actually use convexity in function space to prove that simultaneous gradient descent converges for that particular problem the reason that this breaks down is that we don't actually update the densities directly we update the G and D functions that do the sampling and and the ratio estimation and then on top of that we represent G and D using parametric functions deep neural networks where the actual output values of G and D are very non convex functions of the parameters and so that causes us to lose all of our guarantees for