GAN简介

本文主要是介绍GAN简介，希望对大家解决编程问题提供一定的参考价值，需要的开发者们随着小编来一起学习吧！

1. Auto-Encoder

auto-encoder是一种无监督学习算法，研究的是将一个高维的向量或者矩阵进行encode，然后再decode，使得输出与输出值越接近越好。

训练过程中，encoder和decoder是无法单独训练的，必须两个一起训练，然后训练完成后可以将encoder和decoder单独拿出来。
传统的PCA的过程为:

因此PCA是通过线性变换，将一个高维向量转换成一个低维的$code$，但是这里面只有一个$hidden\hspace{0.2cm}layer$。而$auto-encoder$则允许多个隐藏层：

但是需要用$RBM(Restricted\hspace{0.2cm} Boltzmann\hspace{0.2cm} Machine)$来对权值进行初始化才能训练出比较好的结果。
Auto-Encoder可以用在文本检索上：

图像检索:

将图像压缩至256维，然后比较这256维向量的相似性，比直接比较图像的相似性效果要好。

De-noising auto-encoder：

在图像上面人为加入一些噪声点，这样学习的结果鲁棒性更好。

CNN里面的Auto-Encoder:

就是做unpooling和deconvolution操作，使得两个图片越相似越好。
1. unpooling操作
方法1:

pooling的时候记住最大值的位置，unpooling的时候将其他地方填上0.
方法2:
不记住最大值的位置，unpooling的时候将其他地方填上最大值即可。
2. deconvolution操作
deconvolution操作其实就是convolution操作，由低维变高维的时候，在其他的地方填上0即可。

2. VAE(Variational Auto-Encoder)

通常Auto-Encoder不能直接用来生成，需要用到VAE。

通常在学习$m_1, m_2, m_3, \cdots, m_n$之外，还会学习$\sigma_1, \sigma_2, \sigma_3, \cdots, \sigma_n$和$e_1, e_2, e_3,\cdots, e_n$。通常$e_1, e_2, e_3, \cdots, e_n$是从正太分不中取样出来的。
但这种情况最后学习的结果应该让$\sigma_i=0$。因此一般还会加上另一限制函数:

$$l = \sum_{i=1}^n[(e^{\sigma_i}-(1+\sigma_i)+m_i^2)]$$
因此需要使得 $l$越小越好。此时$\sigma_i$会接近0。

3. GAN的基本结构

由于VAE无法GAN类似于一个Auto-Encoder的不断演化的结构。训练过程为：
1. 首先将VAE的decoder拿出来，然后产生一组假的数据，并标记为0。然后取一些真的数据，并标记为1：

2. 训练下一个Generator

此时的Discriminator能够分辨假的数据和真实的数据，因此对于假的数据，会输出一个比较小的分数。然后固定Discriminator的参数不懂，只不断调整Generator的参数，使得Generator产生的图片无法被Discriminator判别为假即可。
3. 将Generator和Discriminator连接成一个网络

$$Generator + Discriminator => GAN$$
4. 注意，每次训练Generator和Discrimator的时候，只能训练一个，另一个必须要固定参数。即训练Generator的时候，Discriminator的参数不能动。训练Discriminator的时候，Generator的参数不能动。

4. GAN的基本原理

首先从现有的数据中挑选出一批数据，组成$P_{data}(x)$，然后训练一个$P_G(x; \theta)$来产生数据(例如一个高斯混合模型)，我们希望产生的数据集$P_G(x; \theta)$与原来的数据$P_{data}(x)$越接近越好，即使得下面的似然函数达到最大值：

$$L=\prod_{i=1}^mP_G(x^i; \theta)$$
因此需要求得的参数为：
$$\theta^*=\arg\max_\theta\prod_{i=1}^mP_G(x^i; \theta)$$
取对数，得：
$$\begin{align} \theta^*&=\arg\max_\theta\sum_{i=1}^m\ln P_G(x^i; \theta) \hspace{1.0cm}\{x^1, x^2, \cdots, x^m\}\in P_{data}(x)\\ &\approx\arg\max_\theta E_{x\sim P_{data}}[\ln P_G(x; \theta)]\\ &=\arg\max_\theta[\int_xP_{data}(x)\ln P_G(x; \theta)dx-\int_xP_{data}(x)\ln P_{data}(x)dx]\\ &=\arg\min_\theta KL(P_{data}(x)||P_G(x; \theta)) \end{align}$$
即最后需要使得 $P_{data}(x)$和 $P_G(x; \theta)$的 $KL \hspace{0.2cm} divergence$达到最小，即这两个分布越相近越好。
在 $GAN$当中， $P_G(x; \theta)$就是一个神经网络， $\theta$就是网络的各种参数：

