本篇blog的内容基于原始论文WassersteinGAN-GP(NIPs2017)和《生成对抗网络入门指南》第五章。

一、权重裁剪的问题（为什么要改进GP）

       WGAN理论中前提条件是1-Liposchitz条件，而对应使用的方法是权重剪裁，希望把网络固定在一个大小范围内。

       但是后来发现权重剪裁有许多问题，所以改进WGAN-GP，使用一种叫做梯度惩罚（gradient penalty）方法来代替权重剪裁。从实验中发现效果更稳定，生成的图像质量也越高。

这里权重裁剪问题有两个：

• 限制了网络的而表现能力，因为大小限制在了固定范围，神经网络很难再模拟出复杂的函数，而只能产生一些比较简单的函数。研究者使用了一些模拟数据重现了这些问题，并对之后的WGAN-GP做了对比，第一排是WGAN结果，第二排是WGAN-GP的结果。可以看到WGAN已经丢失了很多数据分布的高阶矩特征，而WGAN-GP则降低了这个问题。

• 梯度爆炸和梯度消失：因为限制了权值大小，所以会导致梯度爆炸或者梯度消失。如图所属，WGAN三种选择都产生了梯度爆炸或者消失的情况。

二、梯度惩罚（GP）

观察1-Liposchitz条件，所有满足该条件的函数在任意梯度都小于1，可以更换目标函数，在原有的WGAN基础上增加梯度惩罚项 。

       具体实现方法是在真实数据分布和生成数据分布各进行一次采样，然后在两点的连线上再做一次随机采样，就是我们希望的惩罚项采样，且默认超参数取10。另外由于惩罚项无法使用BN，所以①我们不采用BN，发现效果依然很好。这里论文也②推荐使用层归一化（layer normalization）来代替BN。

③这里也重新采用了Adam，不存在WGAN中使用Adam方法稳定性不高的问题。

论文中最后的损失函数：

三、 WGAN-GP伪代码

1. 采样真实数据，隐含变量，以及一个随机数

2. 使用Adam进行训练判别器

3. 从前置随机分布采样

4. 使用Adam训练生成器

完整伪代码

四、WGAN-GP代码

1. 初始化超参数

使用Adam下降模型，但是官方给出的源码还是RMSProp

# Large amount of credit goes to:
# https://github.com/keras-team/keras-contrib/blob/master/examples/improved_wgan.py
# which I've used as a reference for this implementation

from __future__ import print_function, division

from keras.datasets import mnist
from keras.layers.merge import _Merge
from keras.layers import Input, Dense, Reshape, Flatten, Dropout
from keras.layers import BatchNormalization, Activation, ZeroPadding2D
from keras.layers.advanced_activations import LeakyReLU
from keras.layers.convolutional import UpSampling2D, Conv2D
from keras.models import Sequential, Model
from keras.optimizers import RMSprop
from functools import partial

import keras.backend as K

import matplotlib.pyplot as plt

import sys

import numpy as np

class RandomWeightedAverage(_Merge):
    """Provides a (random) weighted average between real and generated image samples"""
    def _merge_function(self, inputs):
        alpha = K.random_uniform((32, 1, 1, 1))
        return (alpha * inputs[0]) + ((1 - alpha) * inputs[1])

class WGANGP():
    def __init__(self):
        self.img_rows = 28
        self.img_cols = 28
        self.channels = 1
        self.img_shape = (self.img_rows, self.img_cols, self.channels)
        self.latent_dim = 100

        # Following parameter and optimizer set as recommended in paper
        self.n_critic = 5
        optimizer = RMSprop(lr=0.00005)

        # Build the generator and critic
        self.generator = self.build_generator()
        self.critic = self.build_critic()

        #-------------------------------
        # Construct Computational Graph
        #       for the Critic
        #-------------------------------

