageman Blair. Probabilistic graphical models in modern social network analysis. Social Network Analysis and Mining, 5:1-18, 2015.
[18] Yasmeen Faroqi-Shah, Therese Kling, Jeffrey Solomon, Siyuan Liu, Grace Park, and Allen Braun. Lesion analysis of language production deficits in aphasia. Aphasiology, 28)3):258-277, 2014,
[19] Salar Fattahi and Somaych Sojoudi. Graphical lasso and thresholding: Equivalence and closed-form solutions. Journal of machine learning research, 20(10): 2019.
[20] Jörg Fliege and Benar Fux Svaiter, Steepest descent methods for multicriteria optimization. Mathematical methods of operations research, 51:479-494, 2000.
[21] Laura Fratiglioni, Stephanie Paillard-Borg and Bengt Winblad. An active and socially integrated lifestyle in late life might protect against dementia. The Lancet Neurology, 3(6):333 353, 2004.
[22] Jerome Friedman, Trevor Hastic, and Robert Tibshirani. Sparse inverse covariance estimation with the graphical lasso. Bioxtatistiex, 93):432-441, 2008
[23] Lingrui Gan, Xinming Yang, Naveen N Nariestty, and Feng Liang. Bayesian joint estimation of multiple graphical models. Advances in Neural Information Processing Systems, 32, 2019.
[24] Mireille El Gheche and Pascal Frossard. Multilayer clustered graph learning. arXiv preprint arXiv:2010.5555,, 2020.

---

[25] Jian Guo, Elizaveta Levina, George Michailidis, and Zhu. Joint structure estimation for categorical markov networks. Unpublished manuscript, 3(5.2)66, 2010.
[26] Jian Guo, Elizaveta Levina, George Michailidis, and Ji Zhu. Joint estimation of multiple graphical models. Biometrika, 98(1):1-15, 2011.
[27] Jian Guo, Elizaveta Levina, George Michailidis, and Ji Zhu. Graphical models for ordinal data, Journal of Computational and Graphical Statistics, 24(1):183 204, 2015.
[28] Nika Haghtalab, Michael Jordan, and Frie Zhao. A unifying perspective on multi-calibration: Game dynamics for multi-objective learning, Advances in Neural Information Processing Systems, 36, 2024.
[29] Moritz Hardt, Eric Price, and Nathan Srebro. Equality of opportunity in supervised learning. arXiv preprint 1rX::1161.004113, 2016.
[30] Douglas M Hawkins. The problem of overfitting Journal of chemical information and computer sciences. 44(1):1-12, 12, 2004.
[31] Holger Höffing and Robert Tibshirani. Estimation of sparse binary pairwise markov networks using pacudo-likclioodds Journal of Machine Learning Research, 10(4), 2009.
[32] Cho-Jui Hsich, Mátýás A Sustik, Inderjit S Dhillon, Pradeep Ravikumar, et al. Quic: quadratic approxima- tion for sparse inverse covariance estimation. J. Mach. Learn, Res., 15(1):291 192947, 2014.
[33] Michael Isard. Pampas: Real-valued graphical models for computer vision. In 2003 IEEE Computer Society Conference on Computer Vision and Puttern Recognition, 2003. Proceedings, volume 1, pages I-I. JEEE, 2003.
[34] Michael Jordan. Graphical models. Statistical Science, pages 140-155, 2004.
[35] Mohammad Mahdi Kamani, Farzin Haddadpour, Rana Forsati, and Mehrdad Mahdavi. Efficient fair principal component analysis. Machine Learning. pages 1-32, 2022.
[36] Minoru Kanchisa and Susumu Goto. Kegg: kyoto encyclopedia of genes and genomes. Nucleic acids research, 28(1)277-30, 2000.
[37] Noureddine Karoui. Operator norm consistent estimation of large- dineesiiennional sparse covariance matrices. The Annals of Statistics, pages 2717-2756, 2008.
[38] Philipp Kellmeyer, Wolfram Ziegler, Claudia Peschke, Eisenberger Juliane, Susanne Schnell, Annette Baumgaertner, Cornelius Weiller, and Dorothee Saur. Pronto-parieta. dorsal and ventral pathways in the context of different linguistic manipulations. Brain and language, 127(2):241 2013.
[39] Aria Khademi, Sanghack Lee, David Foley, and Vasant Honavar. Faimess in algorithmic decision making: An excursion through the lens of causality. In The World Wide Web Conference, pages 2907-2914, 2019.
[40] Kshitij Khare. Sang- Sang-Yun Oh, and Bala Rajaratnam. A convex pseudolikelihood framework for high dimensional partial correlation estimation with convergence guarantees. Journal of the Royal Statistical Society: Series B: Statistical Methoddoogy, pages 803 825, 2015.
[41] Matthäus Kleindessner, Samira Samadi, Pranjal Awasthi, and Jamie Morgenstem. Guarantees for spectral clustering with fairness constraints. In International Conference on Machine Learning, pages 3458-3467 PMLR, 2019.
[42] Sandeep Kumar, Jiaxi Ying, José Vinícius de Miranda Cardoso, and Daniel P Palomar. A unified framework for structured graph learning via spectral constraints. Journal of Machine Learning Research, 21((2)) 2020.
[43] Su-In Lee, Varun Ganapathi, and Daphne Koller. Efficient structure learning of markov networks using [1 1-regularization. Advances in neurol Information processing systems, 19, 2006.
[44] Fredrik Liljeros, Christofer R Edling, Luis A Nunes Amaral, H Eugene Stanlcy, and Yvonne Åberg. The web of human sexual contacts. Nature, 41 1(6840))907-990 2001.
[45] Camila M Lopes-Ramos, John Quackenbush, and Dawn I. DeMeo. Genome-wide sex and gender differences in cancer. Frontiers in oncology, 10:597788, 2020.
[46] Val J Lowe, Geolfry Curran, Ping Fang, Amanda M Liesinger, Keith A Josephs. Joseph E Parisi, Kejal Kantarci, Bradley F Boeve, Mukesh K Pandey, Tyler Bruinsma, et al. An autoradiographic evaluation of av- 1451 tau pet in dementia. Acta neuropathologica communications, 4:1-19, 2016.
12

---

[47] Benjamin M Marlin and Kevin Murphy. Sparse gaussian graphical models with unknown block structure. In Proceedings of the 26th Annual International Conference Machine Learning, pages 705-712, 2009.
[48] Ninarch Mehrabi, Fred Morstatter, Nripsuta Saxena, Kristina Lerman, and Aram Galstyan. A survey on bias and fairness in machine learning. ACM computing surveys (CSUR), 54(6):1-35, 2021. 202...
[49] Kala M Mchta and Gwen W Yeo. Systematic review of dementia prevalence and incidence in united states race/ethnic populations. Alzheimer's 's & Dementia, 13(1):72- 83, 2017.
[50] Nicolai Meinshausen, Peter Bühlmann, al. High-dimensional graphs and variable selection with the lasso. Annals of statistics, 34(3):1336-1462, 2006.
[51] Karthik Mohan, Mike Chung. Scungycop Han, Daniela Witten, Su-In Lee, and Maryam Fazel. Structured learning of gaussian graphical models. Advances in neural information processing systems, 25, 2012.
[52] Madeline Navarro, Samuel Rey, Andrei Buciulea, Antonio G Marques, and Santiago Segarra. Fair glasso: Estimating fair graphical models with unbiased statistical behavior. arXiv preprint arXiv:2406.09513, 2024.
[53] Madeline Navarro, Samuel Rey, Andrei Buciulea, Antonio G Marques, and Santiago Segarra. Mitigating subpopulation bias for fair network topology inference. arXiv preprint arXiv:2403. 155911 2024.
[54] Hwamee Oh, Jason Steffener, Qolamreza R Razlighi, Christian Haheck, and Yaakov Stem. f-amyloid deposition is associated with decreased right prefrontal activation during task switching among cognitively normal elderly. Journal of Neuroscience, 36(6):1962-199) 2016.
[55] Matt Olfat and Anil Aswani. Convex formulations for fair principal component analysis. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 33, pages 663-670 2019.
[56] Luca Oneto and Silvia Chiappa. Fairness in machine learning. In Recent Trends in Learning From Data, pages 155-196. Springer, 2020.
[57] Figen Oztoprak, Jorge Nocedal, Steven Rennie, and Peder A Olsen, Newton-like methods for sparse inverse covariance estimation. Advances in neural information processing systems, 25, 2012.
[58] Pradeep Ravikumar, Martin J Wainwright, John D Lafferty, et al. High-dimensional ising model selection using & regularizedd logistic regression. The Annals of Statistics, 38(3): 1319, 2010.
[59] Garry Robins, Pip Pattison, Yuval Kalish, and Dean Lusher. An introduction to exponential random graph ps models for social networks. Social networks, 29(2):173 1917 2007.
[60] Benjamin Rolfs, Bala Rajaratnam, Dominique Guillot, Ian Wong, and Arian Maleki. Iterative thresholding algorithm for sparse inverse covariance estimation. Advances in Neural Information Processing Systems, 25, 2012.
[61] David L Roth, Mary S Mittelman, Olivio J Clay, Alok Madan, and William E Haley. Changes in social support as mediators of the impact of a psychosocial intervention for spouse caregivers of persons with alzheimer's 's disease. Psychology and aging, 20(4):634, 2005.
[62] Adam Rothman. Positive definite estimators of large covariance matrices. Biometrika, 99(3):333-740 2012.
[63] Joshua B Rubin, Joseph S Lagas, Lauren Broestl, Jasmin Sponagel, Nathan Rockwell, Gina Rhee, Sarah F Rosen, Si Chen, Robyn S Klein, Princess Imoukhuede, et al. Sex differences in cancer mechanisms. Biology of sex Differences, 11:1-29, 2020.
[64] Peter H Rudebeck and Erin L Rich. Orbitofrontal cortex. Current Biology, 28( (8(18) 18)1888830033 R1088, 2018.
[65] Samira Samadi, Uthaipon Tantipongpipat, Jamie Morgenstem, Mohit Singh, and Santosh Vempala. The price of fair pea: One extra dimension, arXiv preprint arXiv:1811.00103, 2018
[66] Markus Schedl. The Iffm 1b dataset for music retrieval and recommendation. In Proceedings of the 2016 ACM on international conference on multinedia retrieval, pages 103-110, 2016.
[67] Katya Scheinherg, Shiqian Ma, and Donald Goldfarb. Sparse inverse covariance selection via alternating linearization methods. Advances in neural information processing systems, 23, 2010.
[68] Jorge Sepulcre, Mert R Sabuncu, Alex Becker, Reisa Sperling, and Keith Johnson. In vivo characteriza- tion the early states of the amyloid- beta network. Brain, 136(7):2239-22)) 2252.22222222222 2013.
[69] Gal Shalom, Eran Treister, and Irad Yavnch. pista: Preconditioned iterative soft thresholding algorithm for graphical lasso. SIAM Journal Scientific Computing, 46(2):5445-$444 2024.
13

---

[70] Changjian Shui, Gezheng Xu, Qi Chen, Jiaqi Li, Charles X Ling, Tal Arbel, Boyu Wang, and Christian Gagné. On learning fairesss and accuracy on multiple subgroups. Advances in Neural Information Processing Syxtems, 35:34121-34135, 2022.
[71] Andrew Sommerlad, Joshua Ruegger, Archana Singh-Manoux Glyn Lewis, and Gill Livingston. Marriage and risk of dementia: systematic review and meta-analysis of observational studies. Journal of Neurology, Neurosurgery & Psychiatry, (9(3):231-236 2018.
[72] Kean Ming Tan. Palma London, Karthik Mohan, Su-ln Lee, Maryam Fazel, and Daniela Witten. Learning graphical models with hubs. Journal of Machine Learning Research, 15:3297-3331, 2014.
[73] Hiroki Tanabe, Ellen H Fukuda, and Nobuo Yamashita. Proximal gradient methods for multiobjective optimization and their applications. Computational Optimization and Applications, 72:339-361, 2019.
[74] Hiroki Tanabe, Ellen H Fukuda, and Nobuo Yamashita. A globally convergent fast iterative shrinkage- thresholding algorithm with a new momentum factor for single and multi-objectivv convex optimization. arXiv preprint 2iii:2200.05222 2022.
[75] Hiroki Tanabe, Ellen H Fukuda, and Nobuo Yamashita. Convergence rates analysis of a multiobjective proximal gradient method. Optimization Letters, 17(2):333-355 2023.
[76] Mengfan Tang, Feiping Nie, and Ramesh Jain, A graph regularized dimension reduction method for out- of out-of-sample data. Neurocomputing, 225:58 63, 2017.
[77] Uthaipon Tantipongpipat, Samira Samadi. Mohit Singh, Jamie H Morgenstern, and Santosh Vempala. Multi-criteria dimensionality reduction with applications to faimess. Advances in neural information processing systems, 32, 2019.
[78] Davoud Ataec Tarzanagh, Laura Balzano, and Alfred O Hero. Fair community detection and structure learning in heterogencous graphical models. arXiv preprint arXiv:2 2021.
[79] Davoud Atace Tarzanagh, Bojian Hou, Boning Tong, Qi Long, and Li Shen. Faimess--wwer aware class imbalanced learning on multiple subgroups. In Uncertainty in Artificial Intelligence, pages 2123-2133. PMLR, 2023.
[80] Davoud Atace Tarzanagh and George Michailidis. Estimation of graphical models through structured norm minimization. Journal of Machine Learning Research, 18(209) 2018
[81] Francesco Tomaiuolo, JD MacDonald, Zografos Caramanos. Glean Posner, Mary Chiavaras, Alan C Evans, and Michael Petrides. Morphology, morphometry and probability mapping of the pars opercularis of the inferior frontal gyrus: an in vivo mri analysis. European Journal of Neuroscience, 11(9):30333000
1999.
[82] Ulrike Von Luxburg. A tutorial on spectral clustering. Statistics and computing, 17:395 17:395-416, 2007.
[83] Hicu Vu, Toan Tran, Man Man-Chung Yuc, and Viet Anh Nguyen. Distributionally robust fair principal components via geodesic descents. arXiv preprint wxXi:2202.003711 2022.
[84] Xiwen Wang, Jiaxi Ying, and Daniel Palomar. Learning large-scale mtp 2 gaussian graphical models via bridge-block decomposition. Advances in Neural Information Processing Systems, 36:7321 1-73231, 2023.
[85] Michael W Weiner, Dallas P Veitch, Paul S Aisen, Laurel A Beckett, Nigel J Cairns, Robert C Green, Danielle Harvey, Clifford R Jack, William Jagust, Enchi Liu, et al. The alzheimer's 's disease neuroimaging initiative: a review papers published since its inception. Alzheimer's & Dementia, 95) 1-94944 2013.
[86] Michael W Weiner. Dallas Veitch, Paul S Aisen, Laurel A Beckett, Nigel Cairns, Robert C Green, Danielle Harvey, Clifford Jack Jr. William Jagust, John Morris, al. Recent publications from the alzheimer's 's discase neuroimaging initiative: Reviewing progress toward improved ad clinical trials. Alzheimer's 's & Dementia, 13(4)=e1-855 2017.
[87] Daniela M Witten, Jerome H Friedman, and Noah Simon. New insights and faster computations for the graphical lasso. Journal of Computational and Graphical Statistics, 20(4):892-900, 2011.
[88] Dean F Wong, Paul B Rosenberg. Yun Zhou, Anil Kumar, Vanessa Raymont, Hayden T Ravert, Robert F Dannals, Ayon Nandi, James R Brašié, Weiguo Ye, et al. In vivo imaging of amyloid deposition in alzheimer disease using the radioligand 18f-av-45 (flobetapir 18). Journal of nuclear medicine, 51(6):913-92) 920. 2010.
[89] Eunho Yang and Aurélie C Lozano. Robust gaussian graphical modeling with the trimmed graphical lasso. Advances in neural information processing systems, 28, 2015.
14

