Abstract: Feedforward neural networks （FF networks） are the most popular and most widely used models in many practical applications The network is divided into layers Given a network, it can be trained using a set of data containing input output pairs With this data a neural network is usually trained by minimizing a given error function, measuring the discrepancy between the neural network output and the desired output In this paper, the structure modality of the error function is discussed and the necessary and sufficient condition about the error function is derived, which ensures that the output of the trained neural network approximates the conditional logarithm expectation of the desired output with training patterns Further analysis shows that the structure modality of the error function existing already is only a kind of special situations obtained in this paper Besides, the structure modality of the error function can overcome effectively the weakness of not being strong for anti jamming ability of the error function existing already A condition for approximating the first quantile of order α so is discussed to minimize the error function This conclusion has more extensive meaning The results offer a good foundation for further research on the artificial neural network