Fix some typos (d2l-ai#2380)

Fixed some typos in gan.md
Fixed a latex equation

chapter_generative-adversarial-networks/gan.md

@@ -1,13 +1,13 @@
 # Generative Adversarial Networks
 :label:`sec_basic_gan`
 
-Throughout most of this book, we have talked about how to make predictions. In some form or another, we used deep neural networks learned mappings from data examples to labels. This kind of learning is called discriminative learning, as in, we'd like to be able to discriminate between photos cats and photos of dogs. Classifiers and regressors are both examples of discriminative learning. And neural networks trained by backpropagation have upended everything we thought we knew about discriminative learning on large complicated datasets. Classification accuracies on high-res images has gone from useless to human-level (with some caveats) in just 5-6 years. We will spare you another spiel about all the other discriminative tasks where deep neural networks do astoundingly well.
+Throughout most of this book, we have talked about how to make predictions. In some form or another, we used deep neural networks to learn mappings from data examples to labels. This kind of learning is called discriminative learning, as in, we'd like to be able to discriminate between photos of cats and photos of dogs. Classifiers and regressors are both examples of discriminative learning. And neural networks trained by backpropagation have upended everything we thought we knew about discriminative learning on large complicated datasets. Classification accuracies on high-res images have gone from useless to human-level (with some caveats) in just 5-6 years. We will spare you another spiel about all the other discriminative tasks where deep neural networks do astoundingly well.
 
 But there is more to machine learning than just solving discriminative tasks. For example, given a large dataset, without any labels, we might want to learn a model that concisely captures the characteristics of this data. Given such a model, we could sample synthetic data examples that resemble the distribution of the training data. For example, given a large corpus of photographs of faces, we might want to be able to generate a new photorealistic image that looks like it might plausibly have come from the same dataset. This kind of learning is called generative modeling.
 
 Until recently, we had no method that could synthesize novel photorealistic images. But the success of deep neural networks for discriminative learning opened up new possibilities. One big trend over the last three years has been the application of discriminative deep nets to overcome challenges in problems that we do not generally think of as supervised learning problems. The recurrent neural network language models are one example of using a discriminative network (trained to predict the next character) that once trained can act as a generative model.
 
-In 2014, a breakthrough paper introduced Generative adversarial networks (GANs) :cite:`Goodfellow.Pouget-Abadie.Mirza.ea.2014`, a clever new way to leverage the power of discriminative models to get good generative models. At their heart, GANs rely on the idea that a data generator is good if we cannot tell fake data apart from real data. In statistics, this is called a two-sample test - a test to answer the question whether datasets $X=\{x_1,\ldots, x_n\}$ and $X'=\{x'_1,\ldots, x'_n\}$ were drawn from the same distribution. The main difference between most statistics papers and GANs is that the latter use this idea in a constructive way. In other words, rather than just training a model to say "hey, these two datasets do not look like they came from the same distribution", they use the [two-sample test](https://en.wikipedia.org/wiki/Two-sample_hypothesis_testing) to provide training signals to a generative model. This allows us to improve the data generator until it generates something that resembles the real data. At the very least, it needs to fool the classifier. Even if our classifier is a state of the art deep neural network.
+In 2014, a breakthrough paper introduced Generative adversarial networks (GANs) :cite:`Goodfellow.Pouget-Abadie.Mirza.ea.2014`, a clever new way to leverage the power of discriminative models to get good generative models. At their heart, GANs rely on the idea that a data generator is good if we cannot tell fake data apart from real data. In statistics, this is called a two-sample test - a test to answer the question whether datasets $X=\{x_1,\ldots, x_n\}$ and $X'=\{x'_1,\ldots, x'_n\}$ were drawn from the same distribution. The main difference between most statistics papers and GANs is that the latter use this idea in a constructive way. In other words, rather than just training a model to say "hey, these two datasets do not look like they came from the same distribution", they use the [two-sample test](https://en.wikipedia.org/wiki/Two-sample_hypothesis_testing) to provide training signals to a generative model. This allows us to improve the data generator until it generates something that resembles the real data. At the very least, it needs to fool the classifier even if our classifier is a state of the art deep neural network.
 
 ![Generative Adversarial Networks](../img/gan.svg)
 :label:`fig_gan`
@@ -27,16 +27,16 @@ In other words, for a given discriminator $D$, we update the parameters of the g
 
 $$ \max_G \{ - (1-y) \log(1-D(G(\mathbf z))) \} = \max_G \{ - \log(1-D(G(\mathbf z))) \}.$$
 
-If the generator does a perfect job, then $D(\mathbf x')\approx 1$ so the above loss near 0, which results the gradients are too small to make a good progress for the discriminator. So commonly we minimize the following loss:
+If the generator does a perfect job, then $D(\mathbf x')\approx 1$, so the above loss is near 0, which results in the gradients that are too small to make good progress for the discriminator. So commonly, we minimize the following loss:
 
 $$ \min_G \{ - y \log(D(G(\mathbf z))) \} = \min_G \{ - \log(D(G(\mathbf z))) \}, $$
 
-which is just feed $\mathbf x'=G(\mathbf z)$ into the discriminator but giving label $y=1$.
+which is just feeding $\mathbf x'=G(\mathbf z)$ into the discriminator but giving label $y=1$.
 
 
 To sum up, $D$ and $G$ are playing a "minimax" game with the comprehensive objective function:
 
-$$min_D max_G \{ -E_{x \sim \text{Data}} log D(\mathbf x) - E_{z \sim \text{Noise}} log(1 - D(G(\mathbf z))) \}.$$
+$$\min_D \max_G \{ -E_{x \sim \text{Data}} \log D(\mathbf x) - E_{z \sim \text{Noise}} \log(1 - D(G(\mathbf z))) \}.$$