Automated Efficient Estimation using Monte Carlo Efficient Influence Functions

Summary

Efficient influence functions (EIFs) for nonparametric estimands are used to construct debiased estimators. Existing methods are primarily estimand-specific and require intricate analytic derivations; existing automated methods don't scale well. This paper proposes general Monte-Carlo estimators of the efficient influence function (MC-EIF) that to be used within a probabilistic programming workflow, shows their convergence to the true efficient influence function, as well as convergence of some estimates based on MC-EIF to the true value. Disclaimer: I had little familiarity with the use of influence functions for constructing estimators before reading this paper.

Strengths

The motivation for the paper is clear, and the developed mechanism is sound, and the convergence results are convincing. The extensions of the use of EIF for estimation that are opened with the proposal of the estimator are intriguing.

Weaknesses

* I found the paper was harder to understand than was necessary due to unclear notation. In the Problem Statement, P and Q are used interchangably to mean the pdf or the probability measure (e.g. in Definition 2, it's a measure with respect to which the L2 space is defined, and a function in said L2 space). * The Assumptions (esp 3.1-3.3) are not easily interpretable and are not discussed; as the paper proposes a practical method, it would be made stronger by a discussion of the assumptions.

Questions

* The maximum eigenvalue term in Theorem 3.8 may grow in N. Is there an assumptions making sure it does not?

Rating

7

Confidence

1

Soundness

3

Presentation

2

Contribution

3

Limitations

* Authors do not discuss if there is value for lifting the assumptions of the method.

Summary

The paper establishes a novel method for estimating the efficient influence functions under mild assumptions, called Monte Carlo Efficient Influence Functions(MC-EIF). The method is easy to apply and flexible in many cases, where it can seamlessly equip an existing/popular efficient estimator.

Strengths

It is a really well-written paper with complete model establishment, theoretical results and fair comparisons between their MC-EIF method and other existing methods. Details are listed, like the different ways to generate the estimator and how it is sensitive to their methods and also the limitations on the dimension of model size are discussed. This MC-EIF is easy and flexible to use in many scenarios, so it gets good prospects in the application area.

Weaknesses

When applying new methods in practice, especially in high-dimensional cases, the time cost is also a necessary aspect that needs to be considered, so it would be good to show the time cost for your methods and how it compares to other existing methods. Some settings in the experiments are not listed clearly, like what is the value of $D$ and $p$ for experiments in Figure 1,2,3 or provided the value of $F$ instead?

Questions

It draws my attention that, from Assumption 3.7 and Theorem 3.8, as the model size $p$ becomes larger, the constant of the bound of error will increase as $\sqrt{p \log p}$. So, when dealing with the high-dimensional problem, it may require people to use a really large number of samples ($M$) to get a desirable accuracy and the cost will be tremendous, which could be a potential drawback of this method, what do you think?

Rating

8

Confidence

4

Soundness

3

Presentation

4

Contribution

4

Limitations

I am not sure what is the $p$ tested here, but from Figure 2, it seems like $p_{\max}$ is only $1000$, which is not a very large model. People should test on much higher dimensional problems to defend their methods. Moreover, it would be worth looking at more challenging problems, other than well-defined Gaussian problems.

Summary

The paper proposes Monte Carlo Efficient Influence Functions (MC-EIF), an automated technique for numerically computing efficient influence functions using existing differentiable probabilistic programming systems. MC-EIF simplifies efficient statistical estimation for high-dimensional models, achieving optimal convergence rates and consistency without the need for complex manual derivations. The approach is validated both theoretically and empirically.

Strengths

The paper is well-written and can be followed easily. The authors introduce Monte Carlo Efficient Influence Functions (MC-EIF), a technique for numerically computing EIFs using existing AD and PPL system quantities. They express EIFs as a product of the gradient of the functional, the inverse Fisher information matrix, and the gradient of the log-likelihood. This method automates the construction of efficient estimators, avoiding manual derivations, and provides accurate, generalizable estimates applicable to various functionals and models. They also provide a non-asymptotic error bound on the quality of their approximation, showing how estimators using MC-EIF achieve the same asymptotic guarantees as using analytic EIFs. Empirical results show MC-EIF outperforms existing approaches without degrading estimation accuracy.

Weaknesses

I am not fully familiar with this line of work, so I am unable to identify a major weakness. However, I have some questions regarding the