A Proofs

GANs, we need to rewrite the objective functions that are easy to calculate derivatives. Proposition 2. F or any continuous and differentiable function f whose domain is X, we have: E Readers are encouraged to refer to the original proof in [57] for more details. Theorem 2. Given the optimal classifier Please see Appendix A.2 for details. Proposition 1. F or any fixed generator, given a data Theorem 3. The objective function for the generator of SSGAN-LA, given the optimal label-augmented discriminator, boils down to: min Theorem 4. At the equilibrium point of DAGAN, the optimal generator implies We first prove the first sentence in this Theorem. We then prove the second sentence in this Theorem.