Loss function in GAN

Since the aim of a Discriminator is to output 1 for real data and 0 for fake data, hence, the aim is to increase the likelihood of true data vs. fake one. In addition, since maximizing the likelihood is equivalent to minimizing the log-likelihood, why are we updating the discriminator by ascending its stochastic gradient as mentioned in Algorithm 1 in https://arxiv.org/pdf/1406.2661.pdf. Shouldn't we update the discriminator by descending its stochastic gradient?

Any help is much appreciated!

In algorithm 1 of the original GAN article (https://arxiv.org/pdf/1406.2661.pdf), the discriminator is said to be updated by "ascending its stochastic gradient". This is referring to equation 1: $$\min_G \max_D V(D, G)= \mathbb{E}_{x\sim p_{data}(x)}[\log D(x)] + \mathbb{E}_{z\sim p_z(z)}[\log(1 - D(G(z)))]$$
When we want to minimize something, we do grandient descent. When we want to maximize something, we do gradient ascent. In this context, we want to maximize $V(D, G)$ with respect to the discriminator $D$, that is, the $\max_D V(D, G)$ part from equation 1.
• Yes you are right, but from a probabilistic point of view, should the process of maximizing the likelihood be equivalent to minimizing the loss, which is log-likelihood? Therefore, I thought that we should minimize the loss, that is the log(D(x)), And hence the probability/likelihood will increase. Correct me if I'm wrong – I. A Aug 8 '17 at 16:27
• Yes, you minimize the following loss: $\mathscr{L}_d = -\dfrac{1}{2}\mathbb{E}_{x \sim p_{data}} \log D(x) -\dfrac{1}{2}\mathbb{E}_z \log (1 - D(G(z)))$ – ncasas Aug 8 '17 at 16:31
• There is a typo in the equation. The second item in the sum should be $\mathbb{E}_{z\sim p_z(z)}[\log(1 - D(G(z)))]$. $z$ is passed to the generator and not $x$ – Joseph Mar 29 '18 at 17:15