Relativistic GAN的部分数学推导

Relativistic GAN的部分数学推导，自己重新打一遍字加深记忆。
在论文中定义non-saturating GAN的损失函数如下：
$$L_D^{SGAN}=-{\mathbb{E}}_{x_r\sim \mathbb{P}}[log(sigmoid(C(x_r)))]-{\mathbb{E}}_{x_f\sim \mathbb{Q}}[log(sigmoid(C(x_f)))]$$
$$L_G^{SGAN}=-{\mathbb{E}}_{x_f\sim \mathbb{Q}}[log(sigmoid(C(x_f)))]$$
则Standard GAN的梯度如下：
$$\begin{array}{rl}{\nabla }_w{L}_D^{SGAN}& =-{\nabla }_w{\mathbb{E}}_{x_r\sim \mathbb{P}}[logD(x_r)]-{\nabla }_w{\mathbb{E}}_{x_f\sim {\mathbb{Q}}_{\theta }}[log(1-D(x_f))]\\ & =-{\nabla }_w{\mathbb{E}}_{x_r\sim \mathbb{P}}[log(\frac{e^{C(x_r)}}{e^{C(x_r)}+1})]-{\nabla }_w{\mathbb{E}}_{x_f\sim {\mathbb{Q}}_{\theta }}[log(1-\frac{e^{C(x_f)}}{e^{C(x_f)}+1})]\\ & =-{\nabla }_w{\mathbb{E}}_{x_r\sim \mathbb{P}}[C(x_r)-log(e^{C(x_r)}+1)]-{\nabla }_w{\mathbb{E}}_{x_f\sim {\mathbb{Q}}_{\theta }}[log(1)-log(e^{C(x_f)}+1)]\\ & =-{\mathbb{E}}_{x_r\sim \mathbb{P}}[{\nabla }_{w}C(x_r)]+{\mathbb{E}}_{x_r\sim \mathbb{P}}[\frac{e^{C(x_r)}}{e^{C(x_r)+1}}{\nabla }_{w}C(x_r)]+{\mathbb{E}}_{x_f\sim {\mathbb{Q}}_{\theta }}[\frac{e^{C(x_f)}}{e^{C(x_f)+1}}{\nabla }_{w}C(x_f)]\\ & =-{\mathbb{E}}_{x_r\sim \mathbb{P}}[{\nabla }_{w}C(x_r)]+{\mathbb{E}}_{x_r\sim \mathbb{P}}[D(x_r){\nabla }_{w}C(x_r)]+{\mathbb{E}}_{x_f\sim {\mathbb{Q}}_{\theta }}[D(x_f){\nabla }_{w}C(x_f)]\\ & =-{\mathbb{E}}_{x_r\sim \mathbb{P}}[(1-D(x_r)){\nabla }_{w}C(x_r)]+{\mathbb{E}}_{x_f\sim {\mathbb{Q}}_{\theta }}[D(x_f){\nabla }_{w}C(x_f)]\end{array}$$

$$\begin{array}{rl}{\nabla }_{\theta }{L}_G^{SGAN}& =-{\nabla }_{\theta }{\mathbb{E}}_{z\sim {\mathbb{P}}_z}[logD(G(z))]\\ & =-{\nabla }_{\theta }{\mathbb{E}}_{z\sim {\mathbb{P}}_z}[log(\frac{e^{C(G(z))}}{e^{C(G(z))}+1})]\\ & =-{\nabla }_{\theta }{\mathbb{E}}_{z\sim {\mathbb{P}}_z}[C(G(z))-log(e^{C(G(z))}+1)]\\ & =-{\mathbb{E}}_{z\sim {\mathbb{P}}_z}[{\nabla }_{x}C(G(z))J_{\theta }G(z)-(\frac{e^{C(G(z))}}{e^{C(G(z))}+1}){\nabla }_{x}C(G(z))J_{\theta }G(z)]\\ & =-{\mathbb{E}}_{z\sim {\mathbb{P}}_z}[(1-D(G(z))){\nabla }_{x}C(G(z))J_{\theta }G(z)]\end{array}$$
IPM_based GAN的数学推导相对容易：
$${\nabla }_w{L}_D^{IPM}=-{\mathbb{E}}_{x_r\sim \mathbb{P}}[{\nabla }_{w}C(x_r)]+{\mathbb{E}}_{x_f\sim {\mathbb{Q}}_{\theta }}[{\nabla }_{w}C(x_f)]$$
$${\nabla }_{\theta }{L}_G^{IPM}=-{\nabla }_{\theta }{\mathbb{E}}_{z\sim {\mathbb{P}}_z}[C(G(z))]=-{\mathbb{E}}_{z\sim {\mathbb{P}}_z}[{\nabla }_{x}C(G(z))J_{\theta }G(z)]$$