Interpretable and Fair Generalized Additive Neural Networks via Multi-objective Learning

Motivation

• NN-based GAMs are selected for their high performance and scalability, but existing work emphasizes predictive accuracy without guaranteeing understandable explanations.GAMs provide global and local explanations, yet interpretability quality is rarely explicitly measured or optimized.Source: §1 · p. 2

Interpretability metrics

• The paper introduces smoothness and monotonicity as explicit quantitative metrics for directly evaluating and improving global interpretability in NN-based GAMs.Unlike implicit regularization, the metrics can be quantified, monitored, and optimized independently, and their effectiveness is experimentally verified.Source: §1 · pp. 2–3

Fairness gap

• Fairness research primarily addresses black-box models, leaving fairness in self-interpretable models, especially GAMs, comparatively underexplored.The paper focuses on group fairness, which requires equal treatment of groups defined by sensitive attributes.Source: §1 · p. 2

MONBM framework

• MONBM formulates trustworthy-model construction as multi-objective learning over accuracy, interpretability, and fairness, using evolutionary optimization and partial retraining of the feature-specific output layer.Retraining only this layer enables efficient evolutionary multi-objective optimization for large-scale networks.Source: §1 · pp. 2–3

Analysis and validation

• MONBM produces NBMs with diverse trade-offs and uses their self-explanations to investigate relationships and underlying causes among trustworthiness objectives.The paper reports desirable multidimensional trade-offs and competitiveness with state-of-the-art self-interpretable and black-box models.Source: §1 · p. 3

2.1. Generalized Additive Model (GAM)

• GAMs model predictions by summing independent univariate shape functions, enabling global visual explanations of feature effects and local explanations of feature contributions.NN-based GAMs provide precise decision descriptions; NBM improves NAM scalability by using one shared MLP basis function and feature-specific linear combinations.Source: §2.1 · pp. 3–4
• Standard GAMs prioritize transparency over extreme predictive performance, while extensions such as GA2M improve interactions at the cost of interpretability and computational simplicity.MONBM remains centered on standard GAM formulations but can be extended to interaction models and specialized NBM-based modalities such as time series and concept-level vision models.Source: §2.1 · pp. 4–5

2.2. Interpretability of NN-based GAM

• NN-based GAM interpretability comprises local explanations of individual predictions and global explanations of feature–target relationships, with both requiring explicit evaluation and improvement methods.Global interpretability is associated with smoothness and monotonicity, while local interpretability concerns explanation complexity.Source: §2.2 · p. 5, §2.2.1 · p. 5, §2.2.2 · p. 6

2.2.1. Local Interpretability (LI)

• Local interpretability is simpler when feature-importance scores concentrate on fewer dominant features, but GAMs lack methods specifically designed to reduce local explanation complexity.Prior µC and G metrics assess local explanation complexity, primarily in post-hoc settings, while feature-selection approaches only indirectly simplify GAM explanations.Source: §2.2.1 · pp. 5–6

2.2.2. Global Interpretability (GI)

• Global interpretability should exhibit smoothness and monotonicity, yet NN-based GAMs generally lack explicit quantitative metrics and direct optimization mechanisms for these properties.Existing regularization may incidentally smooth functions but primarily targets generalization, while monotonicity penalties do not provide independent multi-objective optimization within NN-based GAMs.Source: §2.2.2 · p. 6

2.3. Group Fairness in Machine Learning

• Group fairness is assessed in this paper using demographic parity, while existing fairness methods are difficult to adapt from black-box architectures to additive GAM structures.Kraus et al. addressed fairness in NN-based GAMs by reducing sensitive-attribute influence, but the paper reports limited fairness improvement and instead incorporates fairness explicitly in MONBM.Source: §2.3 · pp. 6–7

2.4. Summary and Research Positioning

• MONBM jointly optimizes accuracy, fairness, local interpretability, and global interpretability, producing Pareto-optimal models that expose trade-offs among these dimensions.Existing GAM studies generally address trustworthiness dimensions separately or through single-objective regularization, whereas MONBM treats them as explicit objectives in multi-objective evolutionary learning.Source: §2.4 · p. 7