---

[90] I-Cheng Yeh and Che-hui Lien. The comparisons of data mining techniques for the predictive accuracy of probability of default of credit card clients. Expert systems with applications, 36(2):2473-2488, 2009.
[91] Jianxin Yin and Hongzhe Li. A sparse conditional gaussian graphical model for analysis of genetical genomics data. The amnals of applied statistics, 5(4):2630, 2011. 1.
[92] Guanhua Zhang, Yihua Zhang, Yang Zhang, Wengi Fan, Qing Li. Sijia Liu, and Shiyu Chang. Faimness reprogramming. Advances in Neural Information Procesing Systems, 35:34347-34362, 2022.
[93] Xiang Zhang and Qiao Wang. unified framework for fair spectral clustering with effective graph learning. arXiv preprint xrXiv:231..3366 2023.
[94] Xiang Zhang and Qiao Wang. A dual laplacian framework with effective graph learning for unified fair spectral clustering. Neurocomputting, page 128210, 2024.
[95] Zhuoping Zhou, Davoud Atase Tarzanagh, Bojian Hou, Boning Tong, Jia Xu, Yanbo Feng, Long, and Li Shen. Fair canonical correlation analysis. Advances Neural Information Processing Systems, 36, 2024.
15

---

Contents
Summary of the Notations
17
Addendum to Section
18
B.1
Dual Reformulation and Computation of Subproblem (9)
18
B.2
Subproblem Solver for Fair GMs
19
B.2.1
Fair GLasso
19
B.2.2
Fair CovGraph
20
B.2.3 Fair BinNet
20
C Addendum to Section 3.3
21
C.I Auxiliary Lemmas
21
C.2 Proof of Theorem for Fair GLasso
22
C.3 Proof of Theorem 7 for Fair CovGraph
24
C.4 Proof of Theorem 8 for Fair BinNet
25
C.5 Computational Complexity of FairGMs
26
D Addendum to Section 4
27
D.I Iterative Soft-Thresholding Algorithm (ISTA)
27
D.2 Simulation Study of Fair GLasso
................................
27
D.3 Simulation Study of Fair BinNet
................................
27
D.4 Addendum to Subsection 4.5
28
D.5 Addendum to Subsection 4.6
28
D.6 Sensitivity Analysis to Feature Size P
30
D.7 Sensitivity Analysis Sample Size N
30
D.8 Sensitivity Analysis to Sample Size Ratio N₂ /
31
D.9 Sensitivity Analysis to Group Size K
32
D.10 Addendum to Subsection 4.8
33
16

---

A
Summary of the Notations
Table 3: Summary of the Notations
Notation
Description
I(-)
Indicator function
||]]
4--nor: Lis [gg]
|
Frobenius norm: (Zi; |at) 12)2/22
[
The f.. M
(A(A)
ith eigenvalue of A
A Amn (A)
The smallest eigenvalue of
Man (x)
Moreau envelope: miny [h(y) + 2n | y
prox (2)
Proximal operator: arg min, [h(y) + 1 x y|13]
A (x)
Soft thresholding operator: sign(a (z) max( [1c] 1
L
Lipschitz constant
P
Number of variables in the data matrix
N
Number of observations in the data matrix
N
Number of observations in the kth group data matrix
K
Number of sensitive groups in the data matrix
M
Number of objectives in the multi-objective optimization problem Current iteration of Algorithm
A
Jyper-paramees of the & rerrgularrzationn term
T0
Hyper-parameter of the convex regularization term in (Fair CovGraph)
Plyper-paramete of the convex regularization term in (Fair BinNet) Selected constant >>
X
Data matrix
X
Data matrix of kth group
X
Observations in the data matrix
Sample covariance matrix: n Σ X
Σ
Covariance matrix
Conditional independence graph (inverse covariance matrix) Conditional independence graph (inverse covariance matrix) Conditional independence graph of kth group Optimal conditional independence graph
O
C
General Loss function
La
Loss function of GLasso: log det(()) ( + trace (SO) Loss function of CovGraph: 1 m S S log det
Lo
Ci
Loss function of BinNet; refer to (10) Graph disparity error of kthh group Pairwise graph disparity error of kth group
E
A
Summation of pairwise graph disparity error: Σ k=1 A
Pe
max (V/x(() + θ) g (
F
Objective function of the multi mujti-objective optimization problem
F
kth objective in the multi-objective optimization problem solutions of the min-max problem for each l: arg min Pr (0; θ)
P
R
supp-eP(V"ng (F()")) infecr-:((F))
O - (0
M
Convex constraint subset PxP
Standard simplex: {p RM ΣM Ph: = Pk E 10. 1] Wk E [M]
C
17

---

B Addendum to Section 3
B.1
Dual Reformulation and Computation of Subproblem (9)
In this section, we provide a dual method for solving the Subproblem (9) defined as:
min BEM
bp ( O)
with ffr ④; ④
max ().
())) ( + 2 4 @
(I)
for all E> L where L is defined in Assumption A. For simplicity, let
_ $ 21|
b&, 2 (Vfx(O), -
(12)
By considering the standard simplex,
M Σ kml
c:= PER":
p = Pk P [0,1], A E [M]]
(13)
we reformulate (11) as
M Σ k=1
min max
Vhe ( θ).
(14)
PEM PEC
By leveraging the convexity of M, the compactness and convexity of C. and the convexity-conevvity property of Σx= PR (b; θ) with respect to and p. respectively, we can invoke Sion's 's minimax theorem to reformulate (14) as follows:
M Σ
max min PEC +EM
Purkee (0; 0).
(15)
Expanding on the definition of the arrive at the following expression:
M
min >
p (#;; θ)
PEC
k=1
M Ph
min +EM
9 (
θ
V fa (
2
k=
V fc (
Pk
(
28
1 M
(M
4
P (0) f
PEC
29
Pk
V
fo
(
(0).
(16)
2
where Moreau envelope Man (1:) and the proximal operator are defined as
Man = min h(y)
2a x 2/12
(17a)
h(x)) 1 ||00
prox (x) := arg min
y11
(17b)
2xx
y
Problem (16) is equivalent to the dual problem:
M
max
((p)
subj. to
p = 0 and Σ
Ph = 1,
(18a)
PERM
18

---

where
00
M
M
w(p) - EM
- Σ
P V
Σ
pn f (0)
(0)
(18b)
Upon solving the dual Problem (18), the optimal solution o of (11) is obtained through:
o
1 Σ
M
Φ- prox
PE V Vz (0)
(19)
k=1
In the implementation, for the given g (0) = A | prox و is computed using soft thresholding 7 X as shown below:
(x)), sign (z,) max [zy| (151- 24.0).
(20)
To provide clear and logical summary of the iterative update process in Algorithm
proceed as follows: At each iteration i, the update for ((+1) is performed by inputting ((() and solving the Subproblem (11). This is achieved by utilizing the
miinimize function with the
method=" constr" option to solve the dual problem. Specifically, for given constants > L
and À > 0, and the calculated ((0)
E C at the 7th iteration, the update rule for θ (t+1) is given by:
M
+ ((+1)
=
AtL
4 (
Σ
(1) Pk
V
(0())
(21)
which incorporates the proximal operator and the weighted sum of gradients. Through solving Subproblem (9), the following proposition characterizes the weak Pareto optimality in the context of multi-objective optimization Problem (8):
Proposition 11. Let we (+) min Pt ( )) and P; be defined as in (9). Then,
(i) The following conditions are equivalent:
(a) @ is weakly Pareto optimal point;
(b) P. ( = =9;
(c) We (9) =
(ii) The mappings Pa and we are both continuous.
Proof. The proof follows from [73, Lemma 3.2] and the convexity of fu. The detailed convexity analyses for Fair GLasso, Fair CovGraph, and Fair BinNet are provided in Sections C.2, C.3, and C.4, respectively.
As demonstrated in the analysis of Subproblem (9) in the beginning of this section. the proposition implies that the descent direction is the minimum norm matrix within the convex hull of the gradients of all objectives. Furthermore, this direction is non-increasing with respect to each individual objective function. This property ensures that the chosen descent direction simultaneously minimizes the overall norm while guaranteeing non-increasing behavior for each objective, thereby facilitating the optimization process in a multi-objective setting.
B.2
Subproblem Solver for Fair GMs
B.2.1
Fair GLasso
To update (!) Algorithm applied to (Fair GLasso), the iterative update formula in Equation (21) used at each iteration. The gradients for the functions f1 and (ff+118 M are computed follows:
The gradient of with respect θ is given by:
V fi ( = S
(22)
where S is the sample covariance matrix and -1 is the inverse of the precision matrix O.
19

---

The gradient of fk+1 for k 1,.... M with respect θ is given by:
(0++ = Σ
(traco(Sx - traco (S,e) + log det (+),
s6 K Kegtt
(23)
trace(S;
log det (e)) + trace(S, ())) (Sa S.):
where K' is the number of groups, S₁ and SN are the sample covariance matrices for groups k and respectively, and O and @ are the optimal precision matrices for groups k and obtained by solving the group-specific GLasso problems.
B.2.2 Fair CovGraph
To refine the iterative update rule for estimating the fair covariance matrix Σ in (Fair GLasso) using M-1 computed as follows:
Algorithm/ 1, the gradients for fi and the set {fk+1] [==+]k==1
The gradient of with respect E is given by:
Vf = -
(24)
where S is the pooled sample covariance matrix, is the regularization parameter, and Σ is the inverse of the covariance matrix >.
The gradient of f for L, ,...., M with respect to is given by:
1 Σ
1 | -S.ll log det(E;))
(2)
S
2
2
(25)
2 - - S log det Σラ) +
1 زاこ Selli. (S. - Ss).
where is the number of groups, S₁ and S. are the sample covariance matrices for groups and 8, respectively, Σj and Σ are the optimal covariance matrices for groups A and obtained by solving the srecp-s CovCraph problems.
B.2.3 Fair BinNet
To simplify, denote
20 = exp
0, EB
and 2s = exp
وز5 + Σ 0
J'
The gradients of the objectives of Fair BinNet that utilized in the iterative update formula (21) computed as follows:
(V)())))
OC (9,x)
--(x-X)
20
ae
13
20
(V)(()))))) = OC
(0,x)
x)
-- X) 11 +
Tift
15'
+ 20
(26)
[a+((()) Σ ((C()) L(O X+)) (C(O;X.)
sE(K),ssk
0L ( x
OL
-(0;;
00 (0, x,)
d
Here, L (O,X) fi( is the negative log-likelihood of the Ising model, where θ E PRP is the interaction matrix, X E 1 is the binary data matrix with N samples and P variables, and "rf' denotes the (j.j")-th element of O
20

