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.
Abstract. While injecting weight noise during training has been proposed for more than a decade to improve the convergence, generalization and fault tolerance of a neural network, not much theoretical work has been done to its convergence proof and the objective function that it is minimizing. By applying the Gladyshev Theorem, it is shown that the convergence of injecting weight noise during training an RBF network is almost sure. Besides, the corresponding objective function is essentially the mean square errors (MSE). This objective function indicates that injecting weight noise during training an radial basis function (RBF) network is not able to improve fault tolerance. Despite this technique has been effectively applied to multilayer perceptron, further analysis on the expected update equation of training MLP with weight noise injection is presented. The performance difference between these two models by applying weight injection is discussed.
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