Answers.Body: <p>What this paper is <em>not</em> saying is that the discriminator, <span class="math-container">$D_{\phi}$</span>, always returns a scalar value of zero. What they are saying is that the generator, <span class="math-container">$G_{\theta}(z)$</span>, has accurately learned the distribution of the input data, and that the discriminator produces the correct answer for each input from the generator. It's a mathematical description of the global optimum for a GAN. This <a href="https://jonathan-hui.medium.com/gan-why-it-is-so-hard-to-train-generative-advisory-networks-819a86b3750b" rel="nofollow noreferrer">Medium article by Jonathan Hui</a> talks more about Nash equilibria, the Kullback-Leibler, and more.</p>
  
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Answers.Text: Still cannot understand. So it means the generator produces images from real distribution and the discriminator label anything from the generator (also from the true images) as fake?
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Body: <p>I am facing difficulty in understanding the bolded portion of the following statement from <a href="https://arxiv.org/pdf/1801.04406.pdf" rel="nofollow noreferrer">this</a> paper</p>
  <blockquote>
  <p>GANs are defined by a min-max two-player game between a discriminative network <span class="math-container">$D_\Psi(x)$</span> and generative network <span class="math-container">$G_\theta(z)$</span>. While the discriminator tries to distinguish between real data point and data points produced by the generator, the generator tries to fool the discriminator. It can be shown that if both the generator and discriminator are powerful enough to approximate any real-valued function, the unique Nash-equilibrium of this two-player game is given by a generator that produces the true data distribution and a discriminator <strong>which is 0 everywhere on the data distribution</strong>.</p>
  </blockquote>
  <p>My understanding is that discriminator gives <span class="math-container">$\dfrac{1}{2}$</span> for any further inputs after training. But, what is the <span class="math-container">$0$</span> mentioned?</p>
  
Tags: <papers><generative-adversarial-networks><discriminator>
Title: Why does this paper say that the Nash-equilibrium of GAN is given by a discriminator which is 0 everywhere on the data distribution?
UserDisplayName: hanugm
ThreadText: # Why does this paper say that the Nash-equilibrium of GAN is given by a discriminator which is 0 everywhere on the data distribution?
  
  (Asked by: username_0 on 2021-03-18)
  
  I am facing difficulty in understanding the bolded portion of the following statement from [this](https://arxiv.org/pdf/1801.04406.pdf) paper
  
  > GANs are defined by a min-max two-player game between a discriminative network $D\_\Psi(x)$ and generative network $G\_\theta(z)$. While the discriminator tries to distinguish between real data point and data points produced by the generator, the generator tries to fool the discriminator. It can be shown that if both the generator and discriminator are powerful enough to approximate any real-valued function, the unique Nash-equilibrium of this two-player game is given by a generator that produces the true data distribution and a discriminator **which is 0 everywhere on the data distribution**.
  
  My understanding is that discriminator gives $\dfrac{1}{2}$ for any further inputs after training. But, what is the $0$ mentioned?
  
  **Tags:** papers, generative-adversarial-networks, discriminator
  
  
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  ## 1 answers
  
  ### Answer by username_1 on 2021-03-18. Score: 1
  
  What this paper is *not* saying is that the discriminator, $D\_{\phi}$, always returns a scalar value of zero. What they are saying is that the generator, $G\_{\theta}(z)$, has accurately learned the distribution of the input data, and that the discriminator produces the correct answer for each input from the generator. It's a mathematical description of the global optimum for a GAN. This [Medium article by Jonathan Hui](https://jonathan-hui.medium.com/gan-why-it-is-so-hard-to-train-generative-advisory-networks-819a86b3750b) talks more about Nash equilibria, the Kullback-Leibler, and more.
  
  > **Comments:**
  > **username_2**: Still cannot understand. So it means the generator produces images from real distribution and the discriminator label anything from the generator (also from the true images) as fake?  
  
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