此时可以选择一个分布 $z$，然后通过一个神经网络产生一组数据$P_G(x; \theta)$，使得 $P_G(x; \theta)$与 $P_{data}(x)$越接近越好，因此：
$$P_G(x)=\int_zP_{prior}(z)I_{[G(z)=x]}dz$$
表示枚举所有可能的 $z$，然后对其就行求积分。其中$P_{prior}(z)$表示点 $z$出现的概率，$I_{[G(z)=x]}$表示当 $G(z)$与 $x$相等时取$1$，否则取 $0$。
但是$\int_zP_{prior}(z)I_{[G(z)=x]}dz$不好计算，因此 $GAN$采用 $Discriminator$来计算 $KL \hspace{0.2cm} divergence$。 $Discriminator$会输出一个数值 $V(G, D)$来表示产生的数据和实际的数据之间的差值，因此其实要求的 $G$为:
$$G^*=\arg\min_G\max_DV(G,D)$$
其中
$$\begin{align} V(G, D)&=E_{x\sim P_{data}}[\ln D(x)]+E_{x\sim P_G}[\ln(1-D(x))]\\ &=\int_x[P_{data}(x)\ln D(x)+P_G(x)\ln (1-D(x))]dx \end{align}$$
解得：
$$D^*=\frac {P_{data(x)}}{P_{data}(x)+P_G(x)}$$
最终，给定 $G$，有：
$$\begin{align} \max_DV(G, D)&=-2\ln2+\int_xP_{data}(x)\ln \frac {P_{data}(x)}{(P_{data}(x)+P_G(x))/2}dx+\\ &\hspace{2.5cm}\int_xP_{G}(x)\ln \frac {P_{G}(x)}{(P_{data}(x)+P_G(x))/2}dx\\ &=-2\ln2+KL(P_{data}(x)||\frac {P_{data}(x)+P_G(x)}{2})+\\ &\hspace{2.5cm}KL(P_G(x)||\frac {P_{data}(x)+P_G(x)}{2})\\ &=-2\ln2+2JSD(P_{data}(x)||P_G(x)) \end{align}$$
因此最优的 $P_G(x)$就是 $P_{data}(x)$,因此：
$$G^*=\arg\min_G\max_DV(G,D)$$
令
$$L(G)=\max_DV(G,D)$$
解决过程为：
$$\theta_G=\theta-\eta\frac{\partial L(G)}{\partial \theta_G}$$
这样得到 $G$即可。
整个算法的过程为：
1. 从真实的数据分布$P_{data}(x)$中取样一些数据 $\{x^1, x^2, \cdots, x^m\}$
2. 从预先的encoder中取出一些数据 $\{z^1, z^2, \cdots, z^m\}$
3. 通过 $\{z^1, z^2, \cdots, z^m\}$得到一些数据 $\{\widetilde x^1, \widetilde x^2, \cdots, \widetilde x^m\}$，其中 $\widetilde x^i=G(z^i)$
4. 改变 $discriminator$的参数：
$$\begin{align} \widetilde V&=\frac 1m\sum_{i=1}^m\ln D(x^i)+\frac 1m\sum_{i=1}^m\ln(1-D(\widetilde x^i))\\ \theta_d&=\theta_d+\eta\nabla \widetilde V(\theta_d) \end{align}$$
使得失眠的 $\widetilde V$到达 最大
5. 重复2中的步骤，然后改变 $generator$的参数，使得下式的 $\widetilde V$到达 最小。
$$\begin{align} \widetilde V&=\frac 1m\sum_{i=1}^m\ln(1-D(\widetilde x^i))\\ \theta_g&=\theta_g-\eta\nabla \widetilde V(\theta_g) \end{align}$$
6. 重复4，5。
注意
上式中的$D(x^i)$表示的是真实图片被判别为1的概率，$D(\widetilde x^i)$表示生成的图片被判别为1的概率。因子Discriminator就是要使得$D(x^i)$接近1，$D(\widetilde x^i)$接近0。而Generator就是要使得$D(\widetilde x^i)$接近1。

