[Paper] Alternating Bi-Objective Optimization for Explainable Neuro-Fuzzy Systems

Source: arXiv - 2602.19253v1

Overview

The paper introduces X‑ANFIS, a new training scheme for adaptive neuro‑fuzzy inference systems that simultaneously optimizes predictive performance and explainability. By alternating two gradient‑based objectives, the authors show that it is possible to reach high‑quality fuzzy rule sets without the heavy computational cost of classic evolutionary multi‑objective methods.

Key Contributions

• Alternating bi‑objective gradient optimization: a novel training loop that decouples the accuracy loss from an explainability loss, updating each in turn.
• Stable Cauchy membership functions: replaces the usual Gaussian/Bell functions with Cauchy shapes that are less sensitive to initialization and improve convergence.
• Differentiable explainability metric: a mathematically tractable objective that quantifies rule “distinguishability” (how easily a human can separate and interpret rules).
• Extensive empirical validation: ~5 000 experiments on nine UCI regression benchmarks demonstrate that X‑ANFIS matches or exceeds the accuracy of state‑of‑the‑art neuro‑fuzzy models while delivering more interpretable rule bases.
• Recovery of non‑convex Pareto solutions: unlike scalarized gradient methods, the alternating scheme can locate solutions outside the convex hull of traditional multi‑objective optimizers.

Methodology

1. Model backbone – ANFIS: The authors start from the classic Adaptive Neuro‑Fuzzy Inference System (ANFIS), which combines a fuzzy rule base with a neural‑network‑like learning engine.
2. Cauchy membership functions: Each fuzzy antecedent uses a Cauchy function ( \mu(x) = \frac{1}{1 + ((x-c)/\gamma)^2} ). Because the denominator grows quadratically, gradients remain well‑behaved even when parameters are far from optimal, reducing the risk of exploding/vanishing updates.
3. Two loss terms
   • Performance loss – standard mean‑squared error (MSE) on the regression target.
   • Explainability loss – a differentiable proxy for rule distinguishability: it penalizes overlapping antecedent memberships and encourages crisp, non‑redundant rule boundaries.
4. Alternating optimization loop
   • Step A: Freeze the explainability parameters, back‑propagate only the MSE, and update the consequent (output) weights.
   • Step B: Freeze the performance parameters, back‑propagate the explainability loss, and adjust the antecedent parameters (centers and spreads of the Cauchy functions).
   • Repeat A/B for a fixed number of epochs or until convergence.
5. Evaluation protocol – For each dataset the authors run multiple random seeds, compare against (i) a single‑objective gradient baseline, (ii) a scalarized multi‑objective gradient method, and (iii) an evolutionary multi‑objective optimizer (NSGA‑II).

Results & Findings

• Predictive accuracy: X‑ANFIS matches or slightly improves RMSE on all nine benchmarks compared with the best single‑objective ANFIS and is within 1‑2 % of the evolutionary MOO front.
• Explainability gain: The distinguishability metric improves by 15‑30 % on average, meaning the fuzzy rules are less overlapping and easier for a human to read.
• Pareto coverage: Visualizing the trade‑off surface shows X‑ANFIS discovers points beyond the convex hull that NSGA‑II cannot reach with the same computational budget.
• Efficiency: Training time is roughly 3‑5× faster than the evolutionary approach (which requires thousands of population evaluations) while needing only a modest increase over the vanilla gradient baseline.

Practical Implications

• Deployable explainable models: Engineers can now embed neuro‑fuzzy controllers in safety‑critical systems (e.g., autonomous driving, industrial process control) and still provide stakeholders with human‑readable rule sets.
• Rapid prototyping: Because X‑ANFIS runs on standard deep‑learning frameworks (PyTorch/TensorFlow) and uses only gradient steps, it integrates seamlessly into existing ML pipelines without the need for custom evolutionary code.
• Regulatory compliance: In domains where model transparency is mandated (finance, healthcare), X‑ANFIS offers a quantifiable “explainability score” that can be reported alongside performance metrics.
• Hybrid AI systems: The alternating scheme can be combined with other differentiable components (e.g., attention mechanisms, graph neural nets) to build larger, partially interpretable pipelines.

Limitations & Future Work

• Scope limited to regression: The experiments focus on continuous targets; extending the method to classification or multi‑label tasks may require redesigning the explainability loss.
• Rule count fixed a priori: X‑ANFIS assumes a predetermined number of fuzzy rules; dynamic rule discovery (adding/removing rules during training) is not addressed.
• Explainability metric is proxy: While differentiable, the distinguishability score does not capture all aspects of human interpretability (e.g., linguistic simplicity). User studies are needed to validate perceived understandability.
• Potential for local minima: Alternating updates can still get trapped in sub‑optimal regions; future work could explore stochastic or curriculum‑based schedules to improve global search.

Overall, X‑ANFIS demonstrates that we can obtain high‑performing, interpretable neuro‑fuzzy models without the heavy cost of evolutionary multi‑objective optimization—a promising step toward truly explainable AI in production environments.

Authors

• Qusai Khaled
• Uzay Kaymak
• Laura Genga

Paper Information

• arXiv ID: 2602.19253v1
• Categories: cs.LG, cs.NE
• Published: February 22, 2026
• PDF: Download PDF