Venue: NeurIPS 2021 Conference

Reviews:

final_score: rating: 4: Ok but not good enough - rejection
review_content: summary: This paper extends a previous work [17] in providing a convergence rate for the case of \alpha<1.
main_review: My main concern is that this work is too close to the previous work [17]. The case of \alpha<1 has been discussed in the experiment of [17] in a similar setting. The proof technique is also similar. More importantly, it is not clear why the case of \alpha<1 is so crucial to the field. The exposition and literary quality in [17] is so much better than the current manuscript. I find it hard to justify a separate publication. I would suggest the authors to simply incorporate this result into [17] for publication.
limitations_and_societal_impact: Yes
review_title: null

final_score: rating: 6: Marginally above the acceptance threshold
review_content: summary: The authors consider the problem of variational inference with Renyi divergence for learning mixture models. They improve the analysis of the iterative method Daudel et al. Specifically, they establish fast $O(1/n)$ convergence of the method when $\alpha < 1$. They then analyze the method when $\alpha = 1$ by taking the limit of the algorithm updates. They call this algorithm Renyi descent and establish that it also converges at an $O(1/n)$ rate. They experimentally compare their approach with entropic mirror descent.
main_review: Overall I found the content of the paper interesting. The authors complete the analysis of Daudel et al. and show fast convergence of the variety of parameter settings. The paper is theoretically interesting.

However I believe the clarity of the paper should be improved. The paper is quite dense and more care can be used to introduce concepts:

(1) The introduction in particular is quite dense. The appendix describes "practical" versions of the approach. I think a more practical introduction can improve the clarity.

(2) The paper discusses mixture models but works with a density kernel. The connection can be more clearly spelled out.

(3) The paper defines Renyi descent and discusses its connection with entropic mirror descent, however the explanation of this connection can be improved.

The experimental section is perhaps sufficient for this type of paper but not particularly interesting. The scale of the problem is quite small, only 16 dimensions. Does the method break down in higher dimensions? The authors could argue the strength of the method in general, rather than just how it compares to entropic mirror descent.

Overall, I lean towards acceptance due to the content of the paper, however I believe the clarity can be improved.
limitations_and_societal_impact: No issues here.
review_title: null

final_score: rating: 6: Marginally above the acceptance threshold
review_content: summary: The authors study the convergence properties of the Power Descent algorithm, which has been recently proposed in [17].
Power Descent (PD) allows one to optimise the weights of a mixture model by means of $\alpha$-divergence minimisation.
In previous [17] work, PD was shown to have convergence rate O(1/N) for $\alpha>1$.
In the current work, the authors prove convergence for the case where $\alpha<1$, under less strict assumptions that in [17].
Although no convergence rate is given for $\alpha<1$, the authors propose Renyi Descent (RD) as an alternative, which is shown to have convergence rate O(1/N).
Also, PD is extended to the case $\alpha=1$, recovering thus Entropic Mirror Descent.
The convergence rate for $\alpha=1$ is shown to be O(1/N).
The convergence properties are demonstrated on a case study that involves approximating a two-component Gaussian mixture model as the target density.

main_review: 
The paper extends the theoretical results of [17] for Power Descent, providing a more complete characterisation of the convergence properties of $\alpha$-divergence methods.
The mixture model framework presented has been already discussed in [17]; in that sense, the contribution of the current paper is somewhat incremental.
Nevertheless, the addition of Renyi Descent as an $\alpha$-divergence optimisation method, as well as its convergence analysis could be of interest to the community.

The experimental evaluation of the paper is weak.
I appreciate that this is a mostly theoretical work, but the empirical examples are limited.
The simulation study of Section 5 could easily be extended to demonstrate the convergence properties of the methods proposed for models of different dimensionality.
For example, it is shown that Entropic Mirror Descent fails to converge for 16 dimensions, but what about a smaller dimensionality? How would it compare with Power descent or Renyi descent?

