Convergence analyses on on-line weight noise injection-based training algorithms for MLPs

Injecting weight noise during training is a simple technique that has been proposed for almost two decades. However, little is known about its convergence behavior. This paper studies the convergence of two weight noise injection-based training algorithms, multiplicative weight noise injection with weight decay and additive weight noise injection with weight decay. We consider that they are applied to multilayer perceptrons either with linear or sigmoid output nodes. Let w(t) be the weight vector, let V(w) be the corresponding objective function of the training algorithm, let α >; 0 be the weight decay constant, and let μ(t) be the step size. We show that if μ(t)→ 0, then with probability one E[||w(t)||2(2)] is bound and lim(t) → ∞ ||w(t)||2 exists. Based on these two properties, we show that if μ(t)→ 0, Σtμ(t)=∞, and Σtμ(t)(2) <; ∞, then with probability one these algorithms converge. Moreover, w(t) converges with probability one to a point where ∇wV(w)=0.

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

Improving fault tolerance of a neural network has been studied for more than two decades. Various training algorithms have been proposed in sequel. The on-line node fault injection-based algorithm is one of these algorithms, in which hidden nodes randomly output zeros during training. While the idea is simple, theoretical analyses on this algorithm are far from complete. This paper presents its objective function and the convergence proof. We consider three cases for multilayer perceptrons (MLPs). They are: (1) MLPs with single linear output node; (2) MLPs with multiple linear output nodes; and (3) MLPs with single sigmoid output node. For the convergence proof, we show that the algorithm converges with probability one. For the objective function, we show that the corresponding objective functions of cases (1) and (2) are of the same form. They both consist of a mean square errors term, a regularizer term, and a weight decay term. For case (3), the objective function is slight different from that of cases (1) and (2). With the objective functions derived, we can compare the similarities and differences among various algorithms and various cases.

Objective functions of online weight noise injection training algorithms for MLPs

Injecting weight noise during training has been a simple strategy to improve the fault tolerance of multilayer perceptrons (MLPs) for almost two decades, and several online training algorithms have been proposed in this regard. However, there are some misconceptions about the objective functions being minimized by these algorithms. Some existing results misinterpret that the prediction error of a trained MLP affected by weight noise is equivalent to the objective function of a weight noise injection algorithm. In this brief, we would like to clarify these misconceptions. Two weight noise injection scenarios will be considered: one is based on additive weight noise injection and the other is based on multiplicative weight noise injection. To avoid the misconceptions, we use their mean updating equations to analyze the objective functions. For injecting additive weight noise during training, we show that the true objective function is identical to the prediction error of a faulty MLP whose weights are affected by additive weight noise. It consists of the conventional mean square error and a smoothing regularizer. For injecting multiplicative weight noise during training, we show that the objective function is different from the prediction error of a faulty MLP whose weights are affected by multiplicative weight noise. With our results, some existing misconceptions regarding MLP training with weight noise injection can now be resolved.

Convergence and objective functions of some fault/noise-injection-based online learning algorithms for RBF networks

In the last two decades, many online fault/noise injection algorithms have been developed to attain a fault tolerant neural network. However, not much theoretical works related to their convergence and objective functions have been reported. This paper studies six common fault/noise-injection-based online learning algorithms for radial basis function (RBF) networks, namely 1) injecting additive input noise, 2) injecting additive/multiplicative weight noise, 3) injecting multiplicative node noise, 4) injecting multiweight fault (random disconnection of weights), 5) injecting multinode fault during training, and 6) weight decay with injecting multinode fault. Based on the Gladyshev theorem, we show that the convergence of these six online algorithms is almost sure. Moreover, their true objective functions being minimized are derived. For injecting additive input noise during training, the objective function is identical to that of the Tikhonov regularizer approach. For injecting additive/multiplicative weight noise during training, the objective function is the simple mean square training error. Thus, injecting additive/multiplicative weight noise during training cannot improve the fault tolerance of an RBF network. Similar to injective additive input noise, the objective functions of other fault/noise-injection-based online algorithms contain a mean square error term and a specialized regularization term.

Analogue VLSI for probabilistic networks and spike-time computation

The history and some of the methods of analogue neural VLSI are described. The strengths of analogue techniques are described, along with residual problems to be solved. The nature of hardware-friendly and hardware-appropriate algorithms is reviewed and suggestions are offered as to where analogue neural VLSI's future lies.

A quantitative study of fault tolerance, noise immunity, and generalization ability of MLPs

An analysis of the influence of weight and input perturbations in a multilayer perceptron (MLP) is made in this article. Quantitative measurements of fault tolerance, noise immunity, and generalization ability are provided. From the expressions obtained, it is possible to justify some previously reported conjectures and experimentally obtained results (e.g., the influence of weight magnitudes, the relation between training with noise and the generalization ability, the relation between fault tolerance and the generalization ability). The measurements introduced here are explicitly related to the mean squared error degradation in the presence of perturbations, thus constituting a selection criterion between different alternatives of weight configurations. Moreover, they allow us to predict the degradation of the learning performance of an MLP when its weights or inputs are deviated from their nominal values and thus, the behavior of a physical implementation can be evaluated before the weights are mapped on it according to its accuracy.