## 1 Introduction
For example, in [16], the authors of this paper have performed a comparison of the NDEKF against the Adam optimizer on the function approximation with multilayer neural networks. The NDEKF has shown faster convergence, in the number of epochs, and has led to higher accuracy. In this research, we are interested in investigating the convergence speed, in terms of the number of epochs, of the FDEKF with deep convolutional neural networks (DCNNs), as well as its performance in scenarios where first-order optimizers struggle to achieve high accuracy. This evaluation aims to provide compelling evidence of the superiority of the FDEKF in accuracy and speed. Furthermore, the high degree of decoupling inherent to the FDEKF enhances its suitability for neural networks with a multitude of synaptic weights. Consequently, the application of FDEKF in the training of DCNNs becomes a promising prospect. The primary objective of this research is to investigate the effectiveness of the FDEKF as a training algorithm for DCNNs in the context of solving image classification problems. To assess our initial hypothesis, we will conduct a comprehensive comparison of the FDEKF algorithm against the three others, which are explained in Section 4: Adam Optimizer [17], sKAdam [18], and Reduced-EKF (REKF) [19]. In past research, the EKF and its variations have been used in multilayer perceptron neural networks that have one or two hidden layers with a limited number of neurons [11; 13; 14; 20]. In this work, for the first time, the FDEKF is applied to DCNNs. In this research, we faithfully apply the FDEKF algorithm following [11], ensuring that it complies with the recursive nature of the EKF. Updating the weights occurs on a sample-by-sample basis, thus maintaining the conditions for the stability analysis outlined in [21]. The data sets selected for our experiments have been used in other image classification works. We develop three experiments to test the performance of the FDEKF against other algorithms and another experiment to apply the FDEKF to terrain recognition to generate a cost map of its traversability. The main contributions of this article are to show the feasibility of the FDEKF for training DCNNs, the superiority of training with FDEKF in the speed of convergence concerning the number of epochs, and the higher precision in the early training phase. These findings are anticipated to spark greater interest in exploring the EKF as a viable training algorithm for DCNNs. With this article, we want to show the scientific community that the FDEKF is a good strategy for DCNNs training. The FDEKF is a second-order algorithm with some advantages over the first-order algorithms, so it can result in faster convergence and lead to higher accuracy due to a higher probability of finding the global optimum. The FDEKF largely preserves the behavior of the EKF, even neglecting the statistical correlation between some pairs of elements of the synaptic weight vector, benefiting from a reduction in computational complexity. The structure of this work is organized as follows: In Section 2, we provide a concise overview of DCNNs in the context of image classification and describe critical aspects of the training process. Section 3 presents an exhaustive description of the FDEKF model, starting from the EKF and including the key considerations for its application in the training of neural networks. Section 4 provides a brief overview of the algorithms within our framework, namely Adam, sKAdam, and REKF, which have been explored in previous related works. In Section 5, we present the results and details of the experiments performed in this study. Finally, we draw our conclusions and provide information on possible avenues for future research.