3.1. Smoothness and Monotonicity Metrics

• Monotonicity is optional for continuous features when supported by domain knowledge, whereas smoothness is generally required for continuous features and neither is imposed on discrete features.Non-monotonic patterns can naturally reflect real data, so monotonicity is not treated as a universal interpretability requirement.Source: §3.1 · p. 8, §3.3.2 · p. 11
• The smoothness metric Ms and monotonicity metrics Mm↑ and Mm↓ explicitly quantify global interpretability from shape-function changes and trends, and lower values are better.Ms uses threshold-based changes, while Mm distinguishes increasing and decreasing monotonicity; interval weights can encode region-specific domain knowledge.Source: §3.1 · pp. 8–9

3.2. MONBM Framework

• MONBM formulates optimization over selected accuracy, interpretability, and fairness metrics, using evolutionary reproduction and environmental selection to promote convergence and diversity.The general framework repeatedly selects models, generates offspring, partially trains them, evaluates metrics, updates the population, and returns the optimized set.Source: §3.2 · p. 9
• MONBM uses evolutionary multi-objective learning to return non-dominated self-interpretable models spanning different accuracy, interpretability, and fairness trade-offs.The framework evaluates populations, generates and partially trains offspring, applies environmental selection, and returns optimized non-dominated solutions.Source: §3.2 · p. 9

3.3.1. Model

• The instantiation pre-trains a high-accuracy NBM, freezes shared basis functions, and evolves only the feature-specific layer through partial retraining and SRA-based selection.This reduces the search space and optimization complexity while combining evolutionary search with partial gradient-based training.Source: §3.3 · p. 10, §3.3.1 · p. 10

3.3.2. Evaluation Metrics as Optimization Objectives

• The instantiations optimize four objectives: cross-entropy for accuracy, DP for fairness, G-mean-based local interpretability, and combined smoothness-monotonicity global interpretability.Global interpretability combines smoothness and monotonicity into one GI objective, while the framework can keep them separable if needed.Source: §3.3.2 · pp. 10–11

3.3.3. Optimizer

• SRA, weight crossover, isotropic Gaussian perturbation, and partial training generate diverse Pareto solutions instead of requiring manually tuned scalarization weights.Four variants optimize CE-DP, CE-LI, CE-GI, or CE-DP-LI-GI, named MONBM-CD, MONBM-CL, MONBM-CG, and MONBM-CDLG.Source: §3.3.3 · pp. 11–12

4.1. Experimental Setup

• Experiments used Adult, Bank, COMPAS, Default, Dutch, and LSAT with common encoding, Z-score normalization, and 4:1 train-test splits.The study compared MONBM variants with fairness, feature-selection, and multi-objective baselines where applicable; global-interpretability variants lacked existing comparison methods.Source: §4.1.1 · p. 13, §4.1.2 · p. 13

4.1.3. Parameter Settings

• The experiments used population size τ = 100, mutation strength ξp = 0.01, at most 200 generations, partial-training hyperparameter N = 3, and 15 random-seed trials.Smoothness tolerance used η = 1.0; smaller η imposed stricter smoothness, while monotonicity requirements were specified only for selected features based on task and stakeholder needs.Source: §4.1.3 · pp. 13–14

4.1.4. Evaluation Criteria

• Results were evaluated using objective metrics, Pareto-set HV, MS, and PD, plus mutual-dominance analyses against population-based and single-solution approaches.HV was the only Pareto-compliant metric and, like MS and PD, was interpreted as better when larger.Source: §4.1.4 · p. 14

4.2.1. Validation of the Proposed Metrics (RQ1)

• As GI improved, shape functions became smoother and more monotonic while preserving primary functional trends, demonstrating feasible high-global-interpretability solutions without strict smoothness or perfect monotonicity.These visual changes consistently aligned with GI values across datasets, supporting the proposed metrics’ validity.Source: §4.2.1 · pp. 14–15, Figure 4 · p. 15

