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For the purposes of this question I am asking about training the generator, assume that training the discriminator is another topic.

My understanding of generative adversarial networks is that you feed random input data to the generator and it generates images. Out of those images, the ones which the discriminator thinks are real are used to train the generator.

For example, I have the random inputs $i_1$, $i_2$, $i_3$, $i_4$... from which the generator produces $o_1$, $o_2$, $o_3$, $o_4$. Say for example, the discriminator thinks that $o_1$ and $o_2$ are real but $o_3$ and $o_4$ are fake, I then throw away input output pairs 3 and 4, but keep 1 and 2, and run back propagation on the generator to tell it that $i_1$ should produce $o_1$, and $i_2$ should produce $o_2$ since these were "correct" according to the discriminator.

The contradiction seems to come from the fact that the generator already generates those outputs from those inputs, so nothing will be gained by running backprop on those input output pairs.

Where is the flaw in my logic here? I seem to have something wrong in my reasoning, or a misunderstanding of how the generator is trained.

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Let's take a look at the images that fooled the D-network. When this happens the binary cross entropy loss of the D-network is important to look at. The network said a fake image was real. As usual, you back propagate to the very beginning of the D-network. What is the back prop doing? It is telling the weights in the D-network to change such that you lessen the loss (remember you are minimizing the loss function by going down hill --> using gradient descent). So you changed the weights of the D-network, but you really don't have to stop there.

The complex nonlinear function the D-network is calculating is really f(Image, weights of D-network). So the output of the D-network is really dependent on the Image fed in and the weights of the D-network. The cost is a function of the output f. So we have c = g(f(Image, weights of D-network)) where g is really the binary cross entropy. Looking at this, we see that the cost is also a function of the image!

We can take the partials with respect to the image just like we take the partials with respect to the weights to update the weights. And so after doing back prop to the image, we don't stop -> we take the partial of the cost function with respect to the image that is fed into the D-network and we update the image. And essentially we don't have the original image when we continue to back prop to the beginning of the G-network. Let me know if you'd like more clarification. Again this is my understanding and I haven't coded a GAN yet, so someone please confirm. Also, I am open to others' interpretations.

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