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Hi there and welcome to the session.

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In this session we are going to look at the Gan loss function and also how Gans have been trained.

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Then finally we'll look at common Gan training problems.

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So for a brief summary of what we've seen already, we'll consider that we have this two distributions

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right here.

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This distribution or this one, uh, and this black dotted lines represents the real data that's similar

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to what we had here.

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So similar to, uh, this real data right here.

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And then this other distribution in green represents the fake data which is going to be generated by

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our generator.

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So, um, getting back here, we have this, uh, we see this here, which is our discriminator.

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So right here we have the discriminator D then we have, as we've said already, the real and the fake

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distributions.

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Now, once we start training, we have this discriminator which sees that most most of the real data

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is, uh, getting a score of one, or it classifies with a probability of one that this data is real.

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And then as we approach this generated data, uh, as we get samples from our generated distribution,

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we find that the discriminator now outputs a zero.

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Now, as we keep on training, the generation gets better and then the generator samples now look a

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bit more like the real data.

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See, they come, they get closer.

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This, uh, distance here becomes smaller as compared to compared to what we had before.

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So, uh, we've trained and we've gotten to this point where now this discriminator here still sees

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the real data to be one, but now confuses or sometimes classifies this fake data to, to be one.

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So you see data around here, see this samples we consider to be ones or classified as one, though

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we'll still have some samples classified as zero.

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And then once we get to convergence, we have this, uh, half here.

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So we have this, um, classifier now which is unable to differentiate between the real and the generated

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or fake data because the distributions look very much alike.

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And so that's it for the recap of the previous session.

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Now we're going to dive into the Gan loss function.

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So right here we have this, um, training algorithm which we could see here.

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Let's expand this.

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We have this training algorithm and if you notice, we have this two loss functions here, one for the

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discriminator and the other for the generator.

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But this could be combined into one equation right here.

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Let's take this here.

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And now we have our two player Minimax game with this value function V.

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Now, when we talk about this Minimax game right here, it makes allusion to what goes on between the

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generator and the discriminator.

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So our two players in this case are our generator and the discriminator.

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Now one thing you could also do with this Gan lab is you could put out your own, uh, distribution.

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So let's say, for example, this distribution we apply and we start the train and the generator now,

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together with the discriminator, start to play this game where at the end or at Convergence, we expect

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the generator to produce outputs which are similar to that of the discriminator.

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Now, coming back to this equations, you'll notice that we have mean and G and Max and Z.

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So to understand this notation, you can consider that we are minimizing this expression right here

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by updating the parameters of G, and then we maximize this expression by updating the parameters of

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Z, which is our discriminator, where G is our generator.

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And then if we try to separate these two, that's to to get for the two to start with minimizing, for

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example, for the generator.

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You'll find that given that in this first expression, you see this, this expression right here, let's

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have that.

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So in this expression, there is actually no G.

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So we could make use of only this when we're trying to minimize this whole expression with respect to

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parameters of G.

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So that's why if you, if you, if you check in the, um, algorithm given to.

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Us right here.

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You find that for a loss for G, you have only that second expression.

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Now for the D, Let's get back here for the d.

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D is in the both sides.

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Or this both this expression and this other expression.

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So that's why we have the combination of the two in this, our algorithm right here.

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Now, to better explain or better understand this in depth, let's consider this here.

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So we've extracted for the D and for the G.

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You should also note that let's get back here.

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When we talk about updating the discriminator or updating the generator is basically our gradient descent

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step.

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Remember, if we have, for example, Tedder and or rather, if we have let's take this off.

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If we have Tedder at a particular step, let's say step I or to get Tedder at I plus one step, then

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we would have tedder I as a previous step or let's say tedder I minus one.

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Um let's take this off so we want to get the I which is equal to the I minus one minus the learning

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rate times the partial derivative of the loss with respect to our Tedder I minus one.

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So this is basically what we have in here.