lan Goodfellow GAN教程

5. GAN的TensorFlow简单实现

import tensorflow as tf #machine learning
import numpy as np #matrix math
import datetime #logging the time for model checkpoints and training
import matplotlib.pyplot as plt #visualize results
%matplotlib inline#Step 1 - Collect dataset
#MNIST - handwritten character digits ~50K training and validation images + labels, 10K testing
from tensorflow.examples.tutorials.mnist import input_data
#will ensure that the correct data has been downloaded to your 
#local training folder and then unpack that data to return a dictionary of DataSet instances.
mnist = input_data.read_data_sets("MNIST_data/")def discriminator(x_image, reuse=False):if (reuse):tf.get_variable_scope().reuse_variables()# First convolutional and pool layers# These search for 32 different 5 x 5 pixel features#We’ll start off by passing the image through a convolutional layer. #First, we create our weight and bias variables through tf.get_variable. #Our first weight matrix (or filter) will be of size 5x5 and will have a output depth of 32. #It will be randomly initialized from a normal distribution.d_w1 = tf.get_variable('d_w1', [5, 5, 1, 32], initializer=tf.truncated_normal_initializer(stddev=0.02))#tf.constant_init generates tensors with constant values.d_b1 = tf.get_variable('d_b1', [32], initializer=tf.constant_initializer(0))#tf.nn.conv2d() is the Tensorflow’s function for a common convolution.#It takes in 4 arguments. The first is the input volume (our 28 x 28 x 1 image in this case). #The next argument is the filter/weight matrix. Finally, you can also change the stride and #padding of the convolution. Those two values affect the dimensions of the output volume.#"SAME" tries to pad evenly left and right, but if the amount of columns to be added is odd, #it will add the extra column to the right,#strides = [batch, height, width, channels]d1 = tf.nn.conv2d(input=x_image, filter=d_w1, strides=[1, 1, 1, 1], padding='SAME')#add the biasd1 = d1 + d_b1#squash with nonlinearity (ReLU)d1 = tf.nn.relu(d1)##An average pooling layer performs down-sampling by dividing the input into #rectangular pooling regions and computing the average of each region. #It returns the averages for the pooling regions.d1 = tf.nn.avg_pool(d1, ksize=[1, 2, 2, 1], strides=[1, 2, 2, 1], padding='SAME')#As with any convolutional neural network, this module is repeated, # Second convolutional and pool layers# These search for 64 different 5 x 5 pixel featuresd_w2 = tf.get_variable('d_w2', [5, 5, 32, 64], initializer=tf.truncated_normal_initializer(stddev=0.02))d_b2 = tf.get_variable('d_b2', [64], initializer=tf.constant_initializer(0))d2 = tf.nn.conv2d(input=d1, filter=d_w2, strides=[1, 1, 1, 1], padding='SAME')d2 = d2 + d_b2d2 = tf.nn.relu(d2)d2 = tf.nn.avg_pool(d2, ksize=[1, 2, 2, 1], strides=[1, 2, 2, 1], padding='SAME')#and then followed by a series of fully connected layers. # First fully connected layerd_w3 = tf.get_variable('d_w3', [7 * 7 * 64, 1024], initializer=tf.truncated_normal_initializer(stddev=0.02))d_b3 = tf.get_variable('d_b3', [1024], initializer=tf.constant_initializer(0))d3 = tf.reshape(d2, [-1, 7 * 7 * 64])d3 = tf.matmul(d3, d_w3)d3 = d3 + d_b3d3 = tf.nn.relu(d3)#The last fully-connected layer holds the output, such as the class scores.# Second fully connected layerd_w4 = tf.get_variable('d_w4', [1024, 1], initializer=tf.truncated_normal_initializer(stddev=0.02))d_b4 = tf.get_variable('d_b4', [1], initializer=tf.constant_initializer(0))#At the end of the network, we do a final matrix multiply and #return the activation value. #For those of you comfortable with CNNs, this is just a simple binary classifier. Nothing fancy.# Final layerd4 = tf.matmul(d3, d_w4) + d_b4# d4 dimensions: batch_size x 1return d4#You can think of the generator as being a kind of reverse ConvNet. With CNNs, the goal is to 
