GAN Least Squares Loss

GAN Least Squares Loss is a least squares loss function for generative adversarial networks. Minimizing this objective function is equivalent to minimizing the Pearson $\chi^{2}$ divergence. The objective function (here for LSGAN) can be defined as:

$\min\_{D}V\_{LS}\left(D\right) = \frac{1}{2}\mathbb{E}\_{\mathbf{x} \sim p\_{data}\left(\mathbf{x}\right)}\left[\left(D\left(\mathbf{x}\right) - b\right)^{2}\right] + \frac{1}{2}\mathbb{E}\_{\mathbf{z}\sim p\_{data}\left(\mathbf{z}\right)}\left[\left(D\left(G\left(\mathbf{z}\right)\right) - a\right)^{2}\right]$

$\min\_{G}V\_{LS}\left(G\right) = \frac{1}{2}\mathbb{E}\_{\mathbf{z} \sim p\_{\mathbf{z}}\left(\mathbf{z}\right)}\left[\left(D\left(G\left(\mathbf{z}\right)\right) - c\right)^{2}\right]$

where $a$ and $b$ are the labels for fake data and real data and $c$ denotes the value that $G$ wants $D$ to believe for fake data.