4.2.2. Analyzing the Relationship Between Dimensions (RQ2)

• Across six datasets, extreme fairness reduced accuracy by 17.06% on Adult, COMPAS, and Dutch but only 2.15% on the other three, whereas moderate fairness reduced accuracy by 0.48% on average.Moderate fairness mainly adjusts the sensitive attribute, while extreme fairness may alter many features and remove their decision influence.Source: §4.2.2 · p. 16, Table 2 · p. 16
• On MONBM-CL, moderate LI improvement (50% reduction from its extremes) decreased test accuracy 2.03% on average, whereas extreme LI decreased it 12.59% on average, as sparsity removed most features’ influence.Source: §4.2.2 · p. 17
• Global interpretability had the weakest accuracy trade-off: moderate improvement changed accuracy by 0.12% and extreme improvement by 0.85% on average.GI mainly modifies unsmooth or non-monotonic regions of continuous features, leaving discrete features and overall predictive performance largely unchanged.Source: §4.2.2 · pp. 18–19, Table 4 · p. 20
• In the four-objective setting, global interpretability affected accuracy least, fairness next, and local interpretability most, while cross-objective effects were generally beneficial but dataset-dependent.On Dutch, pursuing fairness worsened local interpretability, pursuing global interpretability worsened both, and pursuing local interpretability worsened DP by up to 0.928.Source: §4.2.2 · p. 20, Table 5 · p. 20
• Fairness and local interpretability generally reinforce each other on Adult but trade off on Dutch, where retaining sensitive features can produce extremely unfair decisions.Improving DP often reduces feature influence and improves LI on Adult, whereas Dutch retains many important features; improving LI can retain sex as the only important feature, creating extreme unfairness and a persistent negative DP–LI trend.Source: §4.2.2 · pp. 21, 22, 24
• Global interpretability only weakly aligns with local interpretability and fairness, with asymmetric interactions and a Dutch conflict driven primarily by smoothness rather than monotonicity.Better GI can remove irregular shape-function fluctuations and moderately reduce feature influence, but LI rarely improves GI; on Dutch, improving GI can make sex the most important feature and worsen fairness.Source: §4.2.2 · pp. 22–23
• Moderate fairness and local-interpretability improvements have little accuracy cost, but extreme improvements can substantially reduce performance, while global interpretability has the least accuracy impact.The reported accuracy effects are strongest for extreme LI and fairness, whereas GI mainly modifies portions of continuous-feature shape functions that violate smoothness or monotonicity.Source: §4.2.2 · p. 23
• The four-objective MONBM-CDLG generally underperforms two-objective instantiations because additional objectives weaken selection pressure and make exploration–convergence balance harder.Adaptive selection pressure partially mitigates degradation, but specialized many-objective algorithms and adaptive operators remain future work.Source: §4.2.2 · p. 24

4.2.3. Comparison with Baseline Methods (RQ3)

• MONBM-CD and MONBM-CL achieve competitive accuracy, fairness, and interpretability while providing diverse non-dominated trade-offs that single-point baselines lack.MONBM-CD shows better convergence and diversity than FairEMOL, frequently dominates other methods, and remains rarely dominated except by some LPP-NBM solutions; MONBM-CL significantly outperforms SNAM and NPLAM and generally outperforms GAMI-Net.Source: §4.2.3 · pp. 24–26

4.3. Case Study: Threshold-Based Model Selection

• A case study narrows 43 non-dominated Adult solutions to 13 knee points and then to two feasible deployment candidates using accuracy-within-2% and DP-below-0.05 thresholds.Stakeholders can choose between the remaining candidates according to predictive-performance or fairness priorities; the procedure illustrates, rather than prescribes, a universal selection strategy.Source: §4.3 · pp. 26–27

Scope of interpretability conclusions

• Interpretability relationships may generalize from NN-based GAMs to other GAMs and self-interpretable models through analogous global or local explanation complexity.Relevant complexity factors include the number of linear-model terms, decision-tree depth, and number of decision rules.Source: §5 · p. 27