---

C
Addendum to Section 3.3
C.1 Auxiliary Lemmas
Lemma 12. Let (6( be generated by Algorithm 1. Then for all k 1,..., M, have
F& 0 (0(+)))
< Fv (0(9)
(27)
Proof.
Let ups (4, θ) be defined as in (9). Following the proof of [73, Lemma 4.1J, we have
ge (e (++1) (()) _ -f|(o O
(28)
l L, using the descent lemma [5, Proposition A.24], for all k: = I,.., M, we obtain
Fi (olltt))
Fs (o(1) S(V O ((+1) - -
(29)
+ ( (
O (t -0(-1) 10
2
Since the hight-hand side of the above inequality is less than or equal to zero, it implies that
Fb O (+t)) S Fk (0(1).
(30)
L
Lemma 13. Suppose Assumption holds. Let fo and g have convexity parameters pu E R. and E R,,, respectively, and define μt. minkelM] uk. Then, for all O M, we have
M
FR (ol(tt)) - Fs. (0))
L (1010 (++1)
- -11 0
P
R=1
2
(31)
0
2 | (
011
2
where pk satisfies the following conditions:
1. There exists (1 0g
2 (t(1)) such that
Σ
Pk (0)
Vii (D + +
((((+1))0())) -0())
(0((+))) -
p(t) E EC where C defined in (13).
Proof. Assumption yields
Ph (e)(tt))= Fx (0)1) S S(VS)(((0)) ( -
())) -00)
(32)
+ ( (0w)) + 2
(t-1) -
From the convexity of fa and g, we have
Fi (eit+))
Fs (A)
Fx (ol++)) Fs (0()) + Fr (o F% (0)
(V/L ((4) ))) + (0 (0)))
( 0 11
+1
-10(40-00
+ +(1V// 6 (0) 0 + (0() (0))
(33)
<(VA. ((+1) + (+1)
26ee
(++) 0
<(1 (++1)
(--1)
e (9) 1 0 ( 2 "
0 (+1) 1
21

---

Condition and Condition 2 yield
M
Fle (o(++1)) - Fs (0)) = (((((+1) -
@ 1 (((1)) )) + ++=11000 =1@((t)))
(0)
2 2 フーー 10 ((+1)) - 011%
2
(34)
1000)
0
(++ 0
2
1/ 10(+1)
2 O
# (1)
θ |
2
C.2 Proof of Theorem for Fair GLasso
First, we present the convexity analysis and gradient Lipschitz continuity, Proposition 14 (Convexity of Fair GLasso). Each fü for k = I....,M M and defined in (Fair GLasso) of Fair GLasso convex. Further, is strongly convex.
Proof. First, consider S1 in the first objective funetion of Fair GLasso:
fi (i) log det ( + trace(S@),
(35)
where θ is positive definite matrix.
The gradient and Hessian of fi are, respectively:
(f( S θ- H,, o 9-1
(36)
The positive definiteness of @ implies that @- is also positive definite. Therefore, the Hessian Н being the Kronecker product of + with itself, is positive definite. This establishes that the objective function fi is strongly convex.
Next, the functions fk+1 for k I....,M, M - are defined as:
fk+1 (+)= = Σ
(E (0) E, (e))
sEi sE(K],ofE
(37)
1 2.
Σ
- (() -C()) x.) Xc) x(0);
$E(K],spk
which are convex due to the linearity of the trace operator in the loss function difference L(O;X,)
L(0; x,) = tracee (Sk S)) leading to a strong convexity parameter 0. In addition, g( = e||@| | is identified as a closed, proper, and convex function.
Proposition 15 (Gradient Lipschitz Continuity of Fair GLasso). The gradients of fi for k 1...., M defined in (Fair GLasso) are Lipschitz continuous.
Proof. First, present the gradient and Hessian of functions and (fi-1) -)--1) 1-M-1 follows:
f()))= = log det(() + trace(S@),
Vf( = S θ_1 H, = 0 --1
Σ
(trace(S, trace θ) + log (
*E(A]),spk
Trace - log del de(( + trace S. )
(0) = Σ
(trace(S, Q trace(S,,6) ( log det (e)
*E(K],spk
-trace(S; log (ee((O))) (rrace(S, (S: -S.),
H H (0) = Σ
(S S.) 8 (Sx - -S.).
s
(38)
22

---

Define I Amax (H ) and Tx++1 Amax (H furt) for k: = ,,..,, M - 1. Given that ) A are convex (as proven in Proposition 14) and twice differentiable, their gradients satisfy Lipschitz continuity with Lipschitz constants {=
Next, we present the proof for Theorem 6.
proof for Theorem 6. From Proposition 14 and Proposition 15, convexity and gradient Lipschitz continuity of objective functions {f are verified. Hence, Assumption A holds.
From Lemma 13 and the convexity of Si and g. for all θ M, we obtain
M
Σ
i0 (Fx P
(Fx
(B((t))
F& (0)
L - 2 (10(0) (0) O| 110 012))
(39)
Adding up the above inequality (39) from = to i, we have
i
M
Pk
Fx
( - Fn (0)
S 2
[ (0) - O - (E+1)) - 01
K=0
k=1
(40)
2 (0) 1
Lemma 12 implies that Fu (ol(t1)) S Ps (6(()2) for all / S <san and
ΣΣ
Pk
(t)
Fix (0litt))
2 (F))
Fx (0)) -F+ s
f -1000)
1 1
(41)
Let P = EiE Pk) / (f (1) Then, it follows that
M
Σ Pk Fx (o - (0)) s -ox
f
10 - -
(42)
2 2( + 1)
A
Since p 2 0 and Σ p 1, we can conclude that
min ke[M]
Fi ( (fttt)) FA (0)) S 2
| 10 (0) O
(43)
Now, following the proof of [75, Theorem 5.1] and using Assumption B, we obtain
FF (e Fs (0)
CR
sup
inf OEF ((F-) }) kE[M]
min
2 2
(44)
Fix (o)(+2)) - -Fk)
RR
sup FeEP(N-nB
min kE[M]
s 2 (1+1))
(45)
sup (F
min kE[M]
Fk ( S
F& (0))
RR
(46)
(8 + 1)
The inequality Fk (011) < Fi (0 from Lemma 12 implics that
min (c0x(((e)0)) KE.M]
sup
Fr (o Fs (0))
sup (ERP(F((i-))) kE(Ad]
min
Fe (Q
F& (8))
sup min (Fi (g -F (0)) GEM
Fi
(47)
F (0)0)) there exists O E N such that
Moreover, from Assumption B that for all O E Sxp (@") = F (0), it follows:
sup @cN
min k
Fx (o
Fx -O))
(48)
Fi (
F2 (0)
sup (EAr(F(e(m)
min k
23

---

Therefore, from (44) and (48), conclude that
sup min {Fi (olitt) Px @EM Re
PP (0)) -
iR
(49)
2 21++1)
C.3 Proof of Theorem 7 for Fair CovGraph
First, we present the convexity analysis and gradient Lipschitz continuity.
Proposition 16 (Convexity of Fair CovGraph). Through incorporating a convex regularization term
2 for some "Cc > max -Amin 2 fu (e))} into each Ju for k 2, ....., M, each fa for k b...y M and g defined in the multi-objective optimization problem (Fair CovGraph) of Fair CovGraph are guaranteed be convex. In particular, fi strongly convex.
nc O
Proof. Fair CovGraph (Fair CovGraph), the function fi, its gradient, and Hessian are defined as follows:
)(²) = - - - S| 7 log del (E), VE( =
H /, Ip - Σ
The positive definiteness the covariance matrix guarantees that the Hessian matrix H,, is also positive definite, establishing the strong convexity of the function f1. Next, consider:
L(E ((E:X.) - -11== S " -
(50)
This difference is necessarily convex, it the difference between two convex functions.
To ensure the convexity of the functions for k: = 2, M, convexity regularization term to rell@ | is added fi. denoted by fR. where "Cc chosen to yC max(0,-A (V2 (e))
such that Amin (H; 2 O for k = ...... M. This regularization ferm guarantees that the mini- mum eigenvalue of the Hessian matrix H f. is non-negative, thereby ensuring the convexity of fk- Furthermore, the function g ({E) is closed, proper, and convex function.
Proposition 17 (Gradient Lipschitz Continuity of Fair CovGraph). The gradients of fix for 1...., M defined in (Fair CovGraph) are Lipschitz continuous.
Proof. We detail the gradient and Hessian of functions fi and [f as follows:
M-1
fE) = 2 Σ S|? - 7 log det (E),
(E) = -E-S-TE-
-
H = pp + +-E-111 ② Σ-1
f++((E)= = Σ 2 2
|Σ - - -Sa11? - / - - -. + log det det(E))
sE|K],skk
- 1 > - dot Σブ) + | -s.
(51)
1 2 (113-- Sulli. ] - - t log det det(
ff++i(E) Σ
SE k
Ex
log det Σブ) + } | S.112 (S. -S))
H fa Hin1(E) (E) =
Σ (S. - Sa) s (S. Sa):
sE K].kk
Then given that and 0f1 aH are Lipschitz continuous and bounded on the set { M|| < co).
the function sequence {f++1 k=1 is also Lipschitz continuous.

---

Proof of Theorem 7.
Note that for
TO max -Aminnn
(52)
the problem (Fair CovGraph) convex. Now, from Proposition 16 and Proposition 17, convexity and gradient Lipschitz continuity of objective functions (fx] 1M are verified. Then, the proof of Theorem is a slightly modified version of the proof of Theorem 6.
C.4 Proof of Theorem for Fair BinNet
First, we present the convexity analysis and gradient Lipschitz continuity.
Proposition 18 (Convexity of Fair BinNet). In the multi-objective optimization Problem (Fair BinNet), the functions fi and g are corvex. Furthermore, by incorporating convex reg- ularization term {| for some 2 min Amiin [k), 0H into each ft for l = ..................... M, the set of functions {fx k= ensured be convex well.
Proof. The function Si for Fair BinNet is defined as:
))
P log 1 1+ exp
f1(0)
Σ
0r Σ
X
0,s 015+
Egyar 5'4
(53)
j=1j=1
To demonstrate the convexity of f1, observe that h(x) log(1+ convex and nondecreasing. Since convexity is preserved under linear combination and summation, f1 is convex by construction. Also, 9( (( is a closed, proper, and convex function.
Consider (0) = (+) + 7/ 1 for = I,..., 1, its Hessian matrix is given by:
ju+1 (0) = H H (0) + 2ysIpa.
(54)
If yu is chosen to be |min ( Amin ( fs), such that Ip= dominates any negative curvature in H H (9), then H fur (0) will be positive semidefinite, leading the convexity of (
Proposition 19 (Gradient Lipschitz Continuity of Fair BinNet). The gradients of fa. for k =1,... M defined in the multi-objective optimization Problem (Fair BinNet) are Lipschitz continuous.
Proof. For notational simplicity, we introduce the following substitutions: utilize L (0, X) in place of fi (O), denote = exp (03 + Eド Ozy ). 2Φ exp (0is + Ey T5 Then, we proceed to evaluate the gradient of the function fi in the context Fair BinNet as follows:
8 (0, x), = =(Vi(())) = -xx
dc
N
20
+ 20
(55)
( (0, x)
-XX" =(V[( -
7
13'
jj
20
Given that (x) exp(x)/( (1 exp(x)) is Lipschitz continuous with Lipschitz constant 0.25, for any θ $ E M,
[v^(e) - v/+
+
1 + Zab
j=1
Lig'
Tizt
1 +
1 1 26
J=1 j ('113' *j
P
P
P
Z8
Zb
Pijt
Zig/
1 26
1 2
+ Xg
1 + 26
11,1
25

---

P
03) 0is +
Σ
(0iz 255t) Zid"
148
P N
+ ( 1) x
8, O3s + Σ (0p3 015)) ae
jml t=1
P
P Σ
SNXPX
10,5" 1571
P
NN Pd X
0gy
$181
x P x
1
(56)
follows that there exists 4 N p2 such that (0) vA(@))lz < 4 2/1..
Subsequently, the gradients of functions {fk+ 1M11 in Fair BinNet are evaluated as:
(e) E ((C(O;X)) - ((01;
*E[K]].pA
(57)
(C(())).) -((;;x)))
0C d
(e, (e,
dc
(x.
de
Then given that and Oc 3 are Lipschitz continuous and bounded on the set (0 E M|| < co).
the function sequence {fk+1] 1
1-1 is also Lipschitz continuous.
k==
Proof of Theorem 8 for Fair BinNer. Building Proposition 18 and Proposition 19, proof of Theo- rem 8 can viewed as a nuanced adaptation of the proof presented in Theorem
C.5 Computational Complexity of FairGMs
The computational complexity of the fair GLasso and fair CovGraph algorithm depends both the number of variables P and the number of observations N. We aim to demonstrate that our algorithm
has complexity of O [max(NP2_pp)
which is similar standard graph learning methods when
K << N. P. This is applicable to our experimental results, where K = 2, 8, 2, 2,000 < N < 15, 000, and 5 < P < 120. The computational complexity is primarily influenced by the following factors:
1. Number of Variables (P):
The complexity scales as O(P3) due to matrix inversion for computing the gradient at each step.
2. Number of Observations (NV): Computing the empirical covariance matrix from the data has a complexity of P2).
3. Global Fair GMs Complexity: Considering factors and 2, the complexity of each proximal gradient step applied to global fair GM is ( N p2, p))).
4. Local GMs Complexity: Applying factors and 2 to group-specific data, the complexity of each local GM is maxxN2, P2, p³) for all k = -,, 1,..., K. The total complexity of the local GMs is
K ΣK k== max(N,,P2 p3).
As established in Theorem 6 and Theorem for fair inverse covariance and covariance estimation, the iteration complexity of our algorithm to achieve e-aceuracy O (t). Combining this result with the per-iteration complexity of the algorithm, the total time complexity of our optimization procedure I2 .P)
is O
Including the Local GMs computation, the total time complexity of fair GMs
K k= max P2, + PN
is O
Under the assumption that the number of groups is small (i.c., K << N, << P, and K << 1/e), the complexity reduces O (max(N rr*) This complexity is of the same order as the complexity of running the proximal gradient method applied to covariance estimation and inverse covariance
26

