Wasserstein Loss or W-Loss solves some problems faced by GANs, like mode claps and vanishing gradients. But for it to work well, there is a special condition that needs to be met by the critic. In this video, you'll see what the continuity condition on the critic neural network means and why that condition is important when using W-Loss for training GANs, and trust me, it's worth it, so stay tuned. W-Loss is a simple expression that computes the difference between the expected values of the critics output for the real examples x and its predictions on the fake examples g(z). The generator tries to minimize this expression, trying to get the generative examples to be as close as possible to the real examples while the critic wants to maximize this expression because it wants to differentiate between the reals and the fakes, it wants the distance to be as large as possible. However, for training GANs using W-Loss, the critic has a special condition. It needs to be something called 1-Lipschitz Continuous or 1-L Continuous for short. This condition sounds more sophisticated than it really is. For a function like the critics neural network to be at 1-Lipschitz Continuous, the norm of its gradient needs to be at most one. What that means is that, the slope can't be greater than one at any point, its gradient can't be greater than one. To check if a function here, for example, this function you see here, f(x) equals x squared, is 1-Lipschitz Continuous, you want to go along every point in this function and make sure its slope is less than or equal to one, or its gradient is less than or equal to one, and what you can do is, you can actually draw two lines, one where the slope is exactly one at this certain point that you're evaluating function, and one where the slope is negative one where you're evaluating our function. You want to make sure that the growth of this function never goes out of bounds from these lines because staying within these lines means that the function is growing linearly. Here this function is not Lipschitz Continuous because it's coming out in all these sections. It's not staying within this green area, which suggests that it's growing more than linearly. Look at another example here. This is a smooth curve functions. You want to again check every single point on this function before you can determine whether or not that this is 1-Lipschitz Continuous. Here it looks fine, function looks good. Here it also looks good, here looks good. Let's say you take every single value and the function never grows more than linearly. This function is 1-Lipschitz Continuous. This condition on the critics neural network is important for W-Loss because it assures that the W-Loss function is not only continuous and differentiable, but also that it doesn't grow too much and maintain some stability during training. This is what makes the underlying Earth Movers Distance valid, which is what W-Loss is founded on. This is required for training both the critic and generators neural networks and it also increases stability because the variation as the GAN learns will be bounded. To recap, the critic, and again that uses W-Loss for training needs to be 1-Lipschitz Continuous in order for its underlying Earth Mover's Distance comparison between the reals and the fakes to be a valid comparison. In order to satisfy or try to satisfy this condition during training, there are multiple different methods. Next, we'll learn about a couple of these methods.