On-line node fault injection training algorithm for MLP networks: objective function and convergence analysis

Regularization Effect of Random Node Fault/Noise on Gradient Descent Learning Algorithm

For decades, adding fault/noise during training by gradient descent has been a technique for getting a neural network (NN) tolerant to persistent fault/noise or getting an NN with better generalization. In recent years, this technique has been readvocated in deep learning to avoid overfitting. Yet, the objective function of such fault/noise injection learning has been misinterpreted as the desired measure (i.e., the expected mean squared error (mse) of the training samples) of the NN with the same fault/noise. The aims of this article are: 1) to clarify the above misconception and 2) investigate the actual regularization effect of adding node fault/noise when training by gradient descent. Based on the previous works on adding fault/noise during training, we speculate the reason why the misconception appears. In the sequel, it is shown that the learning objective of adding random node fault during gradient descent learning (GDL) for a multilayer perceptron (MLP) is identical to the desired measure of the MLP with the same fault. If additive (resp. multiplicative) node noise is added during GDL for an MLP, the learning objective is not identical to the desired measure of the MLP with such noise. For radial basis function (RBF) networks, it is shown that the learning objective is identical to the corresponding desired measure for all three fault/noise conditions. Empirical evidence is presented to support the theoretical results and, hence, clarify the misconception that the objective function of a fault/noise injection learning might not be interpreted as the desired measure of the NN with the same fault/noise. Afterward, the regularization effect of adding node fault/noise during training is revealed for the case of RBF networks. Notably, it is shown that the regularization effect of adding additive or multiplicative node noise (MNN) during training an RBF is reducing network complexity. Applying dropout regularization in RBF networks, its effect is the same as adding MNN during training.

A Limitation of Gradient Descent Learning

Over decades, gradient descent has been applied to develop learning algorithm to train a neural network (NN). In this brief, a limitation of applying such algorithm to train an NN with persistent weight noise is revealed. Let V(w) be the performance measure of an ideal NN. V(w) is applied to develop the gradient descent learning (GDL). With weight noise, the desired performance measure (denoted as J(w) ) is E[V(~w)|w] , where ~w is the noisy weight vector. Applying GDL to train an NN with weight noise, the actual learning objective is clearly not V(w) but another scalar function L(w) . For decades, there is a misconception that L(w) = J(w) , and hence, the actual model attained by the GDL is the desired model. However, we show that it might not: 1) with persistent additive weight noise, the actual model attained is the desired model as L(w) = J(w) ; and 2) with persistent multiplicative weight noise, the actual model attained is unlikely the desired model as L(w) ≠ J(w) . Accordingly, the properties of the models attained as compared with the desired models are analyzed and the learning curves are sketched. Simulation results on 1) a simple regression problem and 2) the MNIST handwritten digit recognition are presented to support our claims.

Learning Algorithm for Boltzmann Machines With Additive Weight and Bias Noise

This brief presents analytical results on the effect of additive weight/bias noise on a Boltzmann machine (BM), in which the unit output is in {-1, 1} instead of {0, 1}. With such noise, it is found that the state distribution is yet another Boltzmann distribution but the temperature factor is elevated. Thus, the desired gradient ascent learning algorithm is derived, and the corresponding learning procedure is developed. This learning procedure is compared with the learning procedure applied to train a BM with noise. It is found that these two procedures are identical. Therefore, the learning algorithm for noise-free BMs is suitable for implementing as an online learning algorithm for an analog circuit-implemented BM, even if the variances of the additive weight noise and bias noise are unknown.

A Novel Pruning Algorithm for Smoothing Feedforward Neural Networks Based on Group Lasso Method

In this paper, we propose four new variants of the backpropagation algorithm to improve the generalization ability for feedforward neural networks. The basic idea of these methods stems from the Group Lasso concept which deals with the variable selection problem at the group level. There are two main drawbacks when the Group Lasso penalty has been directly employed during network training. They are numerical oscillations and theoretical challenges in computing the gradients at the origin. To overcome these obstacles, smoothing functions have then been introduced by approximating the Group Lasso penalty. Numerical experiments for classification and regression problems demonstrate that the proposed algorithms perform better than the other three classical penalization methods, Weight Decay, Weight Elimination, and Approximate Smoother, on both generalization and pruning efficiency. In addition, detailed simulations based on a specific data set have been performed to compare with some other common pruning strategies, which verify the advantages of the proposed algorithm. The pruning abilities of the proposed strategy have been investigated in detail for a relatively large data set, MNIST, in terms of various smoothing approximation cases.

Deterministic convergence of chaos injection-based gradient method for training feedforward neural networks

It has been shown that, by adding a chaotic sequence to the weight update during the training of neural networks, the chaos injection-based gradient method (CIBGM) is superior to the standard backpropagation algorithm. This paper presents the theoretical convergence analysis of CIBGM for training feedforward neural networks. We consider both the case of batch learning as well as the case of online learning. Under mild conditions, we prove the weak convergence, i.e., the training error tends to a constant and the gradient of the error function tends to zero. Moreover, the strong convergence of CIBGM is also obtained with the help of an extra condition. The theoretical results are substantiated by a simulation example.