原文链接：https://arxiv.org/pdf/1606.03657.pdf

简介

背景：从Conditional GAN在输入中加上标签来控制数据生成数据的类别到Deep Conventional GAN发现输入的随机噪声可以控制生成数据的形态，我们已经可以十分肯定，输入的随机噪声可以决定生成的数据，但是它是如何决定的？我们又该如何利用这个原理呢？

核心思想：将输入生成器的向量分为两部分：一为随机噪声z，二为潜在约束编码c；然后通过c来控制生成数据。

引入了约束编码c后，我们势必要考虑一个问题就是：如何使得约束编码c与生成数据相对应呢？此处本文与Conditional GAN的生成器结构十分相似，甚至于将上图中的c换为标签的话，那么Info GAN的生成器就退化成了Conditional GAN的。他们在判别器的设计上有所不同，Conditional GAN是将约束标签与图片一起送入判别器，因为类比标签我们是明确他控制类别的，无需基于它去进行训练。而InfoGAN的约束向量是一个未知的约束条件，是需要训练的。所以，在判别器提取特征之后，使用一个Q网络（FC层）来回归出约束向量，并与生成器输入的约束向量求LOSS来逐步优化。

基础结构

基础概念

信息熵

香农提出了“信息熵”的概念，才解决了对信息的量化度量问题。

在信源中，考虑的不是某一单个符号发生的不确定性，而是要考虑这个信源所有可能发生情况的平均不确定性。若信源符号有n种取值：U1…Ui…Un，对应概率为：P1…Pi…Pn，且各种符号的出现彼此独立。这时，信源的平均不确定性应当为单个符号不确定性-logPi的统计平均值（E），可称为信息熵：

$\dpi{150} H(X) = -\sum_X P(x)logP(x)= -\int_{x}P(x)logP(x)dx = -\mathbb{E}_{x\sim X}[log(p_i)]$

式中对数一般取2为底，单位为比特。但是，也可以取其它对数底，采用其它相应的单位，它们间可用换底公式换算。

互信息

它可以看成是一个随机变量中包含的关于另一个随机变量的信息量，或者说是一个随机变量由于已知另一个随机变量而减少的不肯定性。

$H(X)-H(X|Y) = H(Y)-H(Y|X)$ $H(X,Y) = H(X)+H(Y|X) = H(Y)+H(X|Y)$

$I(X;Y) = H(X) - H(X|Y)$

$=H(X)+H(Y)-H(X,Y)$

$=\sum_xp(x)log \frac{1}{p(x)} + \sum_yp(y)log \frac{1}{p(y)}-\sum-{x,y}p(x,y)log\frac{1}{log(x,y)}$

$=\sum_{x,y}p(x,y)log\frac{p(x.y)}{p(x)p(y)}$

优化流程

从互信息可以看出当p(x)和q(x)独立的时候，I(X;Y)为0，表示无关；若I(X;Y)很高，则可以表示两组关系很大。从生成器中得到了任意x，目标让生成器中的后验Pg(c∣x) ，即潜在编码c具有更小的信息熵；这样才能描述从x到c的回溯过程仍然没有丢失潜在编码的信息：

$\min_G\max_D\mathbb{E}_{x \sim P_{data}}[logD(x)]+\mathbb{E}_{z \sim P_z}[log(1-D(G(z)))]-\lambda I(c;G(z,c))$

此式中前半部分为常规GAN的目标函数，后半部分是以生成器输出与约束编码的互信息作为惩罚项，惩罚系数为 $\lambda$ 。

约束编码的后验分布

由于我们无法得知约束编码的后验分布，于是我们使用添加的Q网络结构来回归约束编码c，通过神经网络训练来拟合出约束编码c的分布。

此处有关于一个关于条件概率的期望证明，即条件概率的期望与其本身的期望是一样的：

$\mathbb{E}_Y[\mathbb{E}_X{X|Y}]=\int_Y\mathbb{E}_x[X|Y]f(y)dy$

$=\int_{Y}[\int_{X}x\frac{f(x,y)}{f(y)}dx]f(y)dy$

$=\int_{Y}\int_{X}x\frac{f(x,y)}{f(y)}dxf(y)dy \ \ f(y)\ \ is \ \ constant \ \ w.r.t \ \ Y$

