the Caputo Fractional-Order Gradient Descent Method (FGDM)¶
Implements the Caputo Fractional-Order Gradient Descent Method (FGDM), training a zero-order Takagi-Sugeno fuzzy model by descending along a Caputo fractional gradient of the loss.
A zero-order Takagi-Sugeno fuzzy model produces an output as a fuzzy-membership-weighted combination of constant consequents, and tuning its premise and consequent parameters with the ordinary integer-order gradient captures only the local slope, which the authors find slow and prone to oscillation. FGDM instead steps against the Caputo fractional derivative of order \(\alpha\in(0,1)\) of the loss, whose non-local memory accumulates gradient history between a base point and the current iterate, giving faster convergence and higher classification accuracy. To avoid the composite-function derivatives a full fractional chain rule would demand, the Caputo derivative is taken about the previous iterate \(\theta_{t-1}\), which yields a \(\lvert\theta_t-\theta_{t-1}\rvert^{1-\alpha}\) memory factor and a \(\Gamma(2-\alpha)\) normalizer; the parameter then steps against this fractional gradient.
where \(\theta\) denotes a premise or consequent parameter of the fuzzy model, \(\eta>0\) is the learning rate, \(\alpha\in(0,1)\) is the fractional order, \(\Gamma(\cdot)\) is the gamma function, \(\frac{\partial E}{\partial \theta_{t-1}}\) is the ordinary partial derivative of the loss \(E\), and \(D^{\alpha}_{\theta} E\) is the simplified Caputo fractional gradient. The factor \(\lvert\theta_t-\theta_{t-1}\rvert^{1-\alpha}\) encodes the fractional memory between consecutive iterates; \(\alpha=1\) recovers integer-order gradient descent.
Reference: Yan Liu, Qiang Shao, Yuanquan Liu, Rui Wang, "A novel gradient learning algorithm based on zero-order Takagi-Sugeno fuzzy model: the caputo fractional-order gradient descent", Applied Soft Computing 188 (2026). https://doi.org/10.1016/j.asoc.2025.114430