Scope of fairness conclusions

• The observed trade-off patterns largely persist across DP, EOP, and EOD, but broader fairness-metric generalizability requires further validation [10].The paper plans to analyze accuracy, interpretability, and multiple fairness metrics simultaneously.Source: §5 · pp. 27–28

Structured-data applicability

• MONBM is currently designed for tabular data and is not directly applicable to raw pixels, but concept extraction could enable extensions to structured modalities.Future work must empirically validate concept-level additive architectures on large-scale structured datasets.Source: §5 · p. 28

Overfitting

• RQ2 solutions showed consistent training- and test-set performance, indicating limited overfitting, although non-dominance inconsistencies may remain.A validation set is proposed to mitigate this residual risk.Source: §5 · pp. 28–29

• The paper introduces smoothness and monotonicity as quantitative interpretability metrics and proposes MONBM to jointly optimize accuracy, fairness, and local and global interpretability.MONBM applies evolutionary multi-objective learning to GAMs and retrains only the feature-specific layer of a pretrained NBM for scalability to large networks.Source: §6 · p. 29
• Fairness, local interpretability, and global interpretability each trade off with accuracy, while their relationships can be complementary or conflicting depending on the dataset and problem.The metrics and MONBM are validated through explanation visualizations and comparisons with baseline methods.Source: §6 · p. 29
• Future work will study non-uniform smoothness control, broader model and fairness-metric validation, high-dimensional scalability, and trade-offs involving robustness and security.Source: §6 · p. 29

Data availability

• The experiments use publicly available, open-source datasets, and the manuscript provides their details and access information.Source: Data availability · p. 29

CRediT authorship contribution statement

• Ziming Wang handled the original draft, methodology, data curation, and validation, while Changwu Huang, Ke Tang, Yew-Soon Ong, and Xin Yao contributed to manuscript review and editing and supervision.Huang, Tang, and Ong also handled funding acquisition; Huang contributed conceptualization, and Yao contributed project administration and resources.Source: CRediT authorship contribution statement · p. 29

Dataset Under Consideration

• The experiments use six datasets widely used in fairness research.Table S-I reports each dataset’s number of samples, total features, discrete features, numerical features, and sensitive attribute.Source: S-I. Dataset Under Consideration · p. 33

S-II. Algorithmic and Computational Properties of MONBM

• The analysis covers MONBM’s computational budget, algorithmic complexity, generalization behavior, robustness, and practical execution time to assess scalability.This subsection frames the computational and practical properties examined for the proposed framework.Source: S-II. Algorithmic and Computational… · p. 33

S-II.1. Computational Budget Analysis

• MONBM retrains approximately 1,500 feature-specific parameters for 200 generations on a pre-trained NBM, compared with 1,200–5,000 epochs or 200 generations for the baselines.The pre-trained NBM is trained for 1,000 epochs, while FairEMOL uses approximately 68,000 parameters.Source: S-II.1. Computational Budget Analysis · p. 34
• Updating only ∼2% of NBM parameters keeps MONBM’s per-model cost relatively low despite optimizing a population of 100 models per generation.Unlike single-point methods, MONBM also produces a diverse set of trade-off solutions in one run.Source: S-II.1. Computational Budget Analysis · p. 34

S-II.2. Algorithmic Complexity

• For k objectives, population size τ, data dimension D, and data size n, Algorithm 2 analyzes reproduction, partial training, indicator evaluation, and environmental selection per generation.Accuracy and fairness evaluations take O(n·D), GI takes O(D), LI takes O(n·D log D), and SRA selection takes O(k · τ^2).Source: S-II.2. Algorithmic Complexity · p. 34
• Per-generation time complexity is O(kτ^2 + τ · n · D log D), with quadratic dependence on population size and near-linear scaling with dataset size and dimensionality.The expression combines reproduction, partial training, fitness evaluation, and SRA-based environmental selection costs.Source: S-II.2. Algorithmic Complexity · p. 34
• MONBM’s overall space complexity is O((τ + n)D), comprising population parameters O(τ · D) and training data O(n · D).Frozen, shared bases enable linear scaling with feature dimension D.Source: S-II.2. Algorithmic Complexity · p. 34

