Online gradient methods are widely used for training feedforward neural networks. We prove in this paper a convergence theorem for an online gradient method with variable step size for backward propagation (BP) neural networks with a hidden layer. Unlike most of the convergence results that are of probabilistic and nonmonotone nature, the convergence result that we establish here has a deterministic and monotone nature.

In this paper, we propose a group Lasso regularization term as a hidden layer regularization method for feedforward neural networks. Adding a group Lasso regularization term into the standard error function as a hidden layer regularization term is a fruitful approach to eliminate the redundant or unnecessary hidden layer neurons from the feedforward neural network structure. As a comparison, a popular Lasso regularization method is introduced into standard error function of the network. Our novel hidden layer regularization method can force a group of outgoing weights to become smaller during the training process and can eventually be removed after the training process. This means it can simplify the neural network structure and it minimizes the computational cost. Numerical simulations are provided by using K-fold cross-validation method with $K=5$ to avoid overtraining and to select the best learning parameters. The numerical results show that our proposed hidden layer regularization method prunes more redundant hidden layer neurons consistently for each benchmark dataset without loss of accuracy. In contrast, the existing Lasso regularization method prunes only the redundant weights of the network, but it cannot prune any redundant hidden layer neurons.

Dropout and DropConnect are two techniques to facilitate the regularization of neural network models, having achieved the state-of-the-art results in several benchmarks. In this paper, to improve the generalization capability of spiking neural networks (SNNs), the two drop techniques are first applied to the state-of-the-art SpikeProp learning algorithm resulting in two improved learning algorithms called SPDO (SpikeProp with Dropout) and SPDC (SpikeProp with DropConnect). In view that a higher membrane potential of a biological neuron implies a higher probability of neural activation, three adaptive drop algorithms, SpikeProp with Adaptive Dropout (SPADO), SpikeProp with Adaptive DropConnect (SPADC), and SpikeProp with Group Adaptive Drop (SPGAD), are proposed by adaptively adjusting the keep probability for training SNNs. A convergence theorem for SPDC is proven under the assumptions of the bounded norm of connection weights and a finite number of equilibria. In addition, the five proposed algorithms are carried out in a collaborative neurodynamic optimization framework to improve the learning performance of SNNs. The experimental results on the four benchmark data sets demonstrate that the three adaptive algorithms converge faster than SpikeProp, SPDO, and SPDC, and the generalization errors of the five proposed algorithms are significantly smaller than that of SpikeProp. Furthermore, the experimental results also show that the five algorithms based on collaborative neurodynamic optimization can be improved in terms of several measures.

This paper presents new theoretical results on the backpropagation algorithm with smoothing $$L_{1/2}$$ regularization and adaptive momentum for feedforward neural networks with a single hidden layer, i.e., we show that the gradient of error function goes to zero and the weight sequence goes to a fixed point as n (n is iteration steps) tends to infinity, respectively. Also, our results are more general since we do not require the error function to be quadratic or uniformly convex, and neuronal activation functions are relaxed. Moreover, compared with existed algorithms, our novel algorithm can get more sparse network structure, namely it forces weights to become smaller during the training and can eventually removed after the training, which means that it can simply the network structure and lower operation time. Finally, two numerical experiments are presented to show the characteristics of the main results in detail.