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Consider the following value function of the initial GAN

$V(D, G) = \mathbb{E}_{x \sim p_{data(x)}} [\log D(x)] + \mathbb{E}_{z \sim p_z(z)} [1- \log D(G(z))]$

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?

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  • $\begingroup$ What are the properties you are referring to? $\endgroup$
    – Raphael Lopez Kaufman
    May 12 at 18:51
  • $\begingroup$ The aforementioned value function ensures Global Optimality of $p_g = p_{data}$. 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 $\endgroup$
    – hanugm
    May 12 at 22:01
  • $\begingroup$ $\min \max V(D, G)$ 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. $\endgroup$
    – nbro
    May 13 at 8:48
  • $\begingroup$ 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). $\endgroup$
    – nbro
    May 13 at 8:53
  • $\begingroup$ @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. $\endgroup$
    – hanugm
    May 13 at 12:31

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