生成器
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鉴别器
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r是真实的数据
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鉴别器损失函数
设为公式（1）
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\begin{aligned} &\quad loss_{D} \left(x \right )\\ &=- E_{x \sim p_{r}\left(x \right )} \left(\log \left(D\left(x \right ) \right ) \right ) - E_{z \sim p_{z}\left(z \right )} \left(\log \left(1- D \left(G\left(z \right ) \right ) \right ) \right ) \\ &= - E_{x \sim p_{r}\left(x \right )} \left(\log \left(D\left(x \right ) \right ) \right ) - E_{x \sim p_{g}\left(x \right )} \left(\log \left(1- D \left(x \right ) \right ) \right ) \end{aligned}
lossD(x)=−Ex∼pr(x)(log(D(x)))−Ez∼pz(z)(log(1−D(G(z))))=−Ex∼pr(x)(log(D(x)))−Ex∼pg(x)(log(1−D(x)))
生成器损失函数
第一种
设为公式（2）
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\begin{aligned} &\quad loss_{G} \left(x \right )\\ &=E_{z \sim p_{z}\left(z \right )} \left(\log \left(1-D \left(G\left(z \right ) \right ) \right ) \right ) \\ &=E_{x \sim p_{g}\left(x \right )} \left(\log \left(1- D \left(x \right ) \right ) \right ) \end{aligned}
lossG(x)=Ez∼pz(z)(log(1−D(G(z))))=Ex∼pg(x)(log(1−D(x)))
第二种
设为公式（3）
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\begin{aligned} &\quad loss_{G} \left(x \right )\\ &=E_{z \sim p_{z}\left(z \right )} \left(-\log \left(D \left(G\left(z \right ) \right ) \right ) \right ) \\ &=E_{x \sim p_{g}\left(x \right )} \left(-\log \left(D \left(x \right ) \right ) \right ) \end{aligned}
lossG(x)=Ez∼pz(z)(−log(D(G(z))))=Ex∼pg(x)(−log(D(x)))
最优鉴别器
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D^{*} \left(x \right ) =\frac{p_{r}\left(x \right )}{p_{r}\left(x \right )+p_{g}\left(x \right )}
D∗(x)=pr(x)+pg(x)pr(x)
对于公式2，加上一个与
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E_{x \sim p_{r}\left(x \right )} \left(\log \left(D\left(x \right ) \right ) \right ) + E_{x \sim p_{g}\left(x \right )} \left(\log \left(1- D \left(x \right ) \right ) \right )
Ex∼pr(x)(log(D(x)))+Ex∼pg(x)(log(1−D(x)))
代入最优鉴别器
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\begin{aligned} &\quad E_{x \sim p_{r}\left(x \right )} \left(\log \left(D^{*}\left(x \right ) \right ) \right ) + E_{x \sim p_{g}\left(x \right )} \left(\log \left(1- D^{*} \left(x \right ) \right ) \right )\\ &=E_{p_{r}}\log \frac{p_{r}\left(x \right )}{p_{r}\left(x \right )+p_{g}\left(x \right )} + E_{p_{g}}\log \left(1- \frac{p_{r}\left(x \right )}{p_{r}\left(x \right )+p_{g}\left(x \right )} \right )\\ &= E_{p_{r}}\log \frac{p_{r}\left(x \right )}{p_{r}\left(x \right )+p_{g}\left(x \right )} + E_{p_{g}}\log \left( \frac{p_{g}\left(x \right )}{p_{r}\left(x \right )+p_{g}\left(x \right )} \right ) \end{aligned}
Ex∼pr(x)(log(D∗(x)))+Ex∼pg(x)(log(1−D∗(x)))=Eprlogpr(x)+pg(x)pr(x)+Epglog(1−pr(x)+pg(x)pr(x))=Eprlogpr(x)+pg(x)pr(x)+Epglog(pr(x)+pg(x)pg(x))
等价于
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\begin{aligned} &\quad E_{p_{r}}\log \frac{p_{r}\left(x \right )}{\frac{p_{r}\left(x \right )+p_{g}\left(x \right )}{2}} + E_{p_{g}}\log \frac{p_{g}\left(x \right )}{\frac{p_{r}\left(x \right )+p_{g}\left(x \right )}{2}}-2\log2\\ &=D_{KL} \left(p_{r} || \frac{p_{r}\left(x \right )+p_{g}\left(x \right )}{2}\right )+ D_{KL} \left(p_{g} || \frac{p_{r}\left(x \right )+p_{g}\left(x \right )}{2}\right ) -2\log2\\ &=2JSD \left(p_{r}|| p_{g} \right ) -2 \log \left(2 \right ) \end{aligned}
Eprlog2pr(x)+pg(x)pr(x)+Epglog2pr(x)+pg(x)pg(x)−2log2=DKL(pr∣∣2pr(x)+pg(x))+DKL(pg∣∣2pr(x)+pg(x))−2log2=2JSD(pr∣∣pg)−2log(2)
所以越训练鉴别器他就越接近最优鉴别器
最小化生成器，
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G的分布就越接近真实分布
但是如果
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G的分布和真实分布几乎没有重叠的部分
生成器的损失就会趋于一个常数
证明：
因为几乎没有重叠，所以，对于
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p_{g}\to 0 ,p_{g}\neq 0
pg→0,pg=0时，
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进而
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JS
JS散度为
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0生成器损失为常数
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−2log2,梯度为
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对于公式3
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\begin{aligned} &\quad D_{KL} \left(p_{g}|| p_{r} \right ) \\ &= E_{p_{g}} \left( \log \frac{p_{g}}{p_{r}} \right )\\ &= E_{p_{g}} \left( \log \frac{\frac{p_{g}}{p_{r}\left(x \right )+p_{g}\left(x \right )}}{\frac{p_{r}}{p_{r}\left(x \right )+p_{g}\left(x \right )}} \right )\\ &= E_{p_{g}} \left(\log \frac{1-D^{*}\left(x \right )}{D^{*}\left(x \right )} \right )\\ &= E_{p_{g}}\left( \log \left(1-D^{*} \left(x \right ) \right ) \right ) - E_{p_{g}} \log \left( D^{*} \left(x \right ) \right ) \end{aligned}
DKL(pg∣∣pr)=Epg(logprpg)=Epg(logpr(x)+pg(x)prpr(x)+pg(x)pg)=Epg(logD∗(x)1−D∗(x))=Epg(log(1−D∗(x)))−Epglog(D∗(x))
所以
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\begin{aligned} &\quad E_{p_{g}} \left(- \log \left(D^{*} \left(x \right ) \right ) \right ) \\ &= D_{KL} \left(p_{g} || p_{r} \right ) - E_{p_{g}} \left( \log \left(1- D^{*} \left(x \right ) \right ) \right )\\ &= D_{KL} \left(p_{g} || p_{r} \right ) - 2JSD\left(p_{r}||p_{g} \right )+2\log \left(2 \right )+E_{p_{r}} \left( \log \left( D^{*} \left(x \right ) \right ) \right ) \end{aligned}
Epg(−log(D∗(x)))=DKL(pg∣∣pr)−Epg(log(1−D∗(x)))=DKL(pg∣∣pr)−2JSD(pr∣∣pg)+2log(2)+Epr(log(D∗(x)))
后面两项训练生成器时相当于常数
所以就等价于前两项
最小化KL散度时JS散度就会变大
这个就矛盾了，又要让他们相似，又要拉远