Gradient-based Fuzzy System Optimisation via Automatic Differentiation -- FuzzyR as a Use Case

TL;DR

The paper addresses the gap where fuzzy systems have lacked gradient-based optimisation, proposing automatic differentiation via $Torch$ for $R$ to enable end-to-end optimisation of fuzzy models in $FuzzyR$ without manual derivative computations. It demonstrates autograd-enabled fuzzy inference by adapting a Mamdani-type model with differentiable membership functions, applied to the Iris dataset. Results show RMSE dropping from $0.2872$ to $0.1421$ and misclassifications decreasing from $22$ to $4$ after training, illustrating practical gradient-based optimisation of fuzzy systems. This work highlights a path toward more flexible, interpretable fuzzy architectures using modern optimisation tooling while outlining future work on broader function support and deeper toolkit integration.

Abstract

Since their introduction, fuzzy sets and systems have become an important area of research known for its versatility in modelling, knowledge representation and reasoning, and increasingly its potential within the context explainable AI. While the applications of fuzzy systems are diverse, there has been comparatively little advancement in their design from a machine learning perspective. In other words, while representations such as neural networks have benefited from a boom in learning capability driven by an increase in computational performance in combination with advances in their training mechanisms and available tool, in particular gradient descent, the impact on fuzzy system design has been limited. In this paper, we discuss gradient-descent-based optimisation of fuzzy systems, focussing in particular on automatic differentiation -- crucial to neural network learning -- with a view to free fuzzy system designers from intricate derivative computations, allowing for more focus on the functional and explainability aspects of their design. As a starting point, we present a use case in FuzzyR which demonstrates how current fuzzy inference system implementations can be adjusted to leverage powerful features of automatic differentiation tools sets, discussing its potential for the future of fuzzy system design.

Figures (3)

• Figure 1: Initial membership functions of the inputs
• Figure 2: Final membership functions of the inputs after optimisation
• Figure 3: The learning curves of Parameters theta1 and RMSE