#transform a 2 or 3 dimensional matrix of pixel values into a single probability. A generator, 
#however, seeks to take a d-dimensional noise vector and upsample it to become a 28 x 28 image. 
#ReLUs are then used to stabilize the outputs of each layer.
#example of CNN blocks http://cs231n.github.io/convolutional-networks/#fc#it takes random inputs, and eventually mapping them down to a [1,28,28] pixel to match the MNIST data shape.  
#Be begin by generating a dense 14×14 set of values, and then run through a handful of filters of
#varying sizes and numbers of channels
#weight matrices get progressively smallerdef generator(batch_size, z_dim):z = tf.truncated_normal([batch_size, z_dim], mean=0, stddev=1, name='z')#first deconv blockg_w1 = tf.get_variable('g_w1', [z_dim, 3136], dtype=tf.float32, initializer=tf.truncated_normal_initializer(stddev=0.02))g_b1 = tf.get_variable('g_b1', [3136], initializer=tf.truncated_normal_initializer(stddev=0.02))g1 = tf.matmul(z, g_w1) + g_b1g1 = tf.reshape(g1, [-1, 56, 56, 1])g1 = tf.contrib.layers.batch_norm(g1, epsilon=1e-5, scope='bn1')g1 = tf.nn.relu(g1)# Generate 50 featuresg_w2 = tf.get_variable('g_w2', [3, 3, 1, z_dim/2], dtype=tf.float32, initializer=tf.truncated_normal_initializer(stddev=0.02))g_b2 = tf.get_variable('g_b2', [z_dim/2], initializer=tf.truncated_normal_initializer(stddev=0.02))g2 = tf.nn.conv2d(g1, g_w2, strides=[1, 2, 2, 1], padding='SAME')g2 = g2 + g_b2g2 = tf.contrib.layers.batch_norm(g2, epsilon=1e-5, scope='bn2')g2 = tf.nn.relu(g2)g2 = tf.image.resize_images(g2, [56, 56])# Generate 25 featuresg_w3 = tf.get_variable('g_w3', [3, 3, z_dim/2, z_dim/4], dtype=tf.float32, initializer=tf.truncated_normal_initializer(stddev=0.02))g_b3 = tf.get_variable('g_b3', [z_dim/4], initializer=tf.truncated_normal_initializer(stddev=0.02))g3 = tf.nn.conv2d(g2, g_w3, strides=[1, 2, 2, 1], padding='SAME')g3 = g3 + g_b3g3 = tf.contrib.layers.batch_norm(g3, epsilon=1e-5, scope='bn3')g3 = tf.nn.relu(g3)g3 = tf.image.resize_images(g3, [56, 56])# Final convolution with one output channelg_w4 = tf.get_variable('g_w4', [1, 1, z_dim/4, 1], dtype=tf.float32, initializer=tf.truncated_normal_initializer(stddev=0.02))g_b4 = tf.get_variable('g_b4', [1], initializer=tf.truncated_normal_initializer(stddev=0.02))g4 = tf.nn.conv2d(g3, g_w4, strides=[1, 2, 2, 1], padding='SAME')g4 = g4 + g_b4g4 = tf.sigmoid(g4)# No batch normalization at the final layer, but we do add# a sigmoid activator to make the generated images crisper.# Dimensions of g4: batch_size x 28 x 28 x 1return g4sess = tf.Session()batch_size = 50
z_dimensions = 100x_placeholder = tf.placeholder("float", shape = [None,28,28,1], name='x_placeholder')
# x_placeholder is for feeding input images to the discriminator#One of the trickiest parts about understanding GANs is that the loss function is a little bit more complex than that
#of a traditional CNN classifiers (For those, a simple MSE or Hinge Loss would do the trick). 
#If you think back to the introduction, a GAN can be thought of as a zero sum minimax game. 
#The generator is constantly improving to produce more and more realistic images, while the discriminator is 
#trying to get better and better at distinguishing between real and generated images.
#This means that we need to formulate loss functions that affect both networks. 
#Let’s take a look at the inputs and outputs of our networks.Gz = generator(batch_size, z_dimensions)