$=\int _{Y}\int_{X}xf(x,y)dxdy$

$= \int_Xx\int_Yf(x,y)dydx \ \ \ \ X \ \ is\ \ constant\ \ w.r.t\ \ Y$

$=\int_Xx[\int_Yf(x,y)dy]dx$

$=\int_Xxf(x)dx$

$=\mathbb{E}_{X}[X]$

上式为等价递推，则我们可以由此知道：

$\mathbb{E}_{x\sim X,y\sim Y|x}[f(x,y)] =\int_xP(x)\int_yP(y|x)f(x,y)dydx$

$=\int _x\int_yP(x,y)f(x,y)dydx$

$=\int_yP(y)\int_xP(x|y)f(x,y)dxdy$

$=\int_yP(y)[\int_{x'}P(x'|y)f(x',y)dx']dy$

$=\int_xP(x)[\int_yP(y|x)[\int_{x'}P(x'|y)f(x',y)dx']dy]dx$

$=\mathbb{E}_{x\sim X,y \sim Y|x,x' \sim X|y}[f(x',y)]$

基于上面这个结论，我们最后可以缩放互信息为：

$I(c;G(z,c))\geq H(c)+\mathbb{E}_{x \sim G(z,c)}[\mathbb{E}_{c'\sim P(c|x)}[logQ(c'|x)]]$

$=H(c)+\mathbb{E}_{c \sim P(c),x\sim P_{G}(x|c)}[\mathbb{E}_{c' \sim P(c|x)}[logQ(c'|x)]]$

$=H(c) + \mathbb{E}_{c \sim P(c),x\sim G(z,c)}[logQ(c|x)]$

$=L_I(G,Q)$

将目标函数简化为：

$\min_G\max_D\mathbb{E}_{x \sim P_{data}}[logD(x)]+\mathbb{E}_{z \sim P_z}[log(1-D(G(z)))]-\lambda L_I(G,Q)$

代码与实践结果

参考链接（https://github.com/eriklindernoren/PyTorch-GAN/blob/master/implementations/infogan/infogan.py）

import argparse
import os
import numpy as np
import math
import itertools

import torchvision.transforms as transforms
from torchvision.utils import save_image

from torch.utils.data import DataLoader
from torchvision import datasets
from torch.autograd import Variable

import torch.nn as nn
import torch.nn.functional as F
import torch

os.makedirs("images/static/", exist_ok=True)
os.makedirs("images/varying_c1/", exist_ok=True)
os.makedirs("images/varying_c2/", exist_ok=True)

parser = argparse.ArgumentParser()
parser.add_argument("--n_epochs", type=int, default=200, help="number of epochs of training")
parser.add_argument("--batch_size", type=int, default=64, help="size of the batches")
parser.add_argument("--lr", type=float, default=0.0002, help="adam: learning rate")
parser.add_argument("--b1", type=float, default=0.5, help="adam: decay of first order momentum of gradient")
parser.add_argument("--b2", type=float, default=0.999, help="adam: decay of first order momentum of gradient")
parser.add_argument("--n_cpu", type=int, default=8, help="number of cpu threads to use during batch generation")
parser.add_argument("--latent_dim", type=int, default=62, help="dimensionality of the latent space")
parser.add_argument("--code_dim", type=int, default=2, help="latent code")
parser.add_argument("--n_classes", type=int, default=10, help="number of classes for dataset")
parser.add_argument("--img_size", type=int, default=32, help="size of each image dimension")
parser.add_argument("--channels", type=int, default=1, help="number of image channels")
parser.add_argument("--sample_interval", type=int, default=400, help="interval between image sampling")
opt = parser.parse_args()
print(opt)

cuda = True if torch.cuda.is_available() else False


def weights_init_normal(m):
    classname = m.__class__.__name__
    if classname.find("Conv") != -1:
        torch.nn.init.normal_(m.weight.data, 0.0, 0.02)
    elif classname.find("BatchNorm") != -1:
        torch.nn.init.normal_(m.weight.data, 1.0, 0.02)
        torch.nn.init.constant_(m.bias.data, 0.0)


def to_categorical(y, num_columns):
    """Returns one-hot encoded Variable"""
    y_cat = np.zeros((y.shape[0], num_columns))
    y_cat[range(y.shape[0]), y] = 1.0

    return Variable(FloatTensor(y_cat))


class Generator(nn.Module):
    def __init__(self):
        super(Generator, self).__init__()
        input_dim = opt.latent_dim + opt.n_classes + opt.code_dim

        self.init_size = opt.img_size // 4  # Initial size before upsampling
        self.l1 = nn.Sequential(nn.Linear(input_dim, 128 * self.init_size ** 2))

