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While injecting fault during training has long been demonstrated as an effective method to improve fault tolerance of a neural network, not much theoretical work has been done to explain these results. In this paper, two different node-fault-injection-based on-line learning algorithms, including (1) injecting multinode fault during training and (2) weight decay with injecting multinode fault, are studied. Their almost sure convergence will be proved and thus their corresponding objective functions are deduced.