Paper:

# A Fast Method for Fuzzy Rules Learning with Derivative-Free Optimization by Formulating Independent Evaluations of Each Fuzzy Rule

## Kiyohiko Uehara^{*} and Kaoru Hirota^{**}

^{*}Ibaraki University

4-12-1 Nakanarusawa-cho, Hitachi, Ibaraki 316-8511, Japan

^{**}Beijing Institute of Technology

No.5 South Street, Zhongguancun, Haidian District, Beijing 100081, China

A method is proposed for evaluating fuzzy rules independently of each other in fuzzy rules learning. The proposed method is named *α-FUZZI-ES* (α-weight-based fuzzy-rule independent evaluations) in this paper. In α-FUZZI-ES, the evaluation value of a fuzzy system is divided out among the fuzzy rules by using the compatibility degrees of the learning data. By the effective use of α-FUZZI-ES, a method for fast fuzzy rules learning is proposed. This is named *α-FUZZI-ES learning* (α-FUZZI-ES-based fuzzy rules learning) in this paper. α-FUZZI-ES learning is especially effective when evaluation functions are not differentiable and derivative-based optimization methods cannot be applied to fuzzy rules learning. α-FUZZI-ES learning makes it possible to optimize fuzzy rules independently of each other. This property reduces the dimensionality of the search space in finding the optimum fuzzy rules. Thereby, α-FUZZI-ES learning can attain fast convergence in fuzzy rules optimization. Moreover, α-FUZZI-ES learning can be efficiently performed with hardware in parallel to optimize fuzzy rules independently of each other. Numerical results show that α-FUZZI-ES learning is superior to the exemplary conventional scheme in terms of accuracy and convergence speed when the evaluation function is non-differentiable.

*J. Adv. Comput. Intell. Intell. Inform.*, Vol.25, No.2, pp. 213-225, 2021.

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