        # Freeze generator's layers while training critic
        self.generator.trainable = False

2. GP采样

在真实数据分布和生成数据分布各进行一次采样，然后在两点的连线上再做一次随机采样，就是我们希望的惩罚项采样

        # Image input (real sample)
        real_img = Input(shape=self.img_shape)

        # Noise input
        z_disc = Input(shape=(100,))
        # Generate image based of noise (fake sample)
        fake_img = self.generator(z_disc)

        # Discriminator determines validity of the real and fake images
        fake = self.critic(fake_img)
        valid = self.critic(real_img)

        # Construct weighted average between real and fake images
        interpolated_img = RandomWeightedAverage()([real_img, fake_img])
        # Determine validity of weighted sample
        validity_interpolated = self.critic(interpolated_img)

        # Use Python partial to provide loss function with additional
        # 'averaged_samples' argument
        partial_gp_loss = partial(self.gradient_penalty_loss,
                          averaged_samples=interpolated_img)
        partial_gp_loss.__name__ = 'gradient_penalty' # Keras requires function names

        self.critic_model = Model(inputs=[real_img, z_disc],
                            outputs=[valid, fake, validity_interpolated])
        self.critic_model.compile(loss=[self.wasserstein_loss,
                                              self.wasserstein_loss,
                                              partial_gp_loss],
                                        optimizer=optimizer,
                                        loss_weights=[1, 1, 10])
        #-------------------------------
        # Construct Computational Graph
        #         for Generator
        #-------------------------------

        # For the generator we freeze the critic's layers
        self.critic.trainable = False
        self.generator.trainable = True

        # Sampled noise for input to generator
        z_gen = Input(shape=(100,))
        # Generate images based of noise
        img = self.generator(z_gen)
        # Discriminator determines validity
        valid = self.critic(img)
        # Defines generator model
        self.generator_model = Model(z_gen, valid)
        self.generator_model.compile(loss=self.wasserstein_loss, optimizer=optimizer)

3. 构造损失函数

①GP损失

    def gradient_penalty_loss(self, y_true, y_pred, averaged_samples):
        """
        Computes gradient penalty based on prediction and weighted real / fake samples
        """
        gradients = K.gradients(y_pred, averaged_samples)[0]
        # compute the euclidean norm by squaring ...
        gradients_sqr = K.square(gradients)
        #   ... summing over the rows ...
        gradients_sqr_sum = K.sum(gradients_sqr,
                                  axis=np.arange(1, len(gradients_sqr.shape)))
        #   ... and sqrt
        gradient_l2_norm = K.sqrt(gradients_sqr_sum)
        # compute lambda * (1 - ||grad||)^2 still for each single sample
        gradient_penalty = K.square(1 - gradient_l2_norm)
        # return the mean as loss over all the batch samples
        return K.mean(gradient_penalty)

②Wasserstein损失

    def wasserstein_loss(self, y_true, y_pred):
        return K.mean(y_true * y_pred)

4. 构造生成器和判别器模型（同普通GAN）

①构造生成器模型

    def build_generator(self):

        model = Sequential()

        model.add(Dense(128 * 7 * 7, activation="relu", input_dim=self.latent_dim))
        model.add(Reshape((7, 7, 128)))
        model.add(UpSampling2D())
        model.add(Conv2D(128, kernel_size=4, padding="same"))
        model.add(BatchNormalization(momentum=0.8))
        model.add(Activation("relu"))
        model.add(UpSampling2D())
        model.add(Conv2D(64, kernel_size=4, padding="same"))
        model.add(BatchNormalization(momentum=0.8))
        model.add(Activation("relu"))
        model.add(Conv2D(self.channels, kernel_size=4, padding="same"))
        model.add(Activation("tanh"))

        model.summary()

        noise = Input(shape=(self.latent_dim,))
        img = model(noise)

        return Model(noise, img)