S-II.3. Robustness and Generalization Analysis

• Across 15 independent trials, consistent training and test performance and small train–test differences indicate stable generalization from limited interactions, restricted optimization, and explicit smoothness regularization.MONBM evolves only approximately 2% of total parameters—the linear-combination weights over frozen nonlinear basis functions—while the smoothness objective discourages high-frequency oscillations.Source: S-II.3. Robustness and Generalization… · p. 34

S-II.4. Wall-Clock Time Performance

• MONBM’s execution takes approximately 12.4 times a single NBM training run, yet one run produces diverse accuracy–fairness–interpretability trade-off solutions, reducing the cumulative cost of repeated scalarized optimization.The comparison uses a population of τ = 100 models evolved for 200 generations and evaluates six benchmark datasets under single-threaded execution.Source: S-II.4. Wall-Clock Time Performance · p. 35

S-III.1. Hyperparameters of Pre-trained NBM

• The pre-trained NBM uses AdamW, cosine annealing, 1000 epochs, batch size 1024, a three-layer 256–128–128 MLP, and basis output B = 100, with dataset-specific settings selected by validation random search.The search varies learning rate, weight decay, output penalties, dropout, and feature dropout over the reported continuous and discrete ranges.Source: S-III.1. Hyperparameters of… · p. 35

S-III.2. Hyperparameters of Baseline Methods

• Fairness baselines use AI Fairness 360 [64] or author-recommended implementations, while FairEMOL is expanded to a 256–128–128 MLP for fair comparison.The configured methods include LFR, reweighting, ROC, LPP, Oracle [37], and FRAPPÉ.Source: S-III.2. Hyperparameters of Baseline… · p. 35
• SNAM and GAMI-Net vary λ and η across ten values to obtain models with different degrees of local interpretability.The tested ranges are λ = {0.01, 0.03, 0.05, 0.08, 0.1, 0.3, 0.5, 0.7, 0.9, 1} and η = {5e-4, 1e-3, 3e-3, 5e-3, 8e-3, 0.01, 0.02, 0.03, 0.04, 0.05}.Source: S-III.2. Hyperparameters of Baseline… · pp. 35–36

S-IV. Sensitivity Analysis of the Smoothness Scaling Factor η

• As η decreases, the smoothness constraint becomes more stringent through a smaller tolerance threshold k, shifting the Adult-dataset Pareto front toward lower Ms values and smoother shape functions.The sensitivity study isolates accuracy (CE) and smoothness (Ms) in a two-objective optimization that excludes monotonicity (Mm), with configurations visualized in Fig. S-1.Source: S-IV. Sensitivity Analysis of the… · pp. 36–37, Figure S-1 · p. 36
• The η sensitivity analysis demonstrates that MONBM lets users calibrate smoothness to transition between higher-accuracy prediction and highly interpretable, smooth global explanations.This controllability follows from the observed relationship between η, the tolerance threshold k, and the Ms-based Pareto front.Source: S-IV. Sensitivity Analysis of the… · pp. 36–37

S-V. Required Monotonic Features

• Monotonicity requirements were specified separately for each dataset, with increasing and decreasing features listed in Table S-IV.The required features include examples such as “hours_per_week” for Adult and “edu_level” for Dutch; settings should ultimately reflect specific practical needs.Source: S-V. Required Monotonic Features · p. 37
• The Adult “hours_per_week” requirement assumes that higher weekly hours increase the probability of earning more than $50,000, but this assumption is intended only to validate the metrics and methodology.In practice, monotonicity settings should be determined according to specific needs.Source: S-V. Required Monotonic Features · p. 37