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And this expression here where we have this, um, reversed triangle that's, uh, this right here is

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actually this section of our gradient descent.

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And so this means that we're finding the partial derivative of our loss.

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This is the loss with respect to the parameters.

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Tedder Then we have the same for the generator.

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Now, let's get back here where we're going to, um, understand in depth what all these different expressions

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actually mean, right here we have this real sample and this fake sample.

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So these two here, we have real and fake.

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And then, uh, with this real sample, we pass it into the discriminator, which we hope or wish we

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will train to see that this is a one.

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And then for the, the fake sample, because this is a person who doesn't exist, whereas this person

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actually exists.

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And we train again, uh, which we'll see later on in this course, uh, to produce this type of outputs.

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So, um, let's get back here.

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We have our discriminator, then we, we from here.

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Let's take, let's instead shift this, um, let's move this, this way.

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So let's take this right here.

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Yeah.

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Okay, so we have that.

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And then here we have this right here.

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Now this is our fake sample.

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And then we have our generator.

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Let's change the color for the generator.

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So our generator produces this.

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And then we have some random noise which we send in here because we just we just want to be able to

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produce this from some random noise.

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So we have this random noise right here which produces this, our G, which produces that, um, fake

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sample.

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And then we have our discriminator who is able to classify or say whether an input is fake or not.

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Okay, so getting back here, let's take this off.

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Getting back here.

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When we want to train our discriminator, we're going to have an input X right here.

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You see this input x?

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Now this input X is this real sample right here.

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So this is our x I, which is which is this here.

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And this is our discriminator, which takes in this input.

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Then once the discriminator takes this input is expected to output a one.

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And if you get back your let's get back here you find you're told update the discriminator by ascending

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its stochastic gradient and then you update the generator by descending stochastic gradient.

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Now generally when we're trying to ascend or ascending or descending actually is different, like what

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we've seen already, where we had theta equal theta minus learning rate, partial derivative of the

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loss with respect to theta is gradient descent.

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Which is actually what we're doing for the generator.

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But when we talk about gradient ascent, it's actually this.

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Instead, we have a plus instead of a minus.

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See that?

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So, um, if we have for, uh, let's plot our loss function here with respect to theta for our gradient

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descent, that's classical gradient descent.

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What we want to do is to minimize this loss.

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But for our gradient ascent, we want to instead maximize this loss.

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And then getting back here before we move on, we'll consider this plot of the log function.

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So here if we have X and we have log of x, then we have a plot which looks like this.

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Oops, let's have this.

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There we go.

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And this is one.

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So when x equal one, the log is zero.

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And then as X takes smaller values approaching zero, the log goes towards negative infinity.

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So as we as X goes towards zero, obviously from the right in this direction, going in this direction,

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then.

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The log of X goes towards negative infinity.

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So that's it.

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Now let's get back here.

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For the discriminator, as we've seen, we're having a gradient ascent.

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So we're trying to maximize this.

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Remember, from our minimax loss function, we we try to maximize our discriminator.

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So when we when the discriminator takes in a real sample, um, it, it gives us an output of one and

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with an output of one, the log of one, as we'll see here, is going to give us zero.

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So the log of one is zero.

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So this will give us a zero.

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Now, you should note that all the outputs of our discriminator range between 0 and 1, so our discriminator

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is our usual classifier.

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So it's going to output values between 0 and 1.

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And so getting a value of zero is the highest you could get as an output after passing through the log.

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So let me explain.

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We have this discriminator which outputs a value between 0 and 1 or let's plot it out.

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Let's replot it again here.

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So the discriminator outputs values between 0 and 1.

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See that?

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So this is what the discriminator outputs.

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And so when, when you when you have the log of this values between 0 and 1, then the maximum value

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for the log will be a zero.

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You see that the maximum value for all these values between 0 and 1 will be a zero.

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So when you have a zero, it means you've maximized this and that falls in line with what you expect

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because we want to maximize this, um, expression we have right here.