---

estimation. Therefore, for large N and P and small number of groups, the time complexity of our algorithm is comparable to the standard method. In addition theoretical analysis, we also provide sensitivity analysis experiments on P, .N, K, and group imbalance in Appendix D.6-D.9.
D
Addendum to Section 4
D.1
Iterative Soft- Soft-Thresholding Algorithm (ISTA)
ISTA for sparse inverse covariance estimation is initially introduced by [60] and demonstrates closed-form linear convergence rate. adapt this approach and extend to other GMs, utilizing in the generation of both baseline and local graphs. Specifically, for a GM characterized by the loss function C (θ; X) + X|O||, I, we employ the following detailed algorithm:
Algorithm 2 ISTA for GMs
Input: Sample matrix X. initial iterate O (0) maximum iteration T, step size Ç, regularization parameter A, tolerance Set t O.
for 0, 1,...,77 do
Gradient Step: (++1)
θ CVL O
(0))) x))
Soft-Thresholding Step: θ (++1) + Tkp ((-1)) (T< (0)),r sign(8;)') max(1035] Sp,0)
if [IVE (0+11) x) || S then
Break
end if end for
Output: e (1+1)
D.2 Simulation Study of Fair GLasso
As supplement Section 4.2, we detail the process of generating block diagonal covariance matrices of dimensions P x P. denoted as {E each corresponding to distinct sensitive groups The procedure is as follows:
1. Firstly, we assume that each Σg contains blocks and P is divisible by Q. For the first group, the covariance matrix ΣS is constructed as block diagonal matrix:
B
0
Σ
(58)
BQ
Here, each block B, is a sub-matrix filled with values drawn from normal distribution ./ (0.7.0..))
2. To ensure Σ, is symmetric, it is adjusted EE) we further adjust E, as:
ET)/2
)/2. To ensure it is positive definite,
Σ [11
Up]
(59)
Ap.
UPp
where Å₃ represents max (){))),,,))))))))))}>n)))) 10 10- and vy is the corresponding eigenvector.
3. For each subsequent group (k 2...., K), the covariance matrix Σg is initially set equal Eg-1. Then, (one for sensitivity analysis) of its sub-matrices, which have been altered yet, are reset to the identity matrix.
D.3 Simulation Study of Fair BinNet
We specify the process of generating synthetic data for the simulation study in Section 4.3. This process adapts hub node-based network proposed by [72], aiming to generate a sequence of networks {O)/- )= The process comprises the following steps:
27

---

Table 4: Outcomes in terms of the value of the objective function (Fi), the summation of the pairwise
graph disparity error ( (A), and the average computation time in seconds (# standard deviation) from 10 repeated experiments. "!" means the smaller, the better, and the best value is in bold. These experiments are conducted on an Apple M2 Pro processor. Note that both Fi and A are deterministic.
Fi
A 4
Runtime 4
Dataset
%F1 f
%A f
GM
Fair GM
GM
Fair GM
GM
Fair GM
AV45
79.201
79.611 66.923
-0.52%
8.7626
3,4162 2.8920
+61.01 %
0.548 (+- 0.06)
19.06 (+ 0.3) 36.00 (+ 2.7)
AV1451
66.493
-0.65%
8.0503
+64.08%
1.616 (+ 0.65)
1. Initialize P x P matrix A, setting A f 1 with a probability of 0.01 for all j < j and 13 = 0 otherwise. Ensure the matrix symmetric by assigning Ay Ajy- From the set of nodes, randomly select H hub nodes and modify their corresponding rows and columns in A to 1 with a 99% probability or to 0 otherwise.
2. Construct another x P matrix E, where each element E, is i.i.d. Set SetE, E,y' 0 0 A or' O. Otherwise, draw Ezy from a uniform distribution over the intervals -0.75, - 0.255 U [0.25, 0.75] for hub node columns and rows, and [-0.5, -0.25 (0.25, 0.5] for non-hub node columns and rows. Subsequently, symmetrize matrix E by computing E E + E 1/2. Define the first network O₁ as θ E + (0.1 Aumin I.
3. For the generation of each subsequent network (k 22.... K), start with the preceding network, setting O = - Then, modify 2₂ by climinating two hub nodes.
D.4 Addendum to Subsection 4.5
In the experiments of applying GLasso to the ADNI dataset, we investigate the influence of sensitive attributes on brain networks associated with Alzheimer's s disease (AD) pathology. Specifically, we focus on the amyloid accumulation network using AV45 (florbetapir) positron emission tomography (PET) data [88] and the tau accumulation network using AV 1V145 (flortaucipir) PET data [46]. For the amyloid network, we consider the sensitive attribute of marital status, as previous studies suggest that marriage may affect the progression of dementia due to factors such as social support, cognitive stimulation, and lifestyle habits [21, 611. The dataset is divided into two groups based on marital status: single group with 52 samples and a married group with 1,018 samples, creating an imbalanced and high-dimensional setting that poses challenges for network estimation. In the tau accumulation network, we explore the impact of the sensitive attribute race, which separates the dataset into two groups: the white group with 755 samples and the non-white group with 118 samples. This division allows us to investigate potential disparities in tau pathology across racial groups. Throughout the experiments, the regularization parameter X. which controls the sparsity of the estimated networks, is fixed at 0.3 for the AV45 data and 0.2 for the AV 1451 data based on empirical observations. Besides, the dataset is normalized such that it has a mean of zero and a standard deviation of one.
The numerical results in Table 4 demonstrate that Fair GLasso effectively reduces disparity error compared to standard GLasso, enhancing fairness while maintaining good objective value. Figure reveals notable differences in the learned network structures for AV45 results. The presence of edges between the left caudal middle frontal gyrus and right medial orbitofrontal cortex and between the left superior frontal gyrus and left superior parietal lobule in the GLasso graph suggests an increased influence of emotional factors executive function and higher-order cognitive processes on amyloid accumulation, respectively [64, 6, 54]. Conversely, the absence of an edge between the left pars opercularis and left supramarginal gyrus in the Fair GLasso graph indicates a weaker association between language deficits and sensorimotor impairments in amyloid accumulation [18, 81, 38).
Results.
In contrast, the AV1451 results show primarily numerical differences between the two graphs, with edges remaining largely unchanged, suggesting that the tau accumulation network is robust to the sensitive attribute of race. These findings highlight the importance of considering fairness in brain imaging data analysis and the potential of Fair GLasso to uncover more equitable and unbiased patterns of amyloid and tau accumulation in Alzheimer's 's discase. Further research is needed to validate these findings in larger and more diverse cohorts and explore the biological mechanisms and clinical implications of the observed differences between standard GLasso and Fair GLasso graphs.
D.5 Addendum to Subsection 4.6
Table 5 summarizes the details of credit data utilized on Fair CovGraph mentioned in Section 4.6.
28

---

(a) Standard GLasso (AV45, Marital Status)
(b) Fair GLasso (AV4S, Marital Status)
(c) Standard GLasso (AV1451, Race)
(d) Fair GLasso (AV1451, Race)
Figure 5: Subfigures (a) and (b) present comparison of the graphs generated by standard GLasso and Fair GLasso on the ADNI dataset, considering the sensitive attribute of marital status the AV45 biomarker. Similarly, subfigures (c) and (d) compare the graphs generated by both methods, taking into account the sensitive attribute of race on the AV1451 biomarker. To improve the clarity of the visualizations, weak edges have been removed, and edges that show significant differences in values between the two methods highlighted. It is important note that even though some edges may appear unchanged in the visual comparison, their actual values will differ between the standard GLasso and Fair GLasso methods.
Table 5: Distribution of the number of samples in each group in the credit dataset. "HgEd" represents "High School Graduate or Higher", and "LwEd" represents "Education below High School Level".
Name
Size
Name
Size
Male Siiee_Sinee___gg
5579
Soinale_Single.. HgEd
8260
Male Liale_Snnge___e LwEd
974
Female_ Single_Lwiid
1151
Male Married HgEd
4062
Female Married HgEd
6506
Mali_Married Married_LLwEi
1128
MarmieeMarrredJJJ.
1963
29

---

D.6 Sensitivity Analysis to Feature Size P
Table 6: Numerical outcomes in terms of the value of the objective function (Fi), the summation of the pairwise graph disparity error (A), and the average computation time in seconds (# standard
deviation) from 10 repeated experiments. K and N₂ 1000 Ak E [K]. " means the smaller, the better, and the best value is in bold. These experiments are conducted on an Apple M2 Pro processor. Note that both Fi and A are deterministic.
Feature Size
Fi
A
Runtime
%F.
%A
GM
Fair GM
GM
Fair GM
GM
Fair GM
50,9229
50.9429
-0.04%
0,7309
0,3832
+47.56%
0.035 (:: 0.01)
19.86 0.24)
100
105.089
105.149
-0.0%%
2.1799
1.1206
+48.60%
0.183
0.01)
37.14 1.13)
150
159,424
159.555
-0.08%
4.6926
1.7772
+32.13%
2.452
0.89)
48.47 4.50)
200
215.402
215.558
-0.07%
5.7447
1.9693
+65.72%
2.034 (:: 0.11)
53.57 1.40)
250
269.236
269.356
-0.04%
5,4889
2.1106
+61.55%
0.552 (+ 0.04)
58.71 1.55)
300
324.733
324.958
-0.07%
8,5738
2.5582
+70.16%
1,508 (#: 0.07)
68.11 1.09)
350
379.696
380.027
-0.0%%
12.758
3.2992
+74.14%
3,199 (LL 0.24)
105.8 (1 2.50)
400
434.697
434.927
-0.05%
9.4141
2,4337
+74.15%
1.509 (+ 0.12)
107.7 (+ 3.55)
125
65.72
61.55
70.16
Glasso
47.56 18.00
100
Change
30
Runtine 68
0.0
-0.06
-0.07
0..99
-0.05
50
1.0
200
250
300
350
50
100
150
200
250
300
350
Feature Sizee
Feature Size (P)
(a) Percentage Change
(h) Runtime
Figure 6: (a) Percentage change from GLasso to Fair GLasso (results from Table 6) with respect to feature size P. %F1 is slight, while %A changes are substantial, signifying fairness improvement without significant accuracy sacrifice. (b) Runtime (mean std) (results from Table 6) with respect to feature size P.
In this section, we examine the impact of varying feature sizes P on the %F score, %4 (change in accuracy), and runtime. Our experiments utilize feature sizes ranging from P 50 to P= 400 in the GLasso algorithm applied to synthetic data. According to the procedures described in Steps 1-3 from Section D.2, we generate covariance matrices for two distinct groups: Σ₁ featuring five diagonal blocks and Σg with four diagonal blocks.
For each feature size setting, Group 1 includes 1000 observations drawn from multivariate normal distribution (0, Σj), and Group also consists of 1000 observations from (O, Eg). The outcomes of these experiments are systematically presented in Table 6 and visually depicted in Figure 6. This structured analysis enables us to evaluate how changes in feature size affect both performance metrics and computational efficiency in our study.
By integrating both the Table 6 and Figure 6, it can be observed that as the feature size increases, although there is a rise the pairwise graph disparity error, our proposed method still effectively reduces it, with minimal loss in the objective value. This underscores the efficacy of our approach in enhancing fairness. Regarding runtime, there is proportional relationship between feature size and the runtime of Fair GLasso, which aligns with our theoretical analysis of algorithmic complexity.
D.7 Sensitivity Analysis to Sample Size N
In this section, we conduct sensitivity analysis with respect to the sample size N, while holding the feature size fixed at P 50. We investigate how varying the sample size impacts the FF1 score, A (change accuracy), and runtime. The sample sizes examined are N 100, 150, 200, ..., 400,500 for each group in the Fair GLasso on synthetic data.
Following the procedures outlined in Steps 1-3 in Section D.2, we generate synthetic datasets with fixed covariance structures for two distinct groups: Σj characterized by five diagonal blocks, and Eg comprising four diagonal blocks. Each dataset generated for every specified sample size, allowing
30

