FGDSINN¶
Implements FGDSINN, a Caputo fractional-order gradient learning algorithm for smoothing interval neural networks.
Smoothing interval neural networks (SINNs) represent uncertain inputs as intervals, and training them with the ordinary integer-order gradient tends to be inaccurate and unstable. FGDSINN instead descends along a Caputo fractional-order derivative of the loss, whose non-local memory gives smoother, more accurate parameter updates. To avoid the awkward composite-function derivatives that a full fractional chain rule would require, the method keeps the integer-order chain rule for propagation between layers and applies a simplified Caputo fractional gradient only to the parameters within each layer.
For a weight \(w\) at iteration \(t\), the simplified Caputo fractional derivative of the loss \(E\) is taken about the previous iterate \(w_{t-1}\), which contributes the \(|w_t - w_{t-1}|^{1-\alpha}\) memory factor and a \(\Gamma(2-\alpha)\) normalizer; the parameter then steps against this fractional gradient:
where \(w\) is a within-layer parameter, \(\eta > 0\) is the learning rate, \(\alpha \in (0,1)\) is the fractional order, \(\Gamma(\cdot)\) is the Gamma function, \(\frac{\partial E}{\partial w_{t-1}}\) is the ordinary partial derivative of the loss evaluated through the integer-order chain rule, and \(D^{\alpha}_{w} E\) is the simplified Caputo fractional gradient. The factor \(|w_t - w_{t-1}|^{1-\alpha}\) encodes the fractional memory between consecutive iterates; \(\alpha = 1\) recovers integer-order gradient descent.
Reference: Qiang Shao, Yuanquan Liu, Rui Wang, Yan Liu, "A smoothing interval neural networks-based Caputo fractional-order gradient learning algorithm", International Journal of Machine Learning and Cybernetics 2025. https://doi.org/10.1007/s13042-024-02402-1