Another possibly minor issue is that certain parts of Section 2 bear many similarities to Sections 2.2, 3.1 and 3.2 in [17].
I understand that this is a background section and it has to summarise previous work, but parts of it should be rephrased.


limitations_and_societal_impact: The authors claim that their contribution is mostly theoretical, thus it is unlikely to result in any negative societal impact.
I think this is a fair position.
review_title: null

final_score: rating: 7: Good paper, accept
review_content: summary: The authors explore the connections between power descent and entropic mirror descent for variational inference with $\alpha$-divergences. The authors show that power descent converges under weaker assumptions than previously known when the weights of a mixture model are optimized. They also show a convergence rate of $O(1/N)$ for a variant of power descent, which they refer to as Rényi descent with $\alpha < 1$. A simulation experiment is included indicating the efficacy of the proposed method for reasonably high-dimensional problems, and supporting the theoretical claims.
main_review: ## Summary of Review
The results presented in the paper on convergence of power descent for mixture weights, as well as convergence of Rényi-Descent appear to be interesting and original. However, as I am not familiar with some of the related work, I am hesitant to make an overly strong statement in this regard. Convergence of variational objective functions is certainly of interest to the probabilistic machine learning community. The paper was reasonably clear, though I think it could have been more self-contained. In particular,
- At points notation was used without being introduced e.g.:
    - $\mu f= \int f d\mu$
    -  $\mu K$ is the measure with $\nu$-density $\mu k$
- The paper would benefit from a, perhaps higher-level, discussion of related work on optimization of variational objectives, discussing existing work in this area. 

As I am not familiar with the some of the related work, and it is possible I have misjudged some portion of the paper, I may adjust my score in light of additional information/clarification.

## General comments and questions:
- Can the authors clarify what aspects of the discussion and analysis comparing Power descent and entropic mirror descent are new to this work, and what aspects are previously contained in [17]? Is table 1 in [17] not already claiming something very much like proposition 1? 
- Relatedly, can you clarify why [17, Theorem 3] does not already imply a rate of convergence rate of $O(1/N)$ in the case $\alpha=1$? 
- Are there any interesting cases where $b_{\mu_n,  \alpha}$ can be computed in closed-form (so that the algorithm for which you prove guarantees can actually be implemented)?

## Typos and Minor comments:
- The main paper contains links to the appendix, that are not stated as links to the appendix. As the main paper will be separate (so these links will not work), consider stating they are in the appendix/supplement.
- Renyi $\to$ Rényi 
- I think you may need an absolute continuity assumption in the definition of alpha divergence.
- Line 420: $\zeta_1 R \zeta_2$

limitations_and_societal_impact: The authors address several limitations of the work, notably that practical implementations of the algorithm rely on a stochastic approximation, which is not handled in the theoretical analysis, and that convergence guarantees for power descent are not obtained in the general case. The authors do not discuss societal impacts, which seems reasonable as the contribution theoretical and not attached to a particular application.
review_title: null

Meta-review: Variational approximations are approximations of the posterior in Bayesian statistics, obtained by minimizing a divergence between the approximation and the posterior, where the approximation is constrained to belong to a given set (for example, the set of Gaussian distributions, or the set of mixtures of Gaussian distributions). The divergence chosen is usually the Kullback-Leibler divergence (KL). However, some recent works highlighted the possible benefits of using alpha-divergences instead (note that KL is recovered as the special case alpha=1). [17] proposed an iterative scheme, referred to as the Power Descent algorithm, to minimize the alpha-divergence in practice. A nice property of this scheme is that it leads to a sequence of approximations such that the divergence with respect to the posterior is decreasing.

This paper extends the analysis of the Power Descent algorithm initiated in [17] to approximations by mixtures models. The author(s) then study the convergence of the sequence of approximations to the minimizer of the divergence. In the case alpha>1, this is a direct consequence of Theorem 1 (stated in the paper, but was already proven in [17]). The convergence in the case alpha<1 is given by Theorem 2 (this is a new, and difficult result). Finally, they analyze the case alpha=1 by taking the limit of the algorithm updates. They refer to this algorithm as "Renyi descent", for which they prove a convergence in 1/N (N being the number of iterations, this is Theorem 3). They compare their approach with Entropic Mirror Descent in a set of numerical experiments.

Three Reviewers think that while this is strongly based on [17], there are useful and nontrivial extensions, and I agree with them. One of the members of the Committee already reviewed an earlier version of this paper, and was satisfied to see many of his/her comments taken into account in this version of the paper. One of them points some weaknesses in the empirical evaluation, but it clear to me that the main contribution here is theoretical. I will therefore recommend to accept the paper.

It would be nice if you could improve the writing of the paper before sending the camera ready version. Reviewer ssFh wrote the paper is too dense, Reviewer M13e found some undefined notations and Reviewer B46Q asked you to rephrase parts of Section 2, that explain the context of [17] but that are too close to the original.
Decision: Accept (Poster)