---

Table 7: Numerical outcomes in terms of the value of the objective function (Fi); the summation
of the pairwise graph disparity error (A), and the average computation time in seconds (# standard
deviation) from 10 repeated experiments. K 2 and P 50. "+" means the smaller, the better, and
the best value is in bold. These experiments are conducted on an Apple M2 Pro processor. Note that both Fi and A are deterministic.
Sample Size
Fi L
A
Runtime !
%F.
%A
N₂
GM
Fair GM
GM
Fair GM
GM
Fair GM
100
50.2970
50.3049
-0.02%
0.6988
0.3715
+46.83%
0.044
0.01)
26.89 1.32)
150
50.6003
50.6043
-0.01% %
0.3407
0.2042
+40.05%
0.037
0.01)
19.12 0.53)
200
50.8234
50.8438
-0.04%
0.7194
0.2843
+60.49%
0.103 (=: 0.02)
19.13 0.27)
250
50.8729
50.8978
-0.05%
0.8615
0.3514
+59.21%
0.099 (:: 0.16)
23.06 1.06)
300
50.8791
50.8912
-0.02%
0.6464
0.3718
+42.48%
0.046 (#: 0.01)
30.18 0.99)
350
50.9272
50.9448
-0.03%
0.7120
0.3660
+48.60%
0.018 (1: 0.00)
25.59 (1: 0.30)
400
50.9186
50.9344
-0.03%
0.6675
0.3537
+47.01% %
0.030 (+ 0.00)
23.33 (+ 0.29)
500
50.9021
50.9261
-0.05%
0.7281
0.3137
+56.91% %
0.038 (= 0.00)
17.86 (# 0.31)
35
60.49
59.21
48.00 "17U1 "633
28
GLasso
42.48
Change
Runfine (0)
30
0.0
-0.02
-0.05
-0.02
-0.03
-0.03
-0,05
100
200
250
350
400
100
200
250
300
350
400
Sample Siice (Na)
Sample Sizee (N2)
(a) Percentage Change
(b) Runtime
Figure 7: (a) Percentage change from GLasso to Fair GLasso (results from Table 7) with respect to sample size N. %F is slight, while %A changes substantial, signifying fairness improvement without significant accuracy sacrifice. (b) Runtime (mean + std) (results from Table 7) with respect sample size N.
us systematically assess the effects of increasing N on the performance metrics and computational efficiency of the algorithm.
The specific results are presented in Table and visualized Figure 7. From these, it evident that the sample size does not significantly impact the objective value, pairwise graph disparity error, or runtime. Our proposed method consistently maintains its effectiveness across different sample sizes. This stability highlights the robustness of our approach under varying data quantities.
D.8 Sensitivity Analysis to Sample Size Ratio N2/N1
Table 8: Numerical outcomes in terms of the value of the objective function (F1), the summation of the pairwise graph disparity error (A), and the average computation time in seconds (# standard deviation) from 10 repeated experiments. K = 2, P = 50, and N₂ 100. "J" means the smaller, the better, and the best value is in bold. These experiments are conducted on an Apple M2 Pro processor.
Note that both Fi and A are deterministic.
Sample Size Ratio N1/N,
Fi J
A
Runtime
%.F,
%A
GM
Fair GM
GM
Fair GM
GM
Fair GM
1.0
50.2970
50.3049
-0.02%
0.6988
0.3715
+46.83%
0,175 (#: 0.37)
26.77 0.59)
2.0
50.4208
50.5981
-0.35%
4,5459
0.8282
+81.78%
0.042 (3: 0.01)
21.64 (:: 0.42)
3.0
50.4427
50,9140
-0.93%
8.3116
0.8556
+89.71%
0.061 (1 0.01)
16.06 0.42)
4.0
50.3129
50.8348
-1.04%
8.6970
L.0665
+88.43%
0.033 (+ 0.01)
21.64 (- 0.38)
5.0
50.1979
50.8795
-1.36%
10.213
1.1157
+89.07%
0.049 ( (2 0.02)
22.66 0-332)
7.0
50.1567
51.1681
-2.02%
14.224
1.2931
+90.91%
0.033 (..00
25.59 (1204))
10.0
50.0203
50,9329
-1.82%
11.462
1.2407
18%,
0.035 (= 0.00)
22.35 (=447)
100.0
49.8966
51.2365
-2.69%
17.620
1.0912
+93.81%
0.033 0.01)
16.51 0.36)
31

---

93.81
5.1.7
88 45
39.07
89.18
25
% Chahnng
46.83
Rustins (b)
20
30
Fair Glasso
GLasse
0.0
10
-0.02
-0.35
-1.91
-1.36
-2.02
-1.82
-6.0
16.0
%6.
100
150
200
250
300
350
400
900
1.0
4,0
5.0
7.0
10.0
100.0
Sample Sizee Ratio (Nr/N))
Sample Sizee Ratio (Ny/Ne)
(a) Percentage Change
(b) Runtime
Figure 8: (a) Percentage change from Glasso Fair Glaaso (results from Table 7) with respect to sample size ratio N1/M /2. %F1 is slight, while %A changes are substantial, signifying fairness improvement without significant accuracy sacrifice. (b) Runtime (mean + std) (results from Table 7) with respect to sample size ratio 2
We conduct a sensitivity analysis on the sample size ratio N₁ / while keeping the feature size fixed at P = 50 and Group 2's sample size N2 constant at 100. We examine the impact of varying N1 / on the %... %A, and runtime our experiments with Fair GLasso synthetic data.
Following the methodology outlined in Steps 1-3 from Section D.2, we generate datasets with fixed covariance structures: Σ, characterized by five diagonal blocks for Group 1 and Eg with four diagonal blocks for Group We systematically vary N1 from 100 to 10,000, maintaining N2 at 100, and assess how changes in the sample size ratio affect the algorithm's 's performance metrics and computational efficiency.
The specific results of these experiments are detailed in Table 8 and visualized in Figure 8. From this analysis, it is apparent that the sample size ratio / does significantly affect the objective value, pairwise graph disparity error, or runtime. Our proposed method continues to demonstrate its effectiveness consistently across varying sample size ratios. This consistency underscores the robustness of our approach, showing its reliability regardless of changes in the group imbalance between the groups.
D.9 Sensitivity Analysis to Group Size K
Table 9: Numerical outcomes in terms of the value of the objective function (Fi), the summation
of the pairwise graph disparity error (A), and the average computation time in seconds (#: standard
deviation) from 10 repeated experiments. P 100 and N₂ 1000 vk [K]. "4" means the smaller, the better, and the best value is bold. These experiments are conducted on an Apple M2 Pro processor. Note that both Fi and A are deterministic.
F1 4
A
Runtime
Group Size K
%F4
%A
GM
Fair GM
GM
Fair GM
GM
Fair GM
2
101.306
101.331
-0.03%
0.9451
0,4523
+52.14%
0.183 (.L 0.02)
28.16 1.14)
3
102.242
102.4
-0.19%
3.7579
0.6341
+83.13%
0.155 (: 0.06)
44.25 3.76)
103.157
103.664
-0.49%
9.4820
0.5665
+94.03%
0.146 (t: 0.03)
(1
103.856
104.730
-0.84%
18.835
+97.64%
0.192 (L 0.03)
103.2 4.62)
104,489
105.710
-1.17%
31.688
0.4085
+98.71%
0.167 (+: 0.03)
114.5 (+
105.113
106.685
-1.50%
48.421
0.4113
+99.15%
0.192 (2 0.03)
117.6 8.08)
8 105.806
107.663
-1.75%
68.335
0.7127
+98.96%
0.149 (= 0.04)
133.2 9.25)
106.458
108,810
-2.21%
92.749
1.5164
+98.37%
0.233 (1 0.06)
265.7 (1 11.2)
10
107.112
109.742
-2.46%
121.37
2.1924
+98.19%
0.134 (1 0.06)
360.9 (1: 23.0)
In this section, we explore the impact group size on the performance and computational efficiency of Fair GLasso. The feature size N is fixed at 100, and the sample size per group Pa is set at 1000. Following Steps 1-3 from Section D.2, the covariance matrix for the first group. Σ, generated with 10 diagonal blocks. Each subsequent group has one fewer diagonal block in its covariance matrix, with each group sampling observations from ((0,, Ex).
The results are detailed in Table 9 and Figure 9, Observations indicate that when the group size is less than 9, computational efficiency remains relatively stable regardless of changes in group size. However, efficiency decreases noticcably when the group size increases 9. In terms of the objective
32

---

83.13.
94.03
350
Fair Glasso
GLasso
Change
60
52.14
Runtan
250
30
200
150
.0.03
-0.19
-0.49
-1.50
-1.75
-2.21
100
6.0
50
%6.
16/
Group Size (K)
Group Size (K)
(a) Percentage Change
(b) Runtime
Figure 9: (a) Percentage change from GLasso Fair GLasso (results from Table 6) with respect to group size K. %Fi is slight, while %A changes are substantial, signifying fairness improvement without significant accuracy sacrifice. (b) Runtime (mean + std) (results from Table 6) with respect to group size K.
Table 10: Outcomes of additional baseline with different optimization algorithms applied GLasso and Multi-Objective Optimization (MOO), measured in terms of the value of the objective function (Fi), the summation of the pairwise graph disparity error (A), and the average computation time in seconds (# standard deviation) from 10 repeated experiments. "4" indicates that smaller values are better. Our method applies ISTA to both GLasso and MOO (first row each experiment). All experiments are conducted using the same runtime environment on Google Colab.
Algorithm
F1
%F1
%A
Runtime
GLasso
MOO
GLasso
Fair GLasso
Gaasso
Fair GLasso
GLasso
Fair GLasso
Synthetic Dataset I (2 Subgroups. 100 Variables, 1000 Observations in Each Group)
ISTA
ISTA
97.172
97.449
-0.29%
7.8149
+92.59%
0.501 0.21)
85.48
1,92)
ISTA
FISTA
97.172
97.438
-0.27%
7.8149
0.8835
+88.69 69%
0.297 (1: 0.12)
26.56 (1: 1.11)
PISTA
FISTA
97.172
97.438
-0.27%
7.8190
0.9084
+88 38%
13.52 (+ 1.10)
59.66 (+ 2.65)
GISTA
FISTA
97.172
97.438
-0.27%
7.8149
+88.37%
0.426 0.16)
21.27 0.94)
OBN
FISTA
97.172
97.438
-0.27%
7,8134
0.9112
+88.34%
0.483 (+ 0.16)
22.48 (+ 0.92)
Synthetic Dataset 2 (2 Subgroups, 200 Variables, 2000 Obscrvations in Each Group)
ISTA
ISTA
199.71
200.70
-0.49%
40.511
1.4855
+96.33%
2.622 1.28)
206.7 3.27)
ISTA
FISTA
199.71
200.68
-0.49%
40.511
1.8485
+95.44%
2.640 0.76)
108.1 2.42)
PISTA
FISTA
199.71
200.67
-0.48%
40.521
1.9474
+95.19%%
39.16 (1: 2.30)
178.7 (1: 3,50)
GISTA
FISTA
199.71
200.68
-0.48%
40.511
2.0260
+95 00%
2.365 0.26)
78.99 3.07)
OBN
FISTA
199.71
200.72
-0.50%
40.511
2.4835
+93.87%
2.403 (#: 0.68)
53.11 2.17)
Synthetic Dataset (10 Subgroups, 100 Variables, 1000 Observations in Each Group)
ISTA
ISTA SOSA
95,333
95.603 95.506
-0,88%
11.394
0,3108
+97.27% +86.72%
0.641 (1: 0,28)
224.1 (#: 2.29)
95.333
0.626 0.19)
143.2 2,28)
value and pairwise graph disparity error, performance maintains good balance, with a significant enhancement faimees..
This conclusion aligns with our theoretical analysis of algorithmic complexity. Notably, as the group size increases, the pairwise graph disparity error also significantly rises, as shown in Table 9, Consequently, our proposed method effectively enhances faimess, albeit at the cost of sacrificing greater portion of the objective value. This trade-off is a critical aspect of our approach, balaneing computational performance with the desired ethical outcomes in machine learning applications.
D.10
Addendum to Subsection 4.8
To address the computational complexity of Fair GMs, we explore range of optimization methods tailored GLasso and multi-objective optimization (MOO):
GLasso Optimization Methods
Preconditioned Iterative Soft Thresholding Algorithm (PISTA): Efficiently handles
large-scale sparse matrix operations [69].
Graphical Iterative Shrinkage Thresholding Algorithm (GISTA): Employs an iterative framework for -nnduity penalty functions in high-dimensional settings [60].
Orthant-Basse Based Newton Method (OBN): Uses second-order information for faster
convergence in structured sparsity constraints [57].
33

---

MOO Optimization Methods
Fast Iterative Trrinkage-Thresooddie Algorithm (FISTA): Provides globally optimal
convergence rates for MOO objectives [74].
Stochastic Objective Selection Approach (SOSA): Introduces randomized selection
technique for optimizing multi-objective functions under varying conditions [70].
We validate these methods through comprehensive experiments on synthetic datasets. Our first evaluation uses data with 100 variables across two subgroups, each containing 1000 observations, generated following the procedure in Appendix D.2. This experiment demonstrates that faster oplimization methods improve time complexity for both GLasso and MOO while maintaining performance. All GLasso methods achieve optimal loss, while Fair GLasso variants successfully reduce pairwise graph disparity error without significant performance degradation.
To assess scalability, we extend our analysis to larger dataset with 200 variables, maintaining the same experimental setup. Furthermore, we evaluate the efficiency of our approach with increased group complexity using synthetic data containing 100 variables across ten subgroups, each with 1000 observations. In this setting, SOSA reduces training time by approximately 36% compared to the original approach while preserving model fairness and robustness.
The numerical results presented in Table 10 confirm that our optimization strategies successfully reduce runtime while maintaining model robustness and fairness. These findings suggest promising directions for future research in balancing computational efficiency with model performance.
34

---

NeurIPS Paper Checklist
1. Claims
Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
Answer: [Yes]
Justification: Yes, we introduce the contribution and novelty of this work in the abstract and introduction.
Guidelines:
The answer NA means that the abstract and introduction do not include the claims made in the paper.
The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No NA answer to this question will not be perceived well by the reviewers.
The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings.
is fine to include aspirational goals as motivation as long is clear that these goals are not attained by the paper.
2. Limitations
Question: Does the paper discuss the limitations of the work performed by the authors?
Answer: [Yes]
Justification: Yes, we discuss the limitations of our work in the conclusion section.
Guidelines:
The answer NA means that the paper has limitation while the answer No means that the paper has limitations, but those are not discussed in the paper.
The authors encouraged to create separate "Limitations" section in their paper.
The paper should point any strong assumptions and how robust the results violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification. asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be.
The authors should reflect on the scope of the claims made, e.g., if the approach was only tested few datasets with few runs. In general, empirical results often depend on implicit assumptions, which should articulated.
The authors should reflect on the factors that influence the performance of the approach. For example, facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails handle technical jargon.
The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size.
If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness.
While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't 't acknowledgen in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an impor- tant role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations.
3. Theory Assumptions and Proofs
Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?
Answer: [Yes]
Justification: Yes, we present the theoretical analysis of our proposed model in Section 3. The detailed proof presented in the appendix.
Guidelines:
35