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Okay?

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So we get back here for the reals we want to output a one and the log of one will give us the maximum

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possible value, which is in this case is zero.

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Now, for this year we have a Z.

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Remember this X, Is this real image here.

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The Z here is our random noise, which we have seen already here.

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So when this random noise passes through our generator, we output a fake sample.

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See that the generator takes in the random noise and then the generator.

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Let's have this here.

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This generator outputs this fake sample.

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And once he outputs this, our fake sample, we now take this fake sample and pass it into our discriminator.

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And we expect our discriminator to produce instead of a one this time around.

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A zero.

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See that?

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And in our case, the log, the log of one minus a zero is a log of one and the log of one is zero.

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That's the highest possible value we could get when we're dealing with logs in the range 0 to 1.

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So we're maximizing this.

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You see that?

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Now, if that's understood, we could move on to the generator.

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For the generator, We want to instead minimize this expression.

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So since we were trying to minimize this expression, we would expect the output from this to be negative

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infinity as a lowest possible value.

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But let's get let's put in these values and see how we obtain this negative infinity.

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Now we have Z, which is our random noise.

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When the random noise passes through G, it outputs this fake sample again.

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And now this time around, we expect the.

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To consider this, for example, this time around to be like a real sample.

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So what we're saying here is at one instance, we want our discriminator to see this as real.

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Another instance, we want the discriminator to see this as fake.

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And depending on the instance we're going to get or we're going to update our parameters of the corresponding

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network.

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So in the case where we want to see this as a fake, we want to update the parameters of the discriminator

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such that it sees this as fake.

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And then the case where we want this to be seen as a real by the discriminator, one will be updating

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the parameters of the generator.

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So in fact, um, what's going to be happening here is let's change this color when training the discriminator,

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we're going to freeze the generator.

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So we're not going to update these parameters.

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We're going to update the parameters of the discriminator.

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Now, when training, when training.

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Let's get back when training our generator, we're going to freeze this here.

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Oops when training the generator, we're going to freeze this and then update its parameters so that

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it's able to fool the discriminator to think that this is a real sample.

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And when it thinks it's a real sample, it's going to output a one.

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Now log of one minus one is log of zero and log of zero is negative infinity.

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See that?

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And this is the minimum possible value.

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So we're minimizing this expression.

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So getting back here, we have for a number of training iterations that's basically for the number of

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epochs.

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Uh, we're going to do K steps.

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We're going to update the discriminator via K steps.

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See, for K steps do this.

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We take we sample a mini batch of noise.

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We get the noise, uh, mini batch of real samples.

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Then from here we obtain, uh, we get the generator outputs and then we obtain the output loss which

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we use to update the discriminator parameters.

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And then for this k steps, I think in here, let's say we use K equals one, the number of steps we

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apply to use K equal one.

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So you could modify this, although in practice k equals one is fine.

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So, um, getting back here after going through k steps for this, we're going to update now the generator.

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So sample mini batch of M noise again.

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So just like this, then we obtain this output from the generator.

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This time around, we expect it to fool the discriminator.

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So we're going to update the generator by descending this stochastic gradient such that it fools the

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discriminator.

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And that's it for this section.

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In the next section, we are going to get into some practice and see how to get this kind of results

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we had here.

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So this are the real samples and then what we will obtain will be something like this.

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See, we're able to generate this fake outputs or these people who do not actually exist and they look

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pretty realistic.

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Then one point we have to note before we move on is that the type of neural network used in this original

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Gan paper is a classic artificial neural network, and this way the kinds of outputs they got.

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But in the practice, we're going to be making use of the Gan.

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So instead of the simple gan, we'll make use of the DC gan.

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The DC actually stands for Deep Convolutional.

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Um, and that's it.

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DC Deep Convolutional So this again means deep convolutional generative adversarial neural networks.

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Okay, so here we're going to be looking at this, um, neural networks which are convolution based.

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And so with that, see you in the next section.