DIAGNOSING AND ENHANCING VAE MODELS
Although variational autoencoders (VAEs) represent a widely influential deep generative model, many aspects of the underlying energy function remain poorly understood. In particular, it is commonly believed that Gaussian encoder/decoder assumptions reduce the effectiveness of VAEs in generating realistic samples. In this regard, we rigorously analyze the VAE objective, differentiating situations where this belief is and is not actually true. We then leverage the corresponding insights to develop a simple VAE enhancement that requires no additional hyperparameters or sensitive tuning. Quantitatively, this proposal produces crisp samples and stable FID scores that are actually competitive with a variety of GAN models, all while retaining desirable attributes of the original VAE architecture. The code for our model is available at https://github.com/daib13/TwoStageVAE.
Overview:
I thank the authors for their interesting and detailed work in this paper. I believe it has the potential to provide strong value to the community interested in using VAEs with an explicit and simple parameterization of the approximate posterior and likelihood as Gaussian. Gaussianity can be appropriate in many cases where no sequential or discrete structure needs to be induced in the model. I find the mathematical arguments interesting and enlightening. However, the authors somewhat mischaracterize the scope of applicability of VAE models in contemporary machine learning, and don't show familiarity with the broad literature around VAEs outside of this case (that is, where a Gaussian model of the output would be manifestly inappropriate). Since the core of the paper is valuable and salvageable from a clarity standpoint, my comments below are geared towards what changes the authors may make to move this paper into the "pass" category.
Pros: 
- Mathematical insights are well reasoned and interesting. Based on the insight from the analysis in the supplementary materials, the authors propose a two-stage VAE which separate learning the a parsimonious representation of the low-dimensional (lower than the ambient dimension of the input space), and the training a second VAE to learn the unknown approximate posterior. The two-stage training procedure is both theoretically motivated and appears to enhance the output quality of VAEs w.r.t. FID score, making them rival GAN architectures on this metric.
Cons:
- The title and general tone of the paper is too broad: it is only VAE models with Gaussian approximate posteriors and likelihoods. This is hardly the norm for most applications, contrary to the claims of the authors. VAEs are commonly used for discrete random variables, for example. Many cases where VAEs are applied cannot use a Gaussian assumption for the likelihood, which is the key requirement for the proofs in the supplement to be valid (then, the true posterior is also Gaussian, and the KL divergence between that and the approximate posterior can be driven to zero during optimization--clearly a Gaussian approximate posterior will never have zero KL divergence with a non-Gaussian true posterior).
- None of the proofs consider the approximation error garnered by only having access to empirical samples through a sample of the ground truth population. (The ground-truth distribution must be defined with respect to the population rather just the dataset in hand, otherwise we lose all generalizability from a model.) Moreover, the proofs hold asymptotically. Generalization bounds and error from finite time approximations are very pertinent issues and these are ignored by the presented analyses. Such concerns have motivated many of the recent developments in approximate posterior distributions. Overall, the paper contains little evidence of familiarity with the recent advances in approximate Bayesian inference that have occurred over the past two years.
- A central claim of the paper is that the two-stage VAE obviates the need for highly adaptive approximate posteriors. However, no comparison against those models is done in the paper. How does a two-stage VAE compare against one with, e.g., a normalizing flow approximate posterior? I acknowledge that the purpose of the paper was to argue for the Gaussianity assumption as less stringent than previously believed, but all of the mathematical arguments take place in an imagined world with infinite time and unbounded access to the population distribution. This is not really the domain of interest in modern computational statistics / machine learning, where issues of generalization and computational efficiency are paramount.
- While the mathematical insights are well developed, the specifics of the algorithm used to implement the two-stage VAE are a little opaque. Ancestral sampling now takes place using latent samples from a second VAE. An algorithm box is badly needed for reproducibility.
Recommendations / Typos
I noted a few typos and omissions that need correction.
- Generally, the mathematical proofs in section 7 of the supplement are clear. At the top of page 11, though, the paragraph correctly begins by stating that the composition of invertible functions is invertible, but fails to establish that G is also invertible. Clearly it is so by construction, but the explicit reasons should be stated (as a prior sentence promises), and so I assume this is an accidental omission.
- The title of Section 8.1 has a typo: clearly is it is the negative log of p_{theta_t} (x) which approaches its infimum rather than p_{theta_t} (x) approaching negative infinity.
- Equation (4): the true posterior has an x as its argument instead of the latent z.
- Missing parenthesis under Case 2 and wrong indentation. This analysis also seems to be cut off. Is the case r > d relevant here?
* EDIT: I have read the authors' detailed response. It has clarified a few key issues, and convinced me of the value to the community for publication in its present (slightly edited according to the reviwers' feedback) form. I would like to see this published and discussed at ICLR and have revised my score accordingly. *