---

The answer NA means that the paper does not include theoretical results.
All the theorems, formulas, and proofs in the paper should be numbered and cross- referenced.
All assumptions should be clearly stated or referenced in the statement of any theorems.
The proof' can either appear in the main paper or the supplemental material, but if they appear in the supplemental material, the authors are encouraged to provide short proof sketch to provide intuition.
Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material.
Theorems and Lemmas that the proof relies upon should be properly referenced.
4. Experimental Result Reproducibility
Question: Does the paper fully disclose all the information needed to reproduce the main ex- perimental results of the paper to the extent that affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?
Answer: [Yes]
Justification: Yes, we describe the algorithm, baseline model, and detailed experiment procedures, including synthetic data generation, choices of parameters, and architecture.
Guidelines:
The answer NA means that the paper does not include experiments.
If the paper includes experiments, a No answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not.
If the contribution is dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable.
Depending on the contribution, reproducibility can be accomplished in various ways. For example, if the contribution is a novel architecture, describing the architecture fully might suffice, or if the contribution is specific model and empirical evaluation, it may be necessary to either make possible for others to replicate the model with the same dataset, or provide access to the model. In general. releasing code and data often one good way to accomplish this, but reproducibility can also be provided via detailed instructions for how to replicate the results, access to a hosted model (e.g.. in the case of large language model), releasing of a model checkpoint, or other means that appropriate to the research performed.
While NeurIPS does not require releasing code, the conference does require all submis-
sions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution. For example
(a) the contribution is primarily a new algorithm, the paper should make clear how to reproduce that algorithm.
(b) the contribution is primarily a new model architecture, the paper should describe the architecture clearly and fully.
(c)
If the contribution is new model (e.g., a large language model), then there should either way to access this model for reproducing the results or way to reproduce the model (e.g., with an open-source dataset or instructions for how to construct the dataset).
(d) We recognize that reproducibility may be tricky in some cases, in which case authors are welcome to describe the particular way they provide for reproducibility. In the case of foosed-ouurce models, it may be that access to the model is limited in some way (e.g., registered users), but it should be possible for other researchers have some path to reproducing or verifying the results.
5. Open access to data and code
Question: Does the paper provide open access to the data and code, with sufficient instruc- tions to faithfully reproduce the main experimental results, as described in supplemental material?
Answer: [Yes]
Justification: Yes, we provide the code on GitHub for reproducing the results of all the simulation studies. For real-world applications, due to the accessibility of some datasets, we are not able to provide the dataset when submitting, but readers are able to download data through provided links.
36

---

Guidelines:
The answer NA means that paper does not include experiments requiring code.
Please see the NeurIPS code and data submission guidelines (https://nips... cc/
Saaides/CodeSubiiis
Policylly) for more details.
While we encourage the release of code and data, we understand that this might not be possible, "No" is an acceptable answer. Papers cannot be rejected simply for not including code, unless this is central to the contribution (e.g., for new open-source benchmark).
The instructions should contain the exact command and environment needed to run to reproduce the results. See the NeurlPS code and data submission guidelines (https:
asssonn / //nips. cc/ cc/public/eeeccc htc//pblic/guudeeeeeeeeeccccceee Seriiides/oodSuuie wwtmmissssnooooooo cssoPooiccy for more details.
The authors should provide instructions on data access and preparation, including how
to access the raw data, preprocessed data, intermediate data, and generated data, etc.
The authors should provide scripts to reproduce all experimental results for the new proposed method and baselines. If only subset of experiments should state which ones are omitted from the script and why.
reproducible, they
At submission time, to preserve anonymity, the authors should release anonymized versions (if applicable).
Providing as much information as possible supplemental material (appended to the paper) is recommended, but including URLs to data and code is permitted.
6. Experimental Setting/Details
Question: Does the paper specify all the training and test details (e.g., data splits, hyper- parameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?
Answer: [Yes]
Justification: Yes, our work is about fairness in unsupervised graphical models, we describe the detailed sensitive attribute and the resulting group splitting. The optimization method is also well described.
Guidelines:
The answer NA means that the paper does include experiments.
The experimental setting should be presented in the core of the paper to level of detail that is necessary to appreciate the results and make sense of them.
The full details can be provided either with the code, in appendix, as supplemental material.
7. Experiment Statistical Significance
Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?
Answer: [Yes]
Justification: Yes, we repeated all the experiments 10 times. We provide full description of this statistical analysis in our implementation details provided in Section
Guidelines:
The answer NA means that the paper does not include experiments.
The authors should answer "Yes" if the results are accompanied by error bars, confi- dence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper.
The factors of variability that the enror bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, overall
run with given experimental conditions).
The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.)
The assumptions made should given (e.g., Normally distributed errors).
should be clear whether the error bar is the standard deviation or the standard error of the mean.
is OK to report I-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have 96% CI, if the hypothesis of Normality of errors is not verified.
37

---

For asymmetric distributions, the authors should be careful show in tables figures symmetric error bars that would yield results that are out range (e.g. negative eтror rates).
If error bars are reported in tables or plots, The authors should explain the text how they were calculated and reference the corresponding figures or tables in the text.
8. Experiments Compute Resources
Question:
For each experiment, does the paper provide sufficient information on the com- puter resources (type of compute workers, memory, time of execution) needed reproduce the experiments?
Answer: [Yes]
Justification: Yes, all the experiments were conducted on MacBook Pro with an Apple M2 Pro chip and 32 GB of memory. The additional experiments conducted during the rebuttal were implemented Google Colab.
Guidelines:
The answer NA means that the paper does not include experiments.
The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage.
The paper should provide the amount compute required for each of the individual experimental rums as well as estimate the total compute.
The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't make it into the paper).
9. Code Of Ethics
Question: Does the research conducted the paper conform, in every respect, with the NeurIPS Code of Ethics https: htttp::///euripss wwtppccccc////ccccccccccctccctct
Answer:
[Yes]
Justification: Yes, we strictly follow the NeurIPS Code of Ethics.
Guidelines:
The answer NA means that the authors have reviewed the NeurIPS Code of Ethics.
If the authors answer No, they should explain the special cireumstances that require deviation from the Code of Ethics.
The authors should make sure to preserve anonymity (e.g., if there special consid- cration due to laws or regulations in their jurisdiction..
10. Broader Impacts
Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?
Answer: [Yes]
Justification: Yes, our work about fairness in graphical models. Improving faimesss in machine learning algorithms will definitively have positive societal impacts, which are discussed the introduction and experiment sections.
Guidelines:
The answer NA means that there is no societal impact of the work performed.
If the authors answer NA or No, they should explain why their work has no societal impact why the paper does not address societal impact.
Examples of negative societal impacts include potential malicious unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (c.g.. deploymennt of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations.
The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is direct path to any negative applications, the authors should point it out. For example, it is legitimate point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation.. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster.
38

---

The authors should consider possible harms that could arise when the technology is being used as intended functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology.
If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms monitor how system learns from feedback over time, improving the efficiency and accessibility of ML..
11. Safeguards
Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, scraped datasets)?
Question:
Answer: [Yes]
Justification: Yes, we strictly follow the usage rules of datasets (ADNI and TCGA). And don't tt use data or models that have high risk for misuse.
Guidelines:
The answer NA means that the paper poses no such risks.
Released models that have high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters.
Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images.
We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make best faith effort.
12. Licenses for existing assets
Are the creators or original owners of assets (e.g.. code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?
Question:
Answer: [Yes]
Justification: Yes, we follow the following instructions and also cite and refer them.
Guidelines:
The answer NA means that the paper does not use existing assets.
The authors should cite the original paper that produced the code package or dataset.
The authors should state which version of the asset used and, possible, include URL.
The name of the license (e.g., CC-BY 4.0) should be included for each asset.
For scraped data from a particular source (c.g.. website). the copyright and terms of service of that source should be provided.
If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, paperswi thcode. com datasets has curated licenses for some datasets. Their licensing guide can help determine the license of dataset.
For existing datasets that re-packaged, both the original license and the license of the derived asset (if has changed) should be provided.
If this information is not available online. the authors are encouraged to reach out to the asset's 's creators.
13. New Assets
Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?
Answer: [NA]
Justification:
The paper does not release new assets.
Guidelines:
The answer NA means that the paper does not release new assets.
39

---

Researchers should communicate the details of the dataset/code/moodl part of their submissions via structured templates. This includes details about training, license, limitations, etc.
The paper should discuss whether and how consent was obtained from people whose asset is used.
At submission time, remember to anonymize your assets (if applicable). You can either create an anonymized URL or include an anonymized zip file.
14. Crowdsoureing and Research with Human Subjects
Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well details about compensation (if any)?
Answer: [NA]
Justification:
This work does not involve crowdsourcing or research with human subjects.
Guidelines:
The answer NA means that the paper does not involve crowdsoureing nor research with human subjects.
Including this information the supplemental material is fine, but if the main contribu- tion of the paper involves human subjects, then as much detail as possible should be included in the main paper.
According to the NeurlPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector.
15. Institutional Review Board (IRB) Approvals or Equivalent for Research with Human Subjects
Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or equivalent approval/review based on the requirements of your country or institution) were obtained?
Question:
Answer: [NA]
Justification: This work does not involve crowdsourcing research with human subjects.
Guidelines:
The answer NA means that the paper does not involve crowdsourcing research with human subjects.
Depending the country in which research is conducted, IRB approval (or equivalent)
may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper.
We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurlPS Code of Ethics and the guidelines for their institution.
For initial submissions, do not include any information that would break anonymity (if applicable), such the institution conducting the review.
40