# Gz holds the generated images
#g(z)Dx = discriminator(x_placeholder)
# Dx hold the discriminator's prediction probabilities
# for real MNIST images
#d(x)Dg = discriminator(Gz, reuse=True)
# Dg holds discriminator prediction probabilities for generated images
#d(g(z))#So, let’s first think about what we want out of our networks. We want the generator network to create 
#images that will fool the discriminator. The generator wants the discriminator to output a 1 (positive example).
#Therefore, we want to compute the loss between the Dg and label of 1. This can be done through 
#the tf.nn.sigmoid_cross_entropy_with_logits function. This means that the cross entropy loss will 
#be taken between the two arguments. The "with_logits" component means that the function will operate 
#on unscaled values. Basically, this means that instead of using a softmax function to squish the output
#activations to probability values from 0 to 1, we simply return the unscaled value of the matrix multiplication.
#Take a look at the last line of our discriminator. There's no softmax or sigmoid layer at the end.
#The reduce mean function just takes the mean value of all of the components in the matrixx returned 
#by the cross entropy function. This is just a way of reducing the loss to a single scalar value, 
#instead of a vector or matrix.
#https://datascience.stackexchange.com/questions/9302/the-cross-entropy-error-function-in-neural-networksg_loss = tf.reduce_mean(tf.nn.sigmoid_cross_entropy_with_logits(logits=Dg, labels=tf.ones_like(Dg)))#Now, let’s think about the discriminator’s point of view. Its goal is to just get the correct labels 
#(output 1 for each MNIST digit and 0 for the generated ones). We’d like to compute the loss between Dx 
#and the correct label of 1 as well as the loss between Dg and the correct label of 0.
d_loss_real = tf.reduce_mean(tf.nn.sigmoid_cross_entropy_with_logits(logits=Dx, labels=tf.fill([batch_size, 1], 0.9)))
d_loss_fake = tf.reduce_mean(tf.nn.sigmoid_cross_entropy_with_logits(logits=Dg, labels=tf.zeros_like(Dg)))
d_loss = d_loss_real + d_loss_faketvars = tf.trainable_variables()d_vars = [var for var in tvars if 'd_' in var.name]
g_vars = [var for var in tvars if 'g_' in var.name]# Train the discriminator
# Increasing from 0.001 in GitHub version
with tf.variable_scope(tf.get_variable_scope(), reuse=False) as scope:#Next, we specify our two optimizers. In today’s era of deep learning, Adam seems to be the#best SGD optimizer as it utilizes adaptive learning rates and momentum. #We call Adam's minimize function and also specify the variables that we want it to update.d_trainer_fake = tf.train.AdamOptimizer(0.0001).minimize(d_loss_fake, var_list=d_vars)d_trainer_real = tf.train.AdamOptimizer(0.0001).minimize(d_loss_real, var_list=d_vars)# Train the generator# Decreasing from 0.004 in GitHub versiong_trainer = tf.train.AdamOptimizer(0.0001).minimize(g_loss, var_list=g_vars)#Outputs a Summary protocol buffer containing a single scalar value.
tf.summary.scalar('Generator_loss', g_loss)
tf.summary.scalar('Discriminator_loss_real', d_loss_real)
tf.summary.scalar('Discriminator_loss_fake', d_loss_fake)d_real_count_ph = tf.placeholder(tf.float32)
d_fake_count_ph = tf.placeholder(tf.float32)
g_count_ph = tf.placeholder(tf.float32)tf.summary.scalar('d_real_count', d_real_count_ph)
tf.summary.scalar('d_fake_count', d_fake_count_ph)
tf.summary.scalar('g_count', g_count_ph)# Sanity check to see how the discriminator evaluates
# generated and real MNIST images
d_on_generated = tf.reduce_mean(discriminator(generator(batch_size, z_dimensions)))