        self.conv_blocks = nn.Sequential(
            nn.BatchNorm2d(128),
            nn.Upsample(scale_factor=2),
            nn.Conv2d(128, 128, 3, stride=1, padding=1),
            nn.BatchNorm2d(128, 0.8),
            nn.LeakyReLU(0.2, inplace=True),
            nn.Upsample(scale_factor=2),
            nn.Conv2d(128, 64, 3, stride=1, padding=1),
            nn.BatchNorm2d(64, 0.8),
            nn.LeakyReLU(0.2, inplace=True),
            nn.Conv2d(64, opt.channels, 3, stride=1, padding=1),
            nn.Tanh(),
        )

    def forward(self, noise, labels, code):
        gen_input = torch.cat((noise, labels, code), -1)
        out = self.l1(gen_input)
        out = out.view(out.shape[0], 128, self.init_size, self.init_size)
        img = self.conv_blocks(out)
        return img


class Discriminator(nn.Module):
    def __init__(self):
        super(Discriminator, self).__init__()

        def discriminator_block(in_filters, out_filters, bn=True):
            """Returns layers of each discriminator block"""
            block = [nn.Conv2d(in_filters, out_filters, 3, 2, 1), nn.LeakyReLU(0.2, inplace=True), nn.Dropout2d(0.25)]
            if bn:
                block.append(nn.BatchNorm2d(out_filters, 0.8))
            return block

        self.conv_blocks = nn.Sequential(
            *discriminator_block(opt.channels, 16, bn=False),
            *discriminator_block(16, 32),
            *discriminator_block(32, 64),
            *discriminator_block(64, 128),
        )

        # The height and width of downsampled image
        ds_size = opt.img_size // 2 ** 4

        # Output layers
        self.adv_layer = nn.Sequential(nn.Linear(128 * ds_size ** 2, 1))
        self.aux_layer = nn.Sequential(nn.Linear(128 * ds_size ** 2, opt.n_classes), nn.Softmax())
        self.latent_layer = nn.Sequential(nn.Linear(128 * ds_size ** 2, opt.code_dim))

    def forward(self, img):
        out = self.conv_blocks(img)
        out = out.view(out.shape[0], -1)
        validity = self.adv_layer(out)
        label = self.aux_layer(out)
        latent_code = self.latent_layer(out)

        return validity, label, latent_code


# Loss functions
adversarial_loss = torch.nn.MSELoss()
categorical_loss = torch.nn.CrossEntropyLoss()
continuous_loss = torch.nn.MSELoss()

# Loss weights
lambda_cat = 1
lambda_con = 0.1

# Initialize generator and discriminator
generator = Generator()
discriminator = Discriminator()

if cuda:
    generator.cuda()
    discriminator.cuda()
    adversarial_loss.cuda()
    categorical_loss.cuda()
    continuous_loss.cuda()

# Initialize weights
generator.apply(weights_init_normal)
discriminator.apply(weights_init_normal)

# Configure data loader
os.makedirs("../../data/mnist", exist_ok=True)
dataloader = torch.utils.data.DataLoader(
    datasets.MNIST(
        "../../data/mnist",
        train=True,
        download=True,
        transform=transforms.Compose(
            [transforms.Resize(opt.img_size), transforms.ToTensor(), transforms.Normalize([0.5], [0.5])]
        ),
    ),
    batch_size=opt.batch_size,
    shuffle=True,
)

# Optimizers
optimizer_G = torch.optim.Adam(generator.parameters(), lr=opt.lr, betas=(opt.b1, opt.b2))
optimizer_D = torch.optim.Adam(discriminator.parameters(), lr=opt.lr, betas=(opt.b1, opt.b2))
optimizer_info = torch.optim.Adam(
    itertools.chain(generator.parameters(), discriminator.parameters()), lr=opt.lr, betas=(opt.b1, opt.b2)
)

FloatTensor = torch.cuda.FloatTensor if cuda else torch.FloatTensor
LongTensor = torch.cuda.LongTensor if cuda else torch.LongTensor

# Static generator inputs for sampling
static_z = Variable(FloatTensor(np.zeros((opt.n_classes ** 2, opt.latent_dim))))
static_label = to_categorical(
    np.array([num for _ in range(opt.n_classes) for num in range(opt.n_classes)]), num_columns=opt.n_classes
)
static_code = Variable(FloatTensor(np.zeros((opt.n_classes ** 2, opt.code_dim))))


def sample_image(n_row, batches_done):
    """Saves a grid of generated digits ranging from 0 to n_classes"""
    # Static sample
    z = Variable(FloatTensor(np.random.normal(0, 1, (n_row ** 2, opt.latent_dim))))
    static_sample = generator(z, static_label, static_code)
    save_image(static_sample.data, "images/static/%d.png" % batches_done, nrow=n_row, normalize=True)