S-VI. Evaluation Criteria

• MONBM was compared with population-based methods using mutual dominance relationships between all pairs of solutions, following.With 100 solutions from each method, the procedure performs 10,000 pairwise comparisons and reports MONBM’s dominance probability, such as 0.300 when it dominates in 3,000 comparisons.Source: S-VI. Evaluation Criteria · p. 37
• For single-solution methods, the evaluation follows three mutual-dominance metrics from: Dominate, Incomparable, and Dominated.A larger Dominate value means baseline solutions are dominated in more trials by solutions found by MONBM.Source: S-VI. Evaluation Criteria · p. 37
• The Incomparable metric measures the average proportion of MONBM solutions incomparable with baseline solutions, whereas Dominated measures the proportion dominated by baseline solutions.Larger values indicate more incomparable or MONBM-dominated models, respectively, and these metrics align with the population-based proportions of mutually non-dominated and dominated solutions.Source: S-VI. Evaluation Criteria · p. 38

S-VII. Results on the Training Set

• Training-set results supplement the selected solutions from Section 4.2.2 by examining overfitting across MONBM instantiations.Tables S-V and S-VI compare mean ACC with DP or LI for three selected final-solution points over 15 trials.Source: S-VII. Results on the Training Set · pp. 38–39
• Tables S-VII and S-VIII extend training-set comparisons to ACC with GI and to ACC, DP, LI, and GI across selected MONBM-CG and MONBM-CDLG solutions.The comparisons use 15 trials; Table S-VIII includes four extreme points and a Majority Baseline row for constant majority-class prediction.Source: S-VII. Results on the Training Set · pp. 39–40

S-VIII. Sensitivity Analysis to Fairness Metrics: Results on Equalized Odds (EOD)

• A 50% EOD reduction decreases average ACC by only 0.64% across six datasets, whereas EOD ≈0 causes substantial declines on Adult, COMPAS, and Dutch but moderate effects on Bank, Default, and LSAT.These results confirm that the accuracy–fairness trade-off is robust across fairness definitions.Source: S-VIII. Sensitivity Analysis to… · p. 41
• EOD-optimized shape functions and feature-importance shifts closely resemble DP results, although satisfying EOD still requires distinct sensitive-attribute weights to compensate for disparate group error distributions.This supports the generalizability of MONBM’s interpretability profiles and trade-off insights beyond a single fairness metric.Source: S-VIII. Sensitivity Analysis to… · p. 41, Figure S-2 · p. 42

S-IX. Stability and Reliability of Explanations Across Independent Trials

• Across 15 trials, feature-importance standard deviations remain relatively low near “Best CE” solutions but increase at “Middle” and “Extreme” fairness or local-interpretability points.The main trade-off trends remain consistent, but explanation stability is linked to predictive performance and position on the trade-off surface.Source: S-IX. Stability and Reliability of… · pp. 41, 43
• Strong fairness or interpretability constraints can reduce accuracy and flatten the optimization landscape, allowing multiple feature configurations and increasing sensitivity to sampled training data.Because middle points are run-specific knee points on shifting Pareto fronts, they represent varying compromise solutions that contribute to aggregate fluctuations.Source: S-IX. Stability and Reliability of… · pp. 41, 43
• Variation in per-feature importance does not indicate an unreliable method because GAM explanations are 100% faithful to their models, making cross-run shape-function differences primarily a consequence of training-sample variation.The observed variability therefore reflects model changes across trials rather than shortcomings of the explanation framework.Source: S-IX. Stability and Reliability of… · p. 43

S-X. Additional Explanation Results

• Additional global-explanation analyses examine smoothness, monotonicity, fairness effects, and objective trade-offs across MONBM-CG, MONBM-CD, and MONBM-CDLG solutions.Figures compare best, middle, and extreme objective points, including distributions across non-dominated knee-point solutions and pairwise relationships among DP, LI, and GI.Source: S-X. Additional Explanation Results · p. 43, Figure S-3 · p. 43, Figure S-4 · p. 44
• Additional local-explanation analyses report average feature contributions and standard deviations across 15 trials for LI and GI objectives, while comparing four-objective solutions and accuracy–interpretability knee points.The supplementary figures cover Adult, Bank, and COMPAS datasets and include both arbitrary-trial comparisons and multi-trial averages.Source: Figure S-7 · p. 47, Figure S-8 · p. 48, Figure S-11 · p. 51