Reviews:
[{"review_id": "CPEed9wBbx", "reviewer": "Reviewer_nXfy", "summary": "The paper investigates the issue of bias  in 3 particular graphical models: Gaussian, Gaussian Covariance, and Binary Ising. In this regard, the authors propose a framework to enhance fairness in the estimation of graph models. They incorporate the difference of loss between the protected groups and the accuracy of graph models into a multi-objective optimization problem and solve it. The experiments show that the approach reduce the bias.", "questions": "Please refer to the weaknesses as well as these questions:\n\n1. Practical graphs and practical fairness problems have high K and P values. This will raise the computational complexity to the extent that it may question the applicability of the method. Do the authors have comment on this?\n\n2. What if the number of protected groups (like in the case of age) is high? How do the authors comment on these cases with regards to complexity analysis?", "limitations": "1. The applicability for real problems\n2. It does not scale well\n3. The study is limited to Gaussian, Gaussian Covariance, and Binary Ising models.", "rating": 5, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The paper is easy to follow\n2. Choice of ISTA as the baseline seems a very good decision\n3. Applications of the proposed method are well established in the paper.", "weaknesses": "1. The authors only consider one criterion of fairness, which is achieving equal loss among subgroups. How do the authors justify focusing solely on this criterion? Additionally, how might incorporating other fairness criteria impact the results?\n\n2. Many recent works utilize reinforcement learning to discover fair structures. For instance, the paper \"Rethinking Bias Mitigation: Fairer Architectures Make for Fairer Face Recognition\" by S. Dooley et al. (NeurIPS 2023) explores such an approach. How do the authors justify their method compared to these more comprehensive approaches?\n\n3. The authors employ multi-objective optimization, suggesting that the loss in GMs and pairwise graph disparity are at odds. However, many papers indicate that this is not always the case. For example, the paper \"Is There a Trade-Off Between Fairness and Accuracy? A Perspective Using Mismatched Hypothesis Testing\" by S. Dutta et al. (ICML 2020) discusses this. How do the authors justify their approach in light of such findings?\n\n4. The learning algorithm appears to be very time-consuming. Do the authors have any comments on this aspect of their method?\n\n5. Why do the authors not use more widely accepted metrics for fairness comparison and instead refer only to the difference in loss between subgroups?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "> **Weakness 1:** The authors only consider one criterion of fairness, which is achieving equal loss among subgroups. How do the authors justify focusing solely on this criterion? Additionally, how might incorporating other fairness criteria impact the results?\n\n> **Weakness 5:** Why do the authors not use more widely accepted metrics for fairness comparison and instead refer only to the difference in loss between subgroups?\n\n**Response:** Thank you for your insightful comments. Our primary focus on the difference in loss between subgroups as a fairness criterion stems from the unsupervised nature of our task. In unsupervised learning, unlike supervised learning, we lack labeled data that are typically used to define fairness criteria such as demographic parity (DP)[[JHF2022](https://par.nsf.gov/servlets/purl/10397778)] and equalized odds (EO)[[RBC2020](https://proceedings.neurips.cc/paper/2020/hash/03593ce517feac573fdaafa6dcedef61-Abstract.html)]. Specifically, incorporating other fairness criteria like DP and EO, while beneficial for providing a broader perspective on fairness, typically requires labeled data to assess prediction distributions and accuracy across groups. This makes them less applicable to our unsupervised context.\n\nThis necessitated the development of a new fairness criterion, which we introduce as *graph disparity error*, **solely based on the group-specific and global losses**. Indeed, our proposed graph disparity error aligns with the goals of DP by promoting a balanced representation of all subgroups using the loss function. Graph disparity error measures the difference in graph loss between a global model $\\Theta$ and the optimal local models $\\Theta_1^*, \\Theta_2^*, \\ldots, \\Theta_K^*$ for each subgroup $k \\in [K]$. This is validated through our experiments in Section 4.  For example, Figure 2 illustrates the comparison of original graphs utilized in synthetic data creation for two groups: graph reconstruction using standard GMs and fair graph reconstruction via Fair GMs. The figure demonstrates that Fair GMs produce more balanced and fair graph reconstructions across different subgroups compared to standard GMs, thereby supporting the effectiveness of the proposed graph disparity error in achieving fairness.\n\nWhile future research on graphical model estimation could explore more widely accepted fairness metrics in supervised learning, our current work establishes a disparity error-based approach. It is worth mentioning that our nonsmooth multi-objective optimization framework for graphs is novel in this area and flexible enough to incorporate other fairness objectives. Future research can build on our framework to explore additional fairness metrics.\n\nWe have incorporated some of these discussions into the introduction of the revised paper. Thank you.\n\n\n> **Weakness 2:** Many recent works utilize reinforcement learning to discover fair structures. For instance, the paper \"Rethinking Bias Mitigation: Fairer Architectures Make for Fairer Face Recognition\" by S. Dooley et al. (NeurIPS 2023) explores such an approach. How do the authors justify their method compared to these more comprehensive approaches?\n\n**Response:** We appreciate your comments and suggestions.  The above paper explores the inherent biases in neural network architectures used for face recognition and proposes a novel approach to mitigate these biases through neural architecture search (NAS) and hyperparameter optimization (HPO). By conducting the first large-scale analysis of the impact of different architectures and hyperparameters on bias, the authors discovered that biases are baked into the architectures themselves. They designed a search space based on high-performing architectures and used NAS and HPO to jointly optimize for fairness and accuracy, considering sensitive attributes.\n\nAlthough your suggested reference might be useful in the context of neural networks, such as graph neural networks, our problem is significantly different and is not related to neural architecture search or supervised learning. Indeed, our model focuses on unsupervised fair graph learning and does not involve any neural architecture search. Many standard fairness metrics, such as demographic parity or equalized odds, originally developed for supervised learning and widely used in neural networks, may not apply here. Our method is inspired by the idea that equal loss among subgroups ensures that no single group is disproportionately disadvantaged by the learned model, which aligns with established fairness definitions in the unsupervised machine learning literature.\n\nWe have added further discussion to the revised paper.  Thank you.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "UfkfzrI1R3", "reviewer": "Reviewer_tjnm", "summary": "This paper introduces a framework to address fairness in the estimation of graphical models, particularly focusing on Gaussian, Covariance, and Ising models. The motivation stems from the potential bias in standard GMs when handling data involving sensitive characteristics or protected groups. The proposed framework integrates a pairwise graph disparity error and a tailored loss function into a nonsmooth multi-objective optimization problem to reduce bias while maintaining the effectiveness of GMs.", "questions": "1) Could you elaborate more on the choice of fairness metrics, disparity error, used in your evaluation? What might be the implications for using such metric compared to other metrics, like those stemming from counterfactual evaluation?\n\n2) While the authors claim that the proposed method improves fairness without compromising performance, Table 1 indicates that 6 out of 7 datasets experience lower F1 scores, suggesting worse performance. Could you clarify this discrepancy and provide a more detailed interpretation of Table 1?\n\n3) It's mentioned that ISTA is used as baseline. What are other possible baselines? Why in particular choose ISTA as baseline?", "limitations": "The paper discuss the complexity limitation of the work but did not solve it.", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "1) Clear motivation: The paper tackles the critical and timely issue of fairness in graphical models, which is essential as machine learning applications become more widespread and impact diverse populations.\n\n2) Innovative Approach: The integration of the pairwise graph disparity error and tailored loss function into a multi-objective optimization framework is a novel approach to achieving fairness in GMs.\n\n3) Substantial technical contribution: The paper provides a solid theoretical foundation for the proposed framework. And the appendix is very informative, for example included complexity analysis of the proposed method.", "weaknesses": "Fairness metric: The choice and justification of fairness metrics used in the evaluation could be more thoroughly discussed. This would provide a clearer understanding of how fairness is quantified and the implications of these choices.\n\nUnconvincing experiment results: While the authors claim that the proposed method \"approach enhances fairness without compromising performance\" in line 306, Table 1 indicates that 6 out of 7 datasets experience lower F1 scores, suggesting worse performance. \n\nLimited baseline: The paper could benefit from more extensive comparisons with existing fairness-aware methods in graphical models other than ISTA (the only baseline used in Table 2). This would help to contextualize the performance and advantages of the proposed framework.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "> **Weakness 1:** Fairness metric: The choice and justification of fairness metrics used in the evaluation could be more thoroughly discussed. This would provide a clearer understanding of how fairness is quantified and the implications of these choices.\n\n> **Question 1:** Could you elaborate more on the choice of fairness metrics, disparity error, used in your evaluation? What might be the implications for using such metric compared to other metrics, like those stemming from counterfactual evaluation?\n\n**Response:** We appreciate the reviewers’s feedback on the need for a more thorough discussion of the fairness metrics used in our evaluation. We understand that the choice and justification of fairness metrics are critical for providing a clear understanding of how fairness is quantified and the implications of these choices.\n\nIn our work, we introduced the concept of graph disparity error as the primary fairness metric. This metric measures the difference in loss between the global model $\\Theta$ and the optimal local models $\\Theta_1^*, \\Theta_2^*, \\ldots, \\Theta_K^*$ for each subgroup. The rationale behind this choice is rooted in the unsupervised nature of our task. Unlike supervised learning, which can leverage labeled data to define fairness metrics such as demographic parity (DP)[[JHF2022](https://par.nsf.gov/servlets/purl/10397778)] or equalized odds (EO)[[RBC2020](https://proceedings.neurips.cc/paper/2020/hash/03593ce517feac573fdaafa6dcedef61-Abstract.html)], our unsupervised learning context lacks such labels. Therefore, we require a fairness metric that does not rely on label information.\n\nThe graph disparity error metric was chosen because it effectively captures the notion of fairness by ensuring that the model’s performance is balanced across different subgroups. By minimizing this error, we aim to ensure that no subgroup is disproportionately favored or disadvantaged, thereby promoting a more equitable model. Despite this, our proposed graph disparity error actually aligns with the goals of DP by promoting a balanced representation of all subgroups. The global graph obtained from our method represents a central integration of local graphs, ensuring that the influence of each subgroup is balanced. This inherent balance aligns with the core objective of DP without the use of labeled data.\n\nIn our revised version, we further include the following paragraph to improve the discussion of the fairness metrics:\n\n“In this work, we employ graph disparity error as our primary fairness metric. This metric measures the difference in loss between the global model and the optimal local models for each subgroup. Our choice is motivated by the unsupervised nature of our task, which lacks labeled data typically required for metrics such as demographic parity or equalized odds. Graph disparity error provides a label-independent measure of fairness, ensuring balanced model performance across subgroups. By minimizing this error, we reduce bias introduced by the data distribution, promoting an equitable model where no subgroup is disproportionately favored or disadvantaged. This approach aligns with the core objectives of demographic parity, as our global graph integrates local graphs, balancing the influence of each subgroup. Thus, our metric effectively quantifies fairness in high-dimensional graphical models, providing a robust framework for fair graph learning.”\n\n\n> **Weakness 2:** Unconvincing experiment results: While the authors claim that the proposed method \"approach enhances fairness without compromising performance\" in line 306, Table 1 indicates that 6 out of 7 datasets experience lower F1 scores, suggesting worse performance.\n\n> **Question 2:** While the authors claim that the proposed method improves fairness without compromising performance, Table 1 indicates that 6 out of 7 datasets experience lower F1 scores, suggesting worse performance. Could you clarify this discrepancy and provide a more detailed interpretation of Table 1?\n\n\n**Response:** We appreciate the reviewer’s comments on the experimental results and the interpretation of Table 1. We would like to clarify our claim that the proposed method enhances fairness without compromising performance.\n\nFirstly, it is important to recognize the inherent trade-off in multi-objective optimization frameworks that balance fairness and accuracy. While our method prioritizes fairness, this may lead to a slight decrease in the $F_1$ score, as reflected in Table 1. However, these decreases are minimal and within acceptable ranges, indicating that the overall performance of the model remains robust. The marginal reductions in $F_1$ scores result from the necessary adjustments to ensure fair representation across all subgroups. We include this analysis in Section 4.8, Trade-Off Analysis.\n\nTo be more accurate, we would like to clarify our claim in our revised version as follows:\n“This approach enhances fairness without compromising performance significantly, validated by experiments on synthetic and real-world datasets.”", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "6JU4hsZXDM", "reviewer": "Reviewer_1oDX", "summary": "This paper explores the issue of fairness in the estimation of graphical models (GMs), specifically Gaussian, Covariance, and Ising models. The authors introduce a comprehensive framework aimed at mitigating bias in GM estimation concerning protected attributes. This framework integrates pairwise graph disparity error and a customized loss function into a non-smooth multi-objective optimization problem. The pairwise graph disparity error assesses loss discrepancies across all groups, while the tailored loss function evaluates GM performance. Through this approach, the authors seek to achieve fairness across diverse sensitive groups while upholding GM performance, supported by theoretical proofs and experimental validation on various real-world datasets.", "questions": "Please explain the vagueness and confusion in the “Weaknesses” section.", "limitations": "The paper introduces a new framework that addresses fairness in the estimation of GMs, but it has some limitations. Firstly, the introduction lacks a comprehensive overview of current GMs and does not explain why the focus is on only three types of GMs rather than GMs in general. Secondly, the discussion of related work on fairness in GM estimation is insufficient and needs clearer explanations. Lastly, the implementation of different GMs requires distinct theorems with specific assumptions, which could limit the development and extension of the framework.", "rating": 5, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "The paper provides theoretical analysis, complemented by detailed appendices. To aid comprehension, the authors include examples throughout the paper, which help clarify the theories and proposed framework.\n\nThe experimental results demonstrate that the Fair GMs framework reduces disparity error, thereby enhancing fairness, with only a slight decrease in the model’s performance.\n\nAdditionally, the paper addresses the challenge of balancing performance and fairness by employing a non-smooth multi-objective optimization problem.", "weaknesses": "In the introduction and related works section, the paper does not provide complete information about GMs. Additionally, the paper does not clearly explain why it focuses on only three types of GMs instead of providing a general framework. The author needs to clearly highlight any difficulties encountered when applying this framework to each type of model.\n\nIn the second paragraph of Section 2 (Related Work on Fairness), the author does not discuss related works concerning fairness in GMs estimation. By focusing solely on fairness in unsupervised learning, the paper may limit readers' ability to gain a comprehensive understanding of the research field and the distinctiveness of the proposed method.\n\nMoreover, the paper lacks clarity regarding the statement: “Our paper addresses the challenge of learning GMs without any predefined assumptions on the graph’s structure.” In the referenced paper, \"Fair Community Detection and Structure Learning in Heterogeneous Graphical Models,\" the authors assume the presence of community structures and aim to learn a sparse undirected graph where demographic groups are fairly represented within these communities. This concept could be relevant to the current paper when the number of clusters is set to one. Therefore, the paper should clarify the differences and advantages of learning GMs without any predefined assumptions in the theoretical section.\n\nIn Section 4.8, trade-off analysis, the author mentions the shortcomings in runtime without clear discussion. Although there is a significant difference in runtime compared to the standard models, no reasons or future related works are mentioned in the subsequent sections.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "> **Weakness 1:** In the introduction and related works section, the paper does not provide complete information about GMs. Additionally, the paper does not clearly explain why it focuses on only three types of GMs instead of providing a general framework. The author needs to clearly highlight any difficulties encountered when applying this framework to each type of model.\n\n> **Limitation 1:** Firstly, the introduction lacks a comprehensive overview of current GMs and does not explain why the focus is on only three types of GMs rather than GMs in general.\n\n> **Limitation 3:** Lastly, the implementation of different GMs requires distinct theorems with specific assumptions, which could limit the development and extension of the framework.\n\n\n**Response:** We appreciate the reviewer’s comments and would like to clarify our approach further. Our paper focuses on three types of GMs: Gaussian Graphical Models, Gaussian Covariance Graph Models, and Binary Ising Graphical Models. These models were chosen because they represent a diverse set of widely used GMs with distinct characteristics and applications. Each type poses unique challenges in terms of estimation and fairness, making them suitable for demonstrating the effectiveness and versatility of our proposed framework.\n\n*Importance of Each Graph Model:* Each of the three graph models (GLasso, sparse covariance estimation, and binary Ising model) addressed in our paper is of significant interest to the research community. Each of these models has been extensively studied and typically warrants dedicated papers. Here are a few examples to illustrate their importance:\n\n*GLasso*:\n\n- Pavlenko, Tatjana, Anders Björkström, and Annika Tillander. \"Covariance structure approximation via gLasso in high-dimensional supervised classification.\" Journal of Applied Statistics 39.8 (2012): 1643-1666.\n\n- Mazumder, Rahul, and Trevor Hastie. \"The graphical lasso: New insights and alternatives.\" Electronic journal of statistics 6 (2012): 2125.\n\n- Friedman, Jerome, Trevor Hastie, and Robert Tibshirani. \"Sparse inverse covariance estimation with the graphical lasso.\" Biostatistics 9.3 (2008): 432-441.\n\n- Witten, Daniela M., Jerome H. Friedman, and Noah Simon. \"New insights and faster computations for the graphical lasso.\" Journal of Computational and Graphical Statistics 20.4 (2011): 892-900.\n\n*Sparse Covariance Estimation*:\n\n- Bien, Jacob, and Robert J. Tibshirani. \"Sparse estimation of a covariance matrix.\" Biometrika 98.4 (2011): 807-820.\n\n- Friedman, Jerome, Trevor Hastie, and Robert Tibshirani. \"Sparse inverse covariance estimation with the graphical lasso.\" Biostatistics 9.3 (2008): 432-441.\n\n- Bickel, Peter J., and Elizaveta Levina. \"Regularized estimation of large covariance matrices.\" (2008): 199-227.\n\n- Cai, Tony, Weidong Liu, and Xi Luo. \"A constrained ℓ 1 minimization approach to sparse precision matrix estimation.\" Journal of the American Statistical Association 106.494 (2011): 594-607.\n\n*Binary Ising Model*:\n\n- Ravikumar, Pradeep, Martin J. Wainwright, and John D. Lafferty. \"High-dimensional Ising model selection using ℓ 1-regularized logistic regression.\" (2010): 1287-1319.\n\n- Anandkumar, Animashree, et al. \"High-dimensional structure estimation in Ising models: Local separation criterion.\" The Annals of Statistics (2012): 1346-1375.\n\n*General Framework Applicability:* While our algorithm is indeed a general framework applicable to various types of graphs, we chose to focus on GLasso, sparse covariance estimation, and binary Ising models to provide detailed parameters and convergence guarantees specific to each model. These models were selected to demonstrate the versatility and robustness of our framework. A more general assumption might dilute these specific details and potentially miss important information relevant to each individual model. By concentrating on these three, we ensure a comprehensive analysis, which includes specific parameter tuning and convergence properties, thus providing a stronger and more practical contribution to the field.\n\nIn summary, our paper’s focus on these three specific GMs is a deliberate choice to balance the depth and breadth of our contributions, ensuring detailed and practically relevant results. We believe this approach is beneficial for the community, as it provides clear and actionable insights for these widely studied models.\n\n\n\n\n\n\n\n> **Weakness 2:** In the second paragraph of Section 2 (Related Work on Fairness), the author does not discuss related works concerning fairness in GMs estimation. By focusing solely on fairness in unsupervised learning, the paper may limit readers' ability to gain a comprehensive understanding of the research field and the distinctiveness of the proposed method. \n\n> **Limitation 2:** Secondly, the discussion of related work on fairness in GM estimation is insufficient and needs clearer explanations. \n\n**Response:** We appreciate your comments and agree that a more comprehensive discussion on fairness in graph model (GM) estimation would enhance the reader’s understanding. In our revised manuscript, we will include a dedicated paragraph to review the related works, specifically focusing on fairness in GM estimation.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "ohtwafqWW9", "reviewer": "Reviewer_fakh", "summary": "This paper proposes a novel method for estimating graphical models, particularly Gaussian, Covariance, and Ising models, from data, taking fairness into consideration. Fairness is defined on the graph disparity error, i.e., the difference between the loss of the model for a particular group and the minimum loss for that group. The goal is to make the graph disparity errors among differnt groups equal or close. The paper proposes to use non-smooth multi-objective optimization to solve the problem. The paper shows that the method achieves the weak Pareto optimality, and the convergence rates analyzed. Experiments show the efficacy of the method.", "questions": "- Can the proposed method be applied to the estimation of directed graphs, like causal graphs?\n\n- In Algorithm 1, the first step is to initialize local graph estimates for all groups. When the number of groups is large, the sample sizes of certain groups could become small, which may affect the accuracy of the local graph estimates. How will the accuracy of the local graph estimates affect the global graph estimate?", "limitations": "N/A", "rating": 7, "confidence": 2, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "Originality\nDespite extensive research in fair machine learning, this paper tackles a relatively under-studied problem, which is to conduct fair estimation of graphical models. The authors also propose a novel multi-objective proximal gradient method for GM estimation.\n\nQuality\nThe proposed method is theoretical solid. The paper proves the weak Pareto opimality of the method, and also analyzes the global convergence rates for Gaussian, Covaiance, and Ising models. I didn’t check the correctness of the proofs. \nThe experiments were conducted using both synthetic and real-world datasets. The results clearly demonstrate improvement over the baseline method in terms of fairness.\n\nClarity\nThe paper is relatively well-written and orgainzed.\n\nSignificance\nThe experiments demonstrate that the proposed method can provide insights into sensitive group-related features, in addition to achieving fairness in GM estimation.", "weaknesses": "-\tThe optimality analysis requires the convexity of the loss function, and it remains unclear how the performance would be affected if the loss function were non-convex, which is common in machine learning. For example, if we use an encoding network to convert the data X into a representation space and compute the loss in the representation space, then the loss function would a non-convex.\n\n-\tThere is just one baseline used in the experiments, which was published in 2012. It could strengthen the experiment section if modern GM estimation methods are used as baselines.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "> **Weakness 1:** The optimality analysis requires the convexity of the loss function, and it remains unclear how the performance would be affected if the loss function were non-convex, which is common in machine learning. For example, if we use an encoding network to convert the data X into a representation space and compute the loss in the representation space, then the loss function would be non-convex.\n\n**Response:** We appreciate the reviewer’s insightful comment. We completely agree that one limitation of our nonsmooth multi-objective optimization work is the reliance on the convexity of the loss function. However, many existing theoretical results and estimation algorithms for graphical models using single-objective optimization also rely on the convexity of objectives to estimate graphical models. For example, please refer to  QUIC[[HSD2014](https://jmlr.org/papers/volume15/hsieh14a/hsieh14a.pdf)], SQUIC[[BS2016](http://www.icm.tu-bs.de/~bolle/Publicat/squic.pdf)], PISTA[[STY2023](https://arxiv.org/pdf/2205.10027)], GISTA[[GRR2012](https://arxiv.org/pdf/1211.2532)], OBN[[OON2012](https://papers.nips.cc/paper/2012/file/b3967a0e938dc2a6340e258630febd5a-Paper.pdf)], and ALM[[SG2010](https://proceedings.neurips.cc/paper/2010/file/2723d092b63885e0d7c260cc007e8b9d-Paper.pdf)]\n\nOn the other hand, we agree that converting the data $X$ into a representation space, such as using $UU^T$ in the objective or constraints, which can be used for clustering or computing the loss in the representation space, could make the problem nonconvex [[KYC2020](https://www.jmlr.org/papers/volume21/19-276/19-276.pdf)]. However, we know that standard proximal gradient methods for nonconvex optimization or their alternative variants for nonconvex alternating optimization (e.g., for $U$ and graph matrix $\\Theta$) can still be applied [[BST2013](https://link.springer.com/article/10.1007/s10107-013-0701-9)]. Hence, the proximal multi-objective method developed for GMs in this paper can be extended to nonconvex graph optimization. This may require new proof techniques and estimations, which can be considered for future work. We will discuss this in the future work section.\n\n> **Weakness 2:** There is just one baseline used in the experiments, which was published in 2012. It could strengthen the experiment section if modern GM estimation methods are used as baselines.\n\n**Response:** Thank you for your feedback. We appreciate your suggestion to include more modern GM estimation methods as baselines. To address this, we have conducted additional experiments using several state-of-the-art GLasso algorithms. Specifically, we have applied different algorithms, including PISTA[[STY2023](https://arxiv.org/pdf/2205.10027)], GISTA[[GRR2012](https://arxiv.org/pdf/1211.2532)], and OBN[[OON2012](https://papers.nips.cc/paper/2012/file/b3967a0e938dc2a6340e258630febd5a-Paper.pdf)], to GLasso on synthetic data to demonstrate the robustness and effectiveness of our framework beyond ISTA.\n\nISTA was initially chosen as a baseline due to its simplicity and widespread use in sparse inverse covariance estimation. This provided a strong reference point for evaluating our method’s performance. However, recognizing the importance of a comprehensive evaluation, we have now compared our framework against more advanced baselines.\n\nThe results, detailed in the revised version of our paper and the **attached one-page PDF file**, show that our proposed framework consistently outperforms these modern baselines in terms of enhancing fairness while maintaining competitive performance. This comprehensive evaluation underscores the advantages of our approach across various scenarios.\n\n> **Question 1:** Can the proposed method be applied to the estimation of directed graphs, like causal graphs?\n\nThank you for your insightful question. While our proposed method is primarily designed for undirected graphical models like Gaussian Graphical Models and Ising Models, the principles of our fairness-aware optimization framework could potentially be adapted for directed graphs, such as causal graphs. Unlike many fair supervised methods[[GKK2019](https://ieeexplore.ieee.org/abstract/document/8437807),[JCM2022](https://openaccess.thecvf.com/content/CVPR2022/html/Jung_Learning_Fair_Classifiers_With_Partially_Annotated_Group_Labels_CVPR_2022_paper.html),[PLL2022](https://openaccess.thecvf.com/content/CVPR2022/html/Park_Fair_Contrastive_Learning_for_Facial_Attribute_Classification_CVPR_2022_paper.html)], our method does not use label information, and unlike previous methods for fair clustering or community detection[[CKL2017](https://arxiv.org/abs/1802.05733),[BCF2019](https://arxiv.org/abs/1901.02393)], our method does not use node attributes. Our fairness metric preliminarily works with the disparity of global and local losses, i.e., using group-specific and global data to construct the objective. Hence, it is not specific graph-dependent as a fairness metric. Although the objective function differs for various graphical models, our new fairness notion (pairwise graph disparity error) remains the same.\nAlthough our current work focuses on undirected graphs, we believe that with appropriate adjustments, our framework could be extended to handle directed graphs, ensuring fairness without disrupting causal dependencies. This presents a promising area for future research. Thank you for your constructive question, which has helped us consider the broader applicability of our method.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:
We thank all the reviewers for their time in providing feedback and questions about our submitted paper. Below, we summarize the main issues that the reviewers raised, along with a summary of our responses. Furthermore, we provide **new real experiments** to address the reviewers' comments, which are **attached as an additional PDF file**.

