RGAN：GAN中的相对判别器.
GAN的目标函数为：
\[\begin{aligned} \mathop{ \min}_{G} \mathop{\max}_{D} \Bbb{E}_{x \text{~} P_{data}(x)}[\log D(x)] + \Bbb{E}_{x \text{~} P_{G}(x)}[\log(1-D(x))] \end{aligned}\]本文作者提出了Relativistic GAN (RGAN)，其目标函数为：
\[\begin{aligned} \mathop{ \min}_{D} -\Bbb{E}_{x_r \text{~} P_{data}(x), x_f \text{~} P_{G}(x)}[\log \sigma(D(x_r)-D(x_f))] \\ \mathop{ \min}_{G}- \Bbb{E}_{x_r \text{~} P_{data}(x), x_f \text{~} P_{G}(x)}[\log \sigma(D(x_f)-D(x_r))] \end{aligned}\]下面先求判别器的最优解。判别器的目标函数为：
\[L(D) = -\iint_{x_r,x_f} P_{data}(x_r) P_{G}(x_f) \log \sigma(D(x_r)-D(x_f)) dx_r dx_f\]下面求上式的极值。先求\(\frac{\partial L(D(x))}{\partial D}\)：
\[\begin{aligned} & \nabla_{D} P_{data}(x_r) P_{G}(x_f) \log \sigma(D(x_r)-D(x_f)) \\ = & P_{data}(x_r) P_{G}(x_f) \frac{ \nabla_{D} \sigma (D(x_r)-D(x_f))}{\sigma(D(x_r)-D(x_f))} \\ & (\text{according to} \quad \nabla_x \sigma(x) = \sigma(x)\sigma(-x)) \\ =& P_{data}(x_r) P_{G}(x_f) \sigma(D(x_f)-D(x_r)) (\nabla_{D}D(x_r)-\nabla_{D}D(x_f)) \\ =& P_{data}(x_r) P_{G}(x_f) \sigma(D(x_f)-D(x_r)) \nabla_{D}D(x_r) \\ & -P_{data}(x_r) P_{G}(x_f) \sigma(D(x_f)-D(x_r)) \nabla_{D}D(x_f) \\ & (\text{exchange } x_f \text{ and } x_r \text{ in 2nd formula} ) \\ =& P_{data}(x_r) P_{G}(x_f) \sigma(D(x_f)-D(x_r)) \nabla_{D}D(x_r) \\ & -P_{data}(x_f) P_{G}(x_r) \sigma(D(x_r)-D(x_f)) \nabla_{D}D(x_r) \end{aligned}\]极值在\(\frac{\partial L(D(x))}{\partial D}=0\)处求得，此时有：
\[P_{data}(x_r) P_{G}(x_f) \sigma(D(x_f)-D(x_r)) -P_{data}(x_f) P_{G}(x_r) \sigma(D(x_r)-D(x_f)) =0\]整理得：
\[\frac{P_{data}(x_r) P_{G}(x_f)}{P_{data}(x_f) P_{G}(x_r)} = \frac{\sigma(D(x_r)-D(x_f))}{\sigma(D(x_f)-D(x_r))} = e^{\sigma(D(x_r)-D(x_f))}\]代入生成器的目标函数：
\[\begin{aligned} &- \Bbb{E}_{x_r \text{~} P_{data}(x), x_f \text{~} P_{G}(x)}[\log \sigma(D(x_f)-D(x_r))] \\ &=-\iint_{x_r,x_f} P_{data}(x_r) P_{G}(x_f) \log \sigma(D(x_f)-D(x_r)) dx_r dx_f \\ &= -\iint_{x_r,x_f} P_{data}(x_r) P_{G}(x_f) \log \log \frac{P_{data}(x_f) P_{G}(x_r)}{P_{data}(x_r) P_{G}(x_f)} dx_r dx_f \end{aligned}\]上式表示优化目标为\(P_{data}(x_f) P_{G}(x_r)\)和\(P_{data}(x_r) P_{G}(x_f)\)之间的f散度，且$f(x) = \log \log(x)$。
此时RGAN的判别器不是一个二分类器，而是一个相对判别器。对于真实样本$x_r$和伪造样本$x_f$，判别器评估把它们两个交换后\(P_{data}(x_f) P_{G}(x_r)\)的变化程度。假如变化程度较小，说明真实样本$x_r$和伪造样本$x_f$的相似程度较高，判别器无法区分它们。
下面给出由pytorch实现的RGAN的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):
z = torch.randn(real_imgs.shape[0], opt.latent_dim)
gen_imgs = generator(z)
# 训练判别器
optimizer_D.zero_grad()
d_loss = -torch.mean(torch.log(torch.sigmoid(
discriminator(real_imgs)-discriminator(gen_imgs.detach()))))
d_loss.backward()
optimizer_D.step()
# 训练生成器
optimizer_G.zero_grad()
g_loss = -torch.mean(torch.log(torch.sigmoid(
discriminator(gen_imgs)-discriminator(real_imgs))))
g_loss.backward()
optimizer_G.step()
实验表明RGAN能够加快生成器的训练速度：
