Deep learning model is a multi-layered network structure, and the network parameters that evaluate the final performance of the model must be trained by a deep learning optimizer. In comparison to the mainstream optimizers that utilize integer-order derivatives reflecting only local information, fractional-order derivatives optimizers, which can capture global information, are gradually gaining attention. However, relying solely on the long-term estimated gradients computed from fractional-order derivatives while disregarding the influence of recent gradients on the optimization process can sometimes lead to issues such as local optima and slower optimization speeds. In this paper, we design an adaptive learning rate optimizer called AdaGL based on the Grünwald–Letnikov (G–L) fractional-order derivative. It changes the direction and step size of parameter updating dynamically according to the long-term and short-term gradients information, addressing the problem of falling into local minima or saddle points. To be specific, by utilizing the global memory of fractional-order calculus, we replace the gradient of parameter update with G–L fractional-order approximated gradient, making better use of the long-term curvature information in the past. Furthermore, considering that the recent gradient information often impacts the optimization phase significantly, we propose a step size control coefficient to adjust the learning rate in real-time. To compare the performance of the proposed AdaGL with the current advanced optimizers, we conduct several different deep learning tasks, including image classification on CNNs, node classification and graph classification on GNNs, image generation on GANs, and language modeling on LSTM. Extensive experimental results demonstrate that AdaGL achieves stable and fast convergence, excellent accuracy, and good generalization performance.