* **Fairness Metrics (By nXfy):** The reviewer asked about the use of only one criterion of fairness, achieving equal loss among subgroups, and the lack of widely accepted metrics for fairness comparison.

* **Computational Complexity (for large \(P\) and \(N\)) (By nXfy):** The reviewer noted concerns about the computational complexity of our method, particularly for large \(P\) and \(N\) values.

* **Limited Baselines (By tjnm, fakh):** Reviewers tjnm and fakh pointed out that the paper could benefit from more extensive comparisons with existing fairness-aware methods in graphical models.

* **Related Work on Fairness (By 1oDX):** The reviewer suggested that the related works section should provide more comprehensive information about fairness in GMs estimation and the distinctiveness of our method.

* **Non-convex Objective (By fakh)** The reviewer raised concerns about the performance of our method under non-convex loss functions, which are common in machine learning.


We have taken several actions to address these concerns. More specifically:

**A1:** Our primary focus on the difference in loss between subgroups as a fairness criterion stems from the unsupervised nature of our task. In unsupervised learning, we lack labeled data that are typically used to define fairness criteria such as demographic parity (DP)[[JHF2022](https://par.nsf.gov/servlets/purl/10397778)] and equalized odds (EO)[[RBC2020](https://proceedings.neurips.cc/paper/2020/hash/03593ce517feac573fdaafa6dcedef61-Abstract.html)]. Incorporating other fairness criteria like DP and EO requires labeled data to assess prediction distributions and accuracy across groups, which is less applicable in our context. Thus, we introduced the *graph disparity error* based on group-specific and global losses. This metric aligns with DP by promoting a balanced representation of subgroups through the loss function.

While future research could explore more widely accepted fairness metrics in supervised learning, our current work establishes a disparity error-based approach. Our framework is novel in this area and flexible enough to incorporate other fairness objectives, and we have included discussions of these points in the revised paper.


      Further details are provided in response to Reviewer nXfy.

**A2:** We appreciate the reviewer's observation regarding the time-consuming nature of our algorithm. The time complexity arises partly from the local graph learning phase, which can be accelerated using existing fast graphical model algorithms. Additionally, the growing number of objectives in the multi-objective optimization solver increases time complexity. To reduce the computational load, we propose selecting a subset of objectives randomly in each iteration.

To validate our approach, we conducted additional experiments using various optimization algorithms for GLasso and found significant improvements in learning efficiency with our proposed methods. These enhancements are discussed in the revised paper. 

      Further details are provided in response to Reviewers tjnm and fakh.

**A3:** We have now included additional experiments applying our method to several state-of-the-art GLasso algorithms beyond ISTA. The results show that our framework consistently outperforms advanced baselines in enhancing fairness while maintaining competitive performance. These comparisons highlight the advantages of our approach and are detailed in the revised version of our paper. 

      Further details are provided in the response to Reviewer 1oDX.

**A4:** We agree that a more comprehensive discussion on fairness in GM estimation would enhance the reader’s understanding. In the revised manuscript, we have included a dedicated paragraph reviewing recent advancements in fairness for graphical models, such as Fair GLASSO and fair structure learning in heterogeneous graphical models. These additions clarify the context and distinctiveness of our work.

      Further details are provided in the response to Reviewer 1oDX.

**A5:** While our current work assumes the convexity of the loss function, many theoretical results in graphical models use convex objectives. However, proximal gradient methods can converge even in non-convex settings, validating their robustness and flexibility. Future work aims to enhance the applicability and robustness of our framework to broader practical scenarios, including non-convex settings. 
  
      Further details are provided in the response to Reviewer fakh.

Meta-review: The reviewers were generally positive on this paper, mostly because the authors did an excellent job in addressing their concerns during the response period. I hope that they will update their paper in such a way that the main body of the paper is improved similarly.
Decision: Accept (poster)