Venue: ICLR 2022 Track Blog

Reviews:

final_score: rating: 5: Marginally below acceptance threshold
review_content: ## Overall 

This blog post is well written, clear and concise and does a good job at summarizing the key ideas of the paper. However, some of the more subtle ideas of the original paper are quickly brushed through, or omitted perhaps for the sake of clarity. Overall, the blog post essentially reflects the more elementary ideas (PCA, Nash equilibrium) but does not go deeper into the more complex questions (convergence rates, experimental insights, extension to non-linear case). 

## Is there a significant added value in comparison to the cited papers? 

The blog post gives a nice introduction on PCA (which may be elementary for the machine learning community?) and summarizes the ideas of the paper with added illustrations. However, there is not much critical point of view on the work, potential extensions, applications, or other related work, For instance, the (open access) reviews on OpenReview have highlighted related literature which feature a QR decomposition. It would have been interesting to comment on that and perhaps give an opinion? There is also a critical blog post on the paper (https://towardsdatascience.com/deepminds-eigengame-is-not-a-game-f580b3709316) and perhaps the authors of this submission agree or disagree with the proposed content and could have participated in this discussion? 

## Is this added value supported by accurate, convincing, and clear arguments?

Yes, the content is accurate and clear. 
review_title: Pedagogical summary but perhaps a little elementary

final_score: rating: 8: Top 50% of accepted papers, clear accept
review_content: Overall, I think this blog post does a great job of explaining the derivation of EigenGame and providing the necessary background on PCA and Nash equilibrium. I also think the connection to minimizing reconstruction error and illustration of the eigenvector hierarchies are nice additions.

My only substantial comment is that the argument for the Nash equilibrium for $\hat v_{j > i}$ is a bit sparse and could be fleshed out. It might help to sketch out some of the steps or point readers to section L of the supplementary material for the paper to better understand how one arrives at $u(\hat v_i) = \sum_{p \ge i} \Lambda_{pp} w_p^2$. Maybe remind the reader that $\Lambda_{ii} > \Lambda_{pp > ii}$ which implies $w_i = 1$ and $w_{j \ne i} = 0$ is the maximizer and this clearly recovers $\hat v_i = v_i$.

Minor comments (I used bold to point out suggestions for small edits):
- "multiply the project**ion** matrix with the raw data"
- It might help to rename $x$ and $y$ to $x_1$ and $x_2$ and make sure you define $X$ later in section "2-in-1 Objective" to be the table in the previous section
- The largest eigenvector should have all positive entries, i.e., remove the minus sign on the second entry: $[0.443, 0.896]^\top$
- "but the **current** objective penalizes $\hat{v}_1$ for aligning with $\hat{v}_2$" to emphasize that there's an issue with the objective that you are about to explain how to fix
- The first equation under "Utility Function" is missing a bracket in the denominator all the way on the right.
- Maybe it's worth writing explicitly somewhere that $R(\hat{V})_{ij} = \hat{v}_i^\top M \hat{v}_j$
- Where you talk about using the true eigenvectors as a basis, make sure to define $d$ as the dimensionality of the data: $\hat v_{i} = \sum_{p=1}^{d} w_p v_p$
review_title: Comprehensive Derivation of EigenGame

final_score: rating: 8: Top 50% of accepted papers, clear accept
review_content: The blog post neatly summarises the EigenGame paper. The post is well-structured, introduces the concept of PCA, and the relevant notions of game theory, then interprets PCA in a game-theoretical framework, and sketches the proof of the equivalence between the Nash-equilbrium of the game and the principal components of the dataset. 

My overall impression is very positive, I recommend the acceptance of the post.

minor comments:
- in the ‘Background’ section, the first eigenvector is missing a minus sign
- I think more images/animations illustrating the key concepts (especially the formula-heavy ones) would be nice
review_title: Nice summary of EigenGame

Meta-review: We appreciated the quality of the paper summary but we judged that the added value provided (backgroundon PCA and Nash equilibrium) were relatively basic and could be further developped. In particular, we encourage the authors to put the paper in perspective. 
Decision: Reject