Softmax GAN - 郑之杰的个人网站

把生成对抗网络建模为Softmax函数.

paper：Softmax GAN

本文作者把生成对抗网络建模为Softmax函数。具体地，若记共有 $|P|=|P_{data}|+|P_G|$ 个样本，包括 $|P_{data}|$ 个真实样本和 $|P_G|$ 个生成样本。对于每个样本 $x$ ，使用判别器 $D(x)$ 计算logits，并通过Softmax函数进行建模：

P (x) = \frac{e^{- D (x)}}{\sum_{x} e^{- D (x)}} = \frac{e^{- D (x)}}{Z_{P}}

$P(x) = \frac{e^{-D(x)}}{\sum_x e^{-D(x)}} = \frac{e^{-D(x)}}{Z_P}$

对于判别器 $D(x)$ ，希望其能正确地区分真实样本和生成样本。因此判别器将目标概率均等地分配给 $|P|$ 中的所有真实样本，而生成样本的目标概率为 $0$ ；则判别器学习的目标分布为：

T (x) = {\begin{cases} \frac{1}{| P_{d a t a} |}, & if x \in P_{d a t a} (x) \\ 0, & if x \in P_{G} (x) \end{cases}

$T(x) = \begin{cases} \frac{1}{|P_{data}|}, & \text{if } x \in P_{data}(x) \\ 0, & \text{if } x \in P_G(x) \end{cases}$

构造交叉熵损失函数：

\begin{aligned} L_{D} & = - E_{x ~ P (x)} [T (x) \log P (x)] \\ = - E_{x ~ P (x)} [T (x) \log \frac{e^{- D (x)}}{Z_{P}}] \\ = - E_{x ~ P_{d a t a} (x)} [\frac{1}{| P_{d a t a} |} \log \frac{e^{- D (x)}}{Z_{P}}] \\ = \frac{1}{| P_{d a t a} |} E_{x ~ P_{d a t a} (x)} [D (x)] + \log Z_{P} \end{aligned}

$\begin{aligned} L_D &= - \Bbb{E}_{x \text{~} P(x)} [T(x) \log P(x)] \\ &= - \Bbb{E}_{x \text{~} P(x)} [T(x) \log \frac{e^{-D(x)}}{Z_P}] \\ &=- \Bbb{E}_{x \text{~} P_{data}(x)} [\frac{1}{|P_{data}|} \log \frac{e^{-D(x)}}{Z_P}] \\ &= \frac{1}{|P_{data}|}\Bbb{E}_{x \text{~} P_{data}(x)} [ D(x)] + \log Z_P \end{aligned}$

对于生成器 $G(x)$ ，希望其生成的样本足够接近真实样本。因此生成器将概率平均分配给所有样本；则生成器学习的目标分布为一个均匀分布：

T (x) = \frac{1}{| P |} = \frac{1}{| P_{d a t a} | + | P_{G} |}

$T(x) = \frac{1}{|P|}= \frac{1}{|P_{data}|+|P_{G}|}$

构造交叉熵损失函数：

\begin{aligned} L_{G} & = - E_{x ~ P (x)} [T (x) \log P (x)] \\ = - E_{x ~ P (x)} [\frac{1}{| P_{d a t a} | + | P_{G} |} \log \frac{e^{- D (x)}}{Z_{P}}] \\ = \frac{1}{| P_{d a t a} | + | P_{G} |} E_{x ~ P (x)} [D (x)] + \log Z_{P} \\ = \frac{1}{| P_{d a t a} | + | P_{G} |} (E_{x ~ P_{d a t a} (x)} [D (x)] + E_{x ~ P_{G} (x)} [D (x)]) + \log Z_{P} \end{aligned}

$\begin{aligned} L_G &= - \Bbb{E}_{x \text{~} P(x)} [T(x) \log P(x)] \\ &= - \Bbb{E}_{x \text{~} P(x)} [\frac{1}{|P_{data}|+|P_{G}|} \log \frac{e^{-D(x)}}{Z_P}] \\ &= \frac{1}{|P_{data}|+|P_{G}|}\Bbb{E}_{x \text{~} P(x)} [ D(x)] + \log Z_P \\ &= \frac{1}{|P_{data}|+|P_{G}|}(\Bbb{E}_{x \text{~} P_{data}(x)} [ D(x)]+\Bbb{E}_{x \text{~} P_G(x)} [ D(x)] )+ \log Z_P \end{aligned}$

Softmax GAN的完整目标函数如下：

\begin{aligned} min_{D} \frac{1}{| P_{d a t a} |} E_{x ~ P_{d a t a} (x)} [D (x)] + \log Z_{P} \\ min_{G} \frac{1}{| P_{d a t a} | + | P_{G} |} (E_{x ~ P_{d a t a} (x)} [D (x)] + E_{x ~ P_{G} (x)} [D (x)]) + \log Z_{P} \end{aligned}

$\begin{aligned} & \mathop{ \min}_{D} \frac{1}{|P_{data}|}\Bbb{E}_{x \text{~} P_{data}(x)} [ D(x)] + \log Z_P \\ & \mathop{ \min}_{G} \frac{1}{|P_{data}|+|P_{G}|}(\Bbb{E}_{x \text{~} P_{data}(x)} [ D(x)]+\Bbb{E}_{x \text{~} P_G(x)} [ D(x)] )+ \log Z_P \end{aligned}$

Softmax GAN的完整pytorch实现可参考PyTorch-GAN，下面给出损失函数计算和参数更新过程：

for epoch in range(opt.n_epochs):
    for i, real_imgs in enumerate(dataloader):
        batch_size = real_imgs.shape[0]
        # Adversarial ground truths
        g_target = 1 / (batch_size * 2)
        d_target = 1 / batch_size

        z = torch.randn(batch_size, opt.latent_dim) 
        gen_imgs = generator(z)  

        d_real = discriminator(real_imgs)
        d_fake = discriminator(gen_imgs)                  

        # Partition function
        Z = torch.sum(torch.exp(-d_real)) + torch.sum(torch.exp(-d_fake))

        # 训练判别器
        optimizer_D.zero_grad()
        d_loss = d_target * torch.sum(d_real) + log(Z)
        d_loss.backward(retain_graph=True)
        optimizer_D.step()

        # 训练生成器
        optimizer_G.zero_grad()
        g_loss = g_target * (torch.sum(d_real) + torch.sum(d_fake)) + log(Z)
        g_loss.backward()
        optimizer_G.step()

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