Body: <p>Consider the following value function of the <a href="https://arxiv.org/pdf/1406.2661.pdf#page=3" rel="nofollow noreferrer">initial GAN</a></p>
  <p><span class="math-container">$V(D, G) = \mathbb{E}_{x \sim p_{data(x)}} [\log D(x)] + \mathbb{E}_{z \sim p_z(z)} [1- \log D(G(z))]$</span></p>
  <p>The min-max game on the value function: <span class="math-container">$min_{G} max_{D} V(D, G)$</span> ensures global optima. And finally <span class="math-container">$D(x) = 1/2$</span> for all <span class="math-container">$x$</span>.</p>
  <p>The paper provides the proof for attaining the convergence for <span class="math-container">$V(D, G)$</span>.</p>
  <p>After the paper, several GANs have been proposed and are using different value functions. So, I am wondering whether all the new value functions need to obey the mathematical properties of the initial <span class="math-container">$V(D, G)$</span> mentioned above so that the min-max game leads to the convergence point?</p>
  
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2022-05-12T18:51:41.910Z
Comments.Text: What are the properties you are referring to?
Comments.UserDisplayName: Raphael Lopez Kaufman
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2022-05-12T22:01:41.350Z
Comments.Text: The aforementioned value function ensures [**Global Optimality** of $p_g = p_{data}$](https://arxiv.org/pdf/1406.2661.pdf#page=4). Section 4.1 contains the underlying properties I am asking for that ensure convergence.  Many of the papers on GANs introduce new value functions but do not show whether it ensures the global optima. @RaphaelLopezKaufman
Comments.UserDisplayName: hanugm
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2022-05-13T08:48:48.270Z
Comments.Text: $\min \max V(D, G)$ is not the objective function, but the optimization problem. The objective function is $V(D, G)$. Here's an analogy. Let's say you want to find the maximum of $f(x) = x^2$. $y^* = \max_x f(x)$ is your optimization problem and $f$ is the objective function. But it's also possible that some people are referring to the "optimization problem" as an "objective function", but it makes no sense to me.
Comments.UserDisplayName: nbro
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2022-05-13T08:53:58.720Z
Comments.Text: Your question is also problematic because there could be many papers published and nobody probably knows all these papers. I recommend that you focus your comparison between the original objective and another specific objective (not all other objectives).
Comments.UserDisplayName: nbro
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2022-05-13T12:31:14.160Z
Comments.Text: @nbro The property I'm asking for is the sole and soul of the GAN. Else, I think we cannot ensure convergence.  So, either all papers need to accept. But, I got doubts because many papers just use new value functions and do not take the task of proving the convergence. So, one counterexample. is enough.
Comments.UserDisplayName: hanugm
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2022-05-16T20:11:29.923Z
Comments.Text: Can you point out a paper where you think the objective function and optimization problem don't ensure that $p_g = p_{data}$? It will be simpler to start from a practical example
Comments.UserDisplayName: Raphael Lopez Kaufman
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2022-05-17T10:24:24.680Z
Comments.Text: @RaphaelLopezKaufman You can take [this](https://arxiv.org/pdf/2004.04788.pdf#page=4) for example.
Comments.UserDisplayName: hanugm
Tags: <math><generative-adversarial-networks><value-functions>
Title: Does all GAN's in literature need to satisfy the properties of objective function of initial GAN?
UserDisplayName: hanugm
ThreadText: # Does all GAN's in literature need to satisfy the properties of objective function of initial GAN?
  
  (Asked by: username_0 on 2022-05-12)
  
  Consider the following value function of the [initial GAN](https://arxiv.org/pdf/1406.2661.pdf#page=3)
  
  $V(D, G) = \mathbb{E}\_{x \sim p\_{data(x)}} [\log D(x)] + \mathbb{E}\_{z \sim p\_z(z)} [1- \log D(G(z))]$
  
  The min-max game on the value function: $min\_{G} max\_{D} V(D, G)$ ensures global optima. And finally $D(x) = 1/2$ for all $x$.
  
  The paper provides the proof for attaining the convergence for $V(D, G)$.
  
  After the paper, several GANs have been proposed and are using different value functions. So, I am wondering whether all the new value functions need to obey the mathematical properties of the initial $V(D, G)$ mentioned above so that the min-max game leads to the convergence point?
  
  **Tags:** math, generative-adversarial-networks, value-functions
  > **Comments:**
  > **username_1**: What are the properties you are referring to?  
  > **username_0**: The aforementioned value function ensures [**Global Optimality** of $p_g = p_{data}$](https://arxiv.org/pdf/1406.2661.pdf#page=4). Section 4.1 contains the underlying properties I am asking for that ensure convergence.  Many of the papers on GANs introduce new value functions but do not show whether it ensures the global optima. @RaphaelLopezKaufman  
  > **username_2**: $\min \max V(D, G)$ is not the objective function, but the optimization problem. The objective function is $V(D, G)$. Here's an analogy. Let's say you want to find the maximum of $f(x) = x^2$. $y^* = \max_x f(x)$ is your optimization problem and $f$ is the objective function. But it's also possible that some people are referring to the "optimization problem" as an "objective function", but it makes no sense to me.  
  > **username_2**: Your question is also problematic because there could be many papers published and nobody probably knows all these papers. I recommend that you focus your comparison between the original objective and another specific objective (not all other objectives).  
  > **username_0**: @nbro The property I'm asking for is the sole and soul of the GAN. Else, I think we cannot ensure convergence.  So, either all papers need to accept. But, I got doubts because many papers just use new value functions and do not take the task of proving the convergence. So, one counterexample. is enough.  
  > **username_1**: Can you point out a paper where you think the objective function and optimization problem don't ensure that $p_g = p_{data}$? It will be simpler to start from a practical example  
  > **username_0**: @RaphaelLopezKaufman You can take [this](https://arxiv.org/pdf/2004.04788.pdf#page=4) for example.  
  
  ---
  
  ## 0 answers
  
  