Experimental Setup

• The study decomposes MONBM-CDLG into MONBM-CDLS and MONBM-CDLM and evaluates their extreme final-set solutions on Adult and Dutch datasets.The formulations optimize accuracy, fairness, local interpretability, and either smoothness or monotonicity; results are summarized in Tables S-X and S-XI over 15 trials.Source: S-XI. Ablation Analysis of Global… · p. 56

Ablation Findings

• Both smoothness and monotonicity impose limited accuracy degradation relative to fairness and local interpretability, mainly reshaping learned shape-function geometry rather than the overall decision boundary.Extreme global-constraint solutions retain relatively high predictive performance, while extreme demographic-parity optimization causes larger accuracy reductions.Source: S-XI. Ablation Analysis of Global… · p. 56
• Local interpretability and global constraints exhibit a persistent trade-off because sparse feature usage differs mechanistically from smooth or monotonic feature-response structure.Consequently, improving local interpretability cannot always coincide with improving smoothness or monotonicity.Source: S-XI. Ablation Analysis of Global… · p. 56
• On Dutch, monotonicity optimization preserves fairness (DP = 0.054), whereas smoothness optimization produces much larger disparity (DP = 0.576), implicating smoothness in the fairness–interpretability conflict.These results come from the Best Mm solution in Table S-XI and the Best Ms solution in Table S-X.Source: S-XI. Ablation Analysis of Global… · pp. 56–57
• On Dutch, smoothness suppresses local fluctuations and reduces other features’ contributions, allowing the sensitive attribute “sex” to dominate and exacerbate bias.The ablation therefore indicates that global interpretability is heterogeneous, with fairness interactions depending strongly on the data distribution.Source: S-XI. Ablation Analysis of Global… · p. 57

S-XII. Preliminary Exploration on Adaptive Selection Pressure

• Adaptive pressure adjustment is promising for mitigating weakened selection pressure and convergence limitations caused by severe conflicts among four objectives.The study notes that the performance gap relative to lower-dimensional optimization is not fully eliminated, motivating more advanced adaptive mechanisms and many-objective operators.Source: S-XII. Preliminary Exploration on… · pp. 57–58
• The adaptive strategy dynamically increases convergence pressure during optimization, broadening early exploration and strengthening later selection.The stochastic-ranking parameter pc controls the balance between convergence-oriented ranking and diversity preservation; the quadratic schedule changes it from pstart = 0.45 to pend = 0.8.Source: S-XII. Preliminary Exploration on… · p. 57
• Adaptive selection pressure generally improved solution quality, especially for fairness-, local-interpretability-, and global-interpretability-oriented solutions, while maintaining competitive predictive performance.The adaptive variant was evaluated against fixed-pressure MONBM on Adult using four extreme solutions from the final solution set over 15 trials.Source: S-XII. Preliminary Exploration on… · pp. 57–58

S-XIII. Comparing MONBM Instantiations with Baseline Methods on All Four Objectives

• The all-four-objective comparison covered ACC, DP, LI, and GI for MONBM-CD, MONBM-CL, and MONBM-CDLG against their corresponding baselines.MONBM-CDLG was compared with all ten baseline methods, excluding FairEMOL because its black-box design lacks local and global interpretability.Source: S-XIII. Comparing MONBM Instantiations… · p. 58
• Across four-objective comparisons, MONBM-CD and MONBM-CL were less likely to dominate or be dominated by other methods than in their corresponding two-objective comparisons, sometimes reaching 0.0.The reduction reflects conflicts among the added objectives, which were not jointly optimized by the baseline methods; MONBM-CD retained a non-negligible dominance probability from its accuracy and fairness advantages.Source: S-XIII. Comparing MONBM Instantiations… · pp. 58–59
• MONBM-CDLG was never dominated by any of the ten baselines across ACC, DP, LI, and GI, while its primary benefit was producing diverse, competitive compromise solutions across the trustworthiness spectrum.The “Be-Do.” probability was consistently 0.000; against weaker baselines such as LFR-NBM, its mean “Do.” probability reached 0.789, whereas complete dominance was rare against most competitive baselines.Source: S-XIII. Comparing MONBM Instantiations… · pp. 58–59