d_on_real = tf.reduce_mean(discriminator(x_placeholder))tf.summary.scalar('d_on_generated_eval', d_on_generated)
tf.summary.scalar('d_on_real_eval', d_on_real)images_for_tensorboard = generator(batch_size, z_dimensions)
tf.summary.image('Generated_images', images_for_tensorboard, 10)
merged = tf.summary.merge_all()
logdir = "tensorboard/gan/"
writer = tf.summary.FileWriter(logdir, sess.graph)
print(logdir)saver = tf.train.Saver()sess.run(tf.global_variables_initializer())#During every iteration, there will be two updates being made, one to the discriminator and one to the generator. 
#For the generator update, we’ll feed in a random z vector to the generator and pass that output to the discriminator
#to obtain a probability score (this is the Dg variable we specified earlier).
#As we remember from our loss function, the cross entropy loss gets minimized, 
#and only the generator’s weights and biases get updated.
#We'll do the same for the discriminator update. We’ll be taking a batch of images 
#from the mnist variable we created way at the beginning of our program.
#These will serve as the positive examples, while the images in the previous section are the negative ones.gLoss = 0
dLossFake, dLossReal = 1, 1
d_real_count, d_fake_count, g_count = 0, 0, 0
for i in range(50000):real_image_batch = mnist.train.next_batch(batch_size)[0].reshape([batch_size, 28, 28, 1])if dLossFake > 0.6:# Train discriminator on generated images_, dLossReal, dLossFake, gLoss = sess.run([d_trainer_fake, d_loss_real, d_loss_fake, g_loss],{x_placeholder: real_image_batch})d_fake_count += 1if gLoss > 0.5:# Train the generator_, dLossReal, dLossFake, gLoss = sess.run([g_trainer, d_loss_real, d_loss_fake, g_loss],{x_placeholder: real_image_batch})g_count += 1if dLossReal > 0.45:# If the discriminator classifies real images as fake,# train discriminator on real values_, dLossReal, dLossFake, gLoss = sess.run([d_trainer_real, d_loss_real, d_loss_fake, g_loss],{x_placeholder: real_image_batch})d_real_count += 1if i % 10 == 0:real_image_batch = mnist.validation.next_batch(batch_size)[0].reshape([batch_size, 28, 28, 1])summary = sess.run(merged, {x_placeholder: real_image_batch, d_real_count_ph: d_real_count,d_fake_count_ph: d_fake_count, g_count_ph: g_count})writer.add_summary(summary, i)d_real_count, d_fake_count, g_count = 0, 0, 0if i % 1000 == 0:# Periodically display a sample image in the notebook# (These are also being sent to TensorBoard every 10 iterations)images = sess.run(generator(3, z_dimensions))d_result = sess.run(discriminator(x_placeholder), {x_placeholder: images})print("TRAINING STEP", i, "AT", datetime.datetime.now())for j in range(3):print("Discriminator classification", d_result[j])im = images[j, :, :, 0]plt.imshow(im.reshape([28, 28]), cmap='Greys')plt.show()if i % 5000 == 0:print(i)print(i)
#         save_path = saver.save(sess, "models/pretrained_gan.ckpt", global_step=i)
#         print("saved to %s" % save_path)test_images = sess.run(generator(10, 100))
test_eval = sess.run(discriminator(x_placeholder), {x_placeholder: test_images})real_images = mnist.validation.next_batch(10)[0].reshape([10, 28, 28, 1])
real_eval = sess.run(discriminator(x_placeholder), {x_placeholder: real_images})# Show discriminator's probabilities for the generated images,
# and display the images
for i in range(10):print(test_eval[i])plt.imshow(test_images[i, :, :, 0], cmap='Greys')plt.show()# Now do the same for real MNIST images
for i in range(10):print(real_eval[i])plt.imshow(real_images[i, :, :, 0], cmap='Greys')plt.show()

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