    # Get varied c1 and c2
    zeros = np.zeros((n_row ** 2, 1))
    #约束编码分为两部分：一部是零向量zeros，一部分是c_varied
    #交换前后两部分，分别得到不同的生数据
    c_varied = np.repeat(np.linspace(-1, 1, n_row)[:, np.newaxis], n_row, 0)
    c1 = Variable(FloatTensor(np.concatenate((c_varied, zeros), -1)))
    c2 = Variable(FloatTensor(np.concatenate((zeros, c_varied), -1)))
    #采样
    sample1 = generator(static_z, static_label, c1)
    sample2 = generator(static_z, static_label, c2)
    save_image(sample1.data, "images/varying_c1/%d.png" % batches_done, nrow=n_row, normalize=True)
    save_image(sample2.data, "images/varying_c2/%d.png" % batches_done, nrow=n_row, normalize=True)


# ----------
#  Training
# ----------

for epoch in range(opt.n_epochs):
    for i, (imgs, labels) in enumerate(dataloader):

        batch_size = imgs.shape[0]

        # Adversarial ground truths
        valid = Variable(FloatTensor(batch_size, 1).fill_(1.0), requires_grad=False)
        fake = Variable(FloatTensor(batch_size, 1).fill_(0.0), requires_grad=False)

        # Configure input
        real_imgs = Variable(imgs.type(FloatTensor))
        labels = to_categorical(labels.numpy(), num_columns=opt.n_classes)

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

        optimizer_G.zero_grad()

        # Sample noise and labels as generator input
        z = Variable(FloatTensor(np.random.normal(0, 1, (batch_size, opt.latent_dim))))
        label_input = to_categorical(np.random.randint(0, opt.n_classes, batch_size), num_columns=opt.n_classes)
        code_input = Variable(FloatTensor(np.random.uniform(-1, 1, (batch_size, opt.code_dim))))

        # Generate a batch of images
        gen_imgs = generator(z, label_input, code_input)

        # Loss measures generator's ability to fool the discriminator
        validity, _, _ = discriminator(gen_imgs)
        g_loss = adversarial_loss(validity, valid)

        g_loss.backward()
        optimizer_G.step()

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

        optimizer_D.zero_grad()

        # Loss for real images
        real_pred, _, _ = discriminator(real_imgs)
        d_real_loss = adversarial_loss(real_pred, valid)

        # Loss for fake images
        fake_pred, _, _ = discriminator(gen_imgs.detach())
        d_fake_loss = adversarial_loss(fake_pred, fake)

        # Total discriminator loss
        d_loss = (d_real_loss + d_fake_loss) / 2

        d_loss.backward()
        optimizer_D.step()

        # ------------------
        # Information Loss
        # ------------------

        optimizer_info.zero_grad()

        # Sample labels
        sampled_labels = np.random.randint(0, opt.n_classes, batch_size)

        # Ground truth labels
        gt_labels = Variable(LongTensor(sampled_labels), requires_grad=False)

        # Sample noise, labels and code as generator input
        z = Variable(FloatTensor(np.random.normal(0, 1, (batch_size, opt.latent_dim))))
        label_input = to_categorical(sampled_labels, num_columns=opt.n_classes)
        code_input = Variable(FloatTensor(np.random.uniform(-1, 1, (batch_size, opt.code_dim))))

        gen_imgs = generator(z, label_input, code_input)
        _, pred_label, pred_code = discriminator(gen_imgs)

        info_loss = lambda_cat * categorical_loss(pred_label, gt_labels) + lambda_con * continuous_loss(
            pred_code, code_input
        )

        info_loss.backward()
        optimizer_info.step()

        # --------------
        # Log Progress
        # --------------

        print(
            "[Epoch %d/%d] [Batch %d/%d] [D loss: %f] [G loss: %f] [info loss: %f]"
            % (epoch, opt.n_epochs, i, len(dataloader), d_loss.item(), g_loss.item(), info_loss.item())
        )
        batches_done = epoch * len(dataloader) + i
        if batches_done % opt.sample_interval == 0:
            sample_image(n_row=10, batches_done=batches_done)

minist测试数据

训练时的记录通过约束变量c控制粗细通过约束变量c控制大小

Interpretable Representation Learning by Information Maximizing Generative Adversarial Net（小白学GAN 八）

简介

基础结构

基础概念

信息熵

互信息

优化流程

约束编码的后验分布

代码与实践结果

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