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Alternating Bi-Objective Optimization for Explainable Neuro-Fuzzy Systems

Qusai Khaled, Uzay Kaymak, Laura Genga

TL;DR

This work tackles the non-convex nature of the accuracy–explainability trade-off in neuro-fuzzy systems by introducing X-ANFIS, an alternating bi-objective gradient-based optimization that decouples performance and explainability. It leverages Cauchy membership functions for stable training and interleaves a differentiable explainability loss (X-pass) to enforce target distinguishability among adjacent fuzzy sets, updating antecedents while keeping consequents via regularized least squares. Across nine real-world UCI regression datasets, X-ANFIS achieves high semantic distinguishability (around $D \approx 0.50$) with competitive $R^2$, often recovering non-convex Pareto regions inaccessible to scalarized approaches and producing spatially coherent MF partitions. The findings show gradient-based optimization, when structured as alternating objectives, can yield explainable neuro-fuzzy models, and motivate extending these principles beyond Takagi–Sugeno to other fuzzy paradigms such as Mamdani systems.

Abstract

Fuzzy systems show strong potential in explainable AI due to their rule-based architecture and linguistic variables. Existing approaches navigate the accuracy-explainability trade-off either through evolutionary multi-objective optimization (MOO), which is computationally expensive, or gradient-based scalarization, which cannot recover non-convex Pareto regions. We propose X-ANFIS, an alternating bi-objective gradient-based optimization scheme for explainable adaptive neuro-fuzzy inference systems. Cauchy membership functions are used for stable training under semantically controlled initializations, and a differentiable explainability objective is introduced and decoupled from the performance objective through alternating gradient passes. Validated in approximately 5,000 experiments on nine UCI regression datasets, X-ANFIS consistently achieves target distinguishability while maintaining competitive predictive accuracy, recovering solutions beyond the convex hull of the MOO Pareto front.

Alternating Bi-Objective Optimization for Explainable Neuro-Fuzzy Systems

TL;DR

This work tackles the non-convex nature of the accuracy–explainability trade-off in neuro-fuzzy systems by introducing X-ANFIS, an alternating bi-objective gradient-based optimization that decouples performance and explainability. It leverages Cauchy membership functions for stable training and interleaves a differentiable explainability loss (X-pass) to enforce target distinguishability among adjacent fuzzy sets, updating antecedents while keeping consequents via regularized least squares. Across nine real-world UCI regression datasets, X-ANFIS achieves high semantic distinguishability (around ) with competitive , often recovering non-convex Pareto regions inaccessible to scalarized approaches and producing spatially coherent MF partitions. The findings show gradient-based optimization, when structured as alternating objectives, can yield explainable neuro-fuzzy models, and motivate extending these principles beyond Takagi–Sugeno to other fuzzy paradigms such as Mamdani systems.

Abstract

Fuzzy systems show strong potential in explainable AI due to their rule-based architecture and linguistic variables. Existing approaches navigate the accuracy-explainability trade-off either through evolutionary multi-objective optimization (MOO), which is computationally expensive, or gradient-based scalarization, which cannot recover non-convex Pareto regions. We propose X-ANFIS, an alternating bi-objective gradient-based optimization scheme for explainable adaptive neuro-fuzzy inference systems. Cauchy membership functions are used for stable training under semantically controlled initializations, and a differentiable explainability objective is introduced and decoupled from the performance objective through alternating gradient passes. Validated in approximately 5,000 experiments on nine UCI regression datasets, X-ANFIS consistently achieves target distinguishability while maintaining competitive predictive accuracy, recovering solutions beyond the convex hull of the MOO Pareto front.
Paper Structure (15 sections, 20 equations, 3 figures, 6 tables, 1 algorithm)

This paper contains 15 sections, 20 equations, 3 figures, 6 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Explainability of fuzzy systems
  3. Takagi-Sugeno System
  4. Explainability Taxonomies
  5. The Non-Convex Performance-Explainability Trade-off
  6. Methodology
  7. Cauchy Forward Pass
  8. Exploding Gradients for Small Initialization
  9. Vanishing Updates for Large Initialization
  10. Explainability Pass
  11. Evaluation
  12. Parameter Update Trajectory Analysis
  13. Performance and Distinguishability
  14. Heatmap Experiments
  15. Discussion and Conclusion

Figures (3)

  • Figure 1: Effect of MF initialization on ANFIS parameter update trajectories for the Combined Cycle Power Plant dataset, for humidity and vacuum features. Columns correspond to different initialized spread values $\sigma$/$\gamma$ values; left three columns are Gaussian MFs, right three are Cauchy MFs.
  • Figure 2: Training results of 500 models depicted as R² versus mean distinguishability for ANFIS, X-ANFIS and MO-ANFIS models. MO-ANFIS Pareto front shown in red. Classic ANFIS shown in pink. X-ANFIS shown in green.
  • Figure 3: Comparison of X-ANFIS and MO-ANFIS. Top row presents Kernel Density Estimation heatmaps of MF centers for features temperature, pressure, humidity, vacuum, named as F1–F4. Bottom row displays the corresponding MF shapes for F2, with red circles indicating center locations.