[GAN] 2. Detailed explanation of the original GAN paper

written in front

In the previous article: [GAN] 1. Using keras to implement DCGAN to generate handwritten digital images In we used keras to implement a simple DCGAN and generated handwritten digital images. The results of the program let us appreciate the power of GAN, and then we begin to introduce various GAN models step by step. So let's start with the most basic GAN.

---

1. Introduction to GAN

The full name of GAN (Generative Adversarial Network) is called confrontation generation network or generation confrontation network. The concept of GAN was proposed by Ian Goodfellow in 2014 and quickly became a very hot research topic. At present, there are thousands of variants of GAN. Yann LeCun, the winner of the Nobel Prize "Turing Award" in the computer industry in 2019 and one of the pioneers of deep learning, also said: "GAN and its variants have been the foundation of machine learning for decades. The most interesting idea in the field.”

The paper link of the original GAN ​​is: Generative Adversarial Nets

First, let's summarize the original GAN ​​in one sentence. The original GAN ​​consists of two organic ensembles - the generator $G$ and discriminator$D$ , the purpose of the generator is to map the random input Gaussian noise into an image ("false picture"), and the discriminator is to judge the probability of whether the input image comes from the generator, that is, the probability of judging whether the input image is a fake picture.

The training of GAN is also very different from CNN. CNN defines a specific loss function, and then uses gradient descent and its improved algorithm to optimize parameters, and uses the local optimal solution to approach the global optimal solution as much as possible. But the training of GAN is a dynamic process, the generator $G$ and discriminator$D$ The mutual game process between the two. In layman's terms,the purpose of GAN is to create something out of nothing, to confuse the real with the fake. That is, to make the generator$The\; so-called\; "false\; image"\; generated\; by\; G$ fools the discriminator$D$ , then the optimal state is the generator$The\; so-called\; "false\; graph"\; generated\; by\; G$ is in the discriminator$The\; discriminant\; result\; of\; D$ is 0.5, and it is not known whether it is a real picture or a fake picture.

Next we will explain the relevant terms in GAN. The first to give an explanation is the generator $G.$ _ Generator$G$ is used to capture the data distribution of the input Gaussian noise, producing a "false graph". Next is the discriminator. Discriminator$D$ is to evaluate the probability that the input sample is from the training set rather than the generator. Training Generator$G$ is to maximize the discriminator$D$ Probability of making a mistake. The entire framework of the original GAN ​​is the generator$G$ and discriminator$D$ The dynamic process of mutual game between the two.

---

2. GAN training

Next we introduce the training of GAN. First, we give the objective function (loss function) of the original GAN, and the loss function is listed as follows:
$$\underset{G}{\min }\underset{D}{\max }V(D,G)={\mathbb{E}}_{x\sim p_{data}(x)}[\log D(x)]+E_{z\sim p_{data}(z)}[\log (1-D(G(z)))]\text{\hspace{1em}(1)}$$

Among them, $G$ is for generator,$D$ stands for discriminator,$x$ represents real data,$p_{data}$Represents the probability density distribution of real data, $z$ represents random input data, which is random Gaussian noise.

As can be seen from the above formula, from the discriminator $D$ point of view discriminator$D$ hopes to distinguish the real samples as much as possible$x$ and false samples$G\left(z\right)$ , so$D\left(x\right)$ must be as large as possible,$D\left(G\right(z\left)\right)$ is as small as possible, that is,$V(D,G)$ The whole is as large as possible. from generator$From\; the\; perspective\; of\; G$ , the generator$G$ hopes to generate fake data$G\left(z\right)$ can fool the discriminator$D$ , that is, hope$D\left(G\right(z\left)\right)$ is as large as possible, that is,$V(D,G)$ The whole is as small as possible. The two modules of GAN are training against each other, and finally reach the global optimum.

The original paper below also gives a schematic diagram of the training process of GAN. In the figure above, the parallel lines represent the noise $z$ , which maps to$x$ , the blue dotted line represents the discriminator$The\; output\; of\; D$ , the black dot represents the real data distribution$p_{data}$, the green solid line represents the generator GGProbability distribution pg p_gof spurious data for$G$$p_g$. As can be seen from the figure below, during the training process of GAN, the generator $The\; probability\; density\; distribution\; of\; G$ is slowly approaching the probability density distribution of the real data set, and the predicted value of the discriminator is also decreasing. When the situation in the figure (d) below appears,$D(G(z))=0.5\; ,\; that\; is,\; it\; is\; impossible\; to\; distinguish\; whether\; the\; input\; image\; isa$ real image or a fake image forged by the generator$.$

Next, we give the algorithm for GAN training, as shown in the figure below. As can be seen from the training algorithm in the figure below,first use the above objective function combined with gradient ascent to train the K-time discriminator, and then combine the gradient descent to train the 1-time discriminator.

In the paper, the author also gave the mathematical proof of the relevant conclusions of GAN. There are already great gods on CSDN who have given the relevant detailed derivation process. I will not give the relevant proof here. If you are interested, please move to: GAN paperreading—— Original GAN ​​(basic concept and theoretical derivation). However, here is a summary of the related conclusions:

1. Generator probability density distribution $p_g$with real data distribution $p_{data}$When equal, the objective function of GAN achieves the global optimal solution.
2. Optimal discriminator $The\; expression\; of\; D$ is:$D_G^{\ast }(x)=\frac{p_{data}(x)}{p_{data}(x)+p_G(x)}$, then GAN achieves the best $p_g=p_{data}$,，那么$D_G^{\ast }\left(x\right)=0.5$
3. Combining the two points 1 and 2, although in the actual training of GAN, we cannot make $p_g=p_{data}$, but we v must try to approach this result as much as possible, so that the generated "false picture" can be confused with the real one.

---

3. Experimental results

The original GAN ​​experimental results are given below. Firstly, the logarithmic maximum likelihood estimation on the MNIST data set and TFD is given, and the experimental results are shown below.

The next step is to visualize the experimental results, train on the mnist data set, TFD data set and CIFAR-10 data set respectively, and then generate intermediate training results, as shown in the figure below. Among them, Figure a is the training result of the mnist data set, Figure b is the training result of the TFD data set, Figure c is the training result of the GAN using the fully connected network on the CIFAR-10 data set, and Figure d is the use of convolution and deconvolution The training results of the product GAN on the CIFAR-10 dataset.

---

postscript

So far, the second part of the GAN series - the detailed explanation of the original GAN ​​paper is over. In the next blog, we will introduce DCGAN in detail.