text: The authors propose an objective whose Lagrangian dual admits a variety of modern objectives from variational auto-encoders and generative adversarial networks.  They describe tradeoffs between flexibility and computation in this objective leading to different approaches.  Unfortunately, I'm not sure what specific contributions come out,  and the paper seems to meander in derivations and remarks that I didn't understand what the point was. First, it's not clear what this proposed generalization offers.  It's a very nuanced and not insightful construction (eq. 3) and with a specific choice of a weighted sum of mutual informations subject to a combinatorial number of divergence measure constraints, each possibly held in expectation (eq. 5) to satisfy the chosen subclass of VAEs and GANs; and with or without likelihoods (eq. 7).  What specific insights come from this that isn't possible without the proposed generalization? It's also not clear with many GAN algorithms that reasoning with their divergence measure in the limit of infinite capacity discriminators is even meaningful  (e.g., Arora et al., 2017;  Fedus et al., 2017).  It's only true for consistent objectives such as MMD-GANs. Section 4 seems most pointed in explaining potential insights.   However, it only introduces hyperparameters and possible combinatorial choices with no particular guidance in mind.  For example, there are no experiments demonstrating the usefulness of this approach except for a toy mixture of Gaussians and binarized MNIST, explaining what is already known with the beta-VAE and infoGAN.  It would be useful if the authors could make the paper overall more coherent and targeted to answer specific problems in the literature rather than try to encompass all of them. Misc + The "feature marginal" is also known as the aggregate posterior (Makhzani et al., 2015) and average encoding distribution (Hoffman and Johnson, 2016); also see Tomczak and Welling (2017).