②构造判别器模型

    def build_critic(self):

        model = Sequential()

        model.add(Conv2D(16, kernel_size=3, strides=2, input_shape=self.img_shape, padding="same"))
        model.add(LeakyReLU(alpha=0.2))
        model.add(Dropout(0.25))
        model.add(Conv2D(32, kernel_size=3, strides=2, padding="same"))
        model.add(ZeroPadding2D(padding=((0,1),(0,1))))
        model.add(BatchNormalization(momentum=0.8))
        model.add(LeakyReLU(alpha=0.2))
        model.add(Dropout(0.25))
        model.add(Conv2D(64, kernel_size=3, strides=2, padding="same"))
        model.add(BatchNormalization(momentum=0.8))
        model.add(LeakyReLU(alpha=0.2))
        model.add(Dropout(0.25))
        model.add(Conv2D(128, kernel_size=3, strides=1, padding="same"))
        model.add(BatchNormalization(momentum=0.8))
        model.add(LeakyReLU(alpha=0.2))
        model.add(Dropout(0.25))
        model.add(Flatten())
        model.add(Dense(1))

        model.summary()

        img = Input(shape=self.img_shape)
        validity = model(img)

        return Model(img, validity)

5. 训练

    def train(self, epochs, batch_size, sample_interval=50):

        # Load the dataset
        (X_train, _), (_, _) = mnist.load_data()

        # Rescale -1 to 1
        X_train = (X_train.astype(np.float32) - 127.5) / 127.5
        X_train = np.expand_dims(X_train, axis=3)

        # Adversarial ground truths
        valid = -np.ones((batch_size, 1))
        fake =  np.ones((batch_size, 1))
        dummy = np.zeros((batch_size, 1)) # Dummy gt for gradient penalty
        for epoch in range(epochs):

            for _ in range(self.n_critic):

                # ---------------------
                #  Train Discriminator
                # ---------------------

                # Select a random batch of images
                idx = np.random.randint(0, X_train.shape[0], batch_size)
                imgs = X_train[idx]
                # Sample generator input
                noise = np.random.normal(0, 1, (batch_size, self.latent_dim))
                # Train the critic
                d_loss = self.critic_model.train_on_batch([imgs, noise],
                                                                [valid, fake, dummy])

            # ---------------------
            #  Train Generator
            # ---------------------

            g_loss = self.generator_model.train_on_batch(noise, valid)

            # Plot the progress
            print ("%d [D loss: %f] [G loss: %f]" % (epoch, d_loss[0], g_loss))

            # If at save interval => save generated image samples
            if epoch % sample_interval == 0:
                self.sample_images(epoch)

6. 可视化

    def sample_images(self, epoch):
        r, c = 5, 5
        noise = np.random.normal(0, 1, (r * c, self.latent_dim))
        gen_imgs = self.generator.predict(noise)

        # Rescale images 0 - 1
        gen_imgs = 0.5 * gen_imgs + 1

        fig, axs = plt.subplots(r, c)
        cnt = 0
        for i in range(r):
            for j in range(c):
                axs[i,j].imshow(gen_imgs[cnt, :,:,0], cmap='gray')
                axs[i,j].axis('off')
                cnt += 1
        fig.savefig("images/mnist_%d.png" % epoch)
        plt.close()

7. 运行

if __name__ == '__main__':
    wgan = WGANGP()
    wgan.train(epochs=30000, batch_size=32, sample_interval=100)

0 [D loss: 8.065693] [G loss: -0.143634]

100 [D loss: -0.699471] [G loss: -2.357683]

200 [D loss: -0.720496] [G loss: -2.259384]

由于训练时间原因放出前3次训练结果。

WGAN-GP在CIFAR-10的表现明显十分优秀。

五、效果分析

对四种GAN在大量情况下实验对哦比，可以看到DCGAN和LSGAN（最小二乘GAN）大多数条件小之下已经无法很好的生成图像，而WGAN虽然稳定生成，后几组很模糊，但是WGAN-GP都保证了高质量表现。