Venue: NeurIPS 2022 Conference

Reviews:

final_score: rating: 6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.
review_content: summary: This paper studies the problem of learning causal effects with the existence of hidden confounders under a simple linear causal model. The authors provide an analysis of this learning problem in observational settings from the perspective of interpolation and regularization. They drive precise asymptotic risk of the min-norm interpolator and the ridge-regularized regressors. They further propose a new measure of confounding strength to characterise rich behaviours of causal learning under the ICM principle. 
strengths_and_weaknesses: Strengths:
+ The problem studied in this paper is of importance to the community.
+ convincing theoretical analysis and easy to follow

Weaknesses:
+ The presentation of some paragraphs should be improved, since they involve a few unexplained terminology (e.g., U-shaped generalization, double-descent phenomenon, benign causal overfitting, etc), which is unfriendly to those unfamiliar with learning theory. I suggest that the authors add a bit explanations to them to improve readability.   

+ The work only focuses on the simple linear causal models with Gaussian variables. Hence, their theoretical results are also limited to the linear setting. However, this is not the case in practice. 

+ Empirical evaluation is limiting. 
questions: + In the simple linear causal model, why is the dimension of $x$ assumed to be smaller than that of $z$? 

+ If the variables considered in the paper are non-Gaussian, will it change the theoretical results? If so, how?

+ Throughout the paper, only the hard intervention is taken into account. What will happen to the soft intervention?

+ How will these theoretical results change in the nonlinear setting? As we know, under the nonlinear causal model, it is infeasible to represent the causal effect using only one parameter (e.g., $\beta$). In this case, I can imagine the theoretical results would be quite different. 
limitations: The authors discussed the limitations but not the potential negative societal impact of their work.
review_title: null

final_score: rating: 7: Accept: Technically solid paper, with high impact on at least one sub-area, or moderate-to-high impact on more than one areas, with good-to-excellent evaluation, resources, reproducibility, and no unaddressed ethical considerations.
review_content: summary: This paper investigates the risk for a (linear causal) parameter estimator in terms of interpolation and regularization. The authors propose several notions to study the difference between causal and statistical learning behavior and showed that causal risk curve exhibits a double-descent phenomenon. Further authors proved that, under a certain condition, a causal risk can be decreased with a stronger regularization than usual for learning statistical parameters.

strengths_and_weaknesses: 
Strengths
- Highly interesting questions and mathematically rigorous analysis.
- Concepts introduced and theoretical results can be beneficial to further study “causal learning” of parameters in the sense of modern “statistical learning”.


Weaknesses
- Regardless of (strong/weak) regularization, it is hard to accept (in my causal mind) that a regularization “enables” causal learning. Nudging parameters toward a little bit doesn’t seem qualified for the term “causal learning” (e.g., figure 2 is drawn without omega square = squared norm of confounding strength)
- Whether to use strong regularization or not depends on confounding strength measure (and SNR_S), where confounding strength measure would not be typically measured. (I didn’t check out [20, 21] where the authors claimed that they can estimate the confounding strength.) If Gamma can be measured, empirical validation would be desired.

Sorry if there is any misunderstanding. I will correct if the authors can point out any.
questions: My only question is whether learning a statistical parameter with regularization (or through interpolation in overparameter regime) can be referred to as causal learning.

Thanks for an intriguing paper.
limitations: - I would like to see how confounding measure can be estimated, which is then one can practically learn causal parameter. This seems wrong...
review_title: null

final_score: rating: 6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.
review_content: summary: The causal risk (under do operator) of ridge regression estimator and min-norm interpolator have been studied. Intriguing and deep relationship between causal signal strength and regularization for ridge regression is studied. 
strengths_and_weaknesses: + The study of causal risk for linear interpolator is timely and impactful
+ Theoretical analysis is solid
questions: (1) It seems $\||\Gamma\||^2_{\Sigma}$ in line 122 on p.3 should’ve been $\||\Gamma\||_{\Sigma^+}^2$ based on Proposition A.1, in which the variance is $I-A^\top (A A^\top)^+ A$?

(2) Should the variance $y|do(x=x_0)$ be $\sigma_{\tilde \varepsilon}^2+\||\Gamma\||^2_{\Sigma^+}$ in line 132, 154, 155 on p.4, because $do(x=x_0)$ eliminates the dependence of $z$ on $x$ so $z^\top\alpha$ is also regarded as an error term?

(3) Perhaps it would be great to add something to clarify that the $E_{Y|X}$ in $R_X^C(\hat\beta)=E_{Y|X}[R^C(\hat\beta)]$ is taken with respect to $\hat\beta$ in line 214 on p.6.

(4) For line 227 and 228 in Theorem 3.1, to extend the results for ridge regression in Eq. (7) and (8) to min-norm estimator, I think one also needs $d>n$ or $d/n\to\gamma\in(1,\infty)$ in addition to $\lambda\to 0$? 

limitations: None.
review_title: null

final_score: rating: 6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.
review_content: summary: The paper gives a theoretical analysis about the relationship between interpolation and causal generalization. The paper proposes a new measurement of confounding strength \zeta, and discusses under different cases of \zeta, how the optimal strength of the regularization varies. 
strengths_and_weaknesses: I am not expert of the causal learning and theoretical statistical learning. My evaluation is only based on a general view of the paper, not diving into the details (proof, etc). 
Strength:
1. the problem seems important and open
2. the theoretical results see solid and very interesting

Weakness:
1. This is a pure theory paper. While it is unnecessary, demonstrating the results on some real-world datasets/applications will be  promising and even more interesting. 
questions: N/A
limitations: N/A
review_title: null

Meta-review: metareview: 
The consensus view was that the theoretical analysis was important, fundamentally novel and very interesting.  The paper was well written.
confidence: Certain
Decision: Accept