GAN Course 1 - Week1 assignment: Why batchnormalization is not used in discriminator block?

Batch normalization is used in the generator block and its use is intuitive. As per my
understanding, it must have been used as a remedy on covariate shift and internal covariate shift issues. On the similar lines, I would expect to make use batch normalization in discriminator block as well. Discriminator is a neural network with 3 hidden layers, so internal covariate shift issue is a possibility. Requesting you to elucidate the reasoning behind not using batch normalization in discriminator NN.

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I suppose the reason for this is that if you apply BN in the discriminator for both real and fake images, the learned BN variables will aggregate both distributions, which could break or slow down the training since fake images do not contain any useful information at the beginning of the training and their distribution will change over the time, so it will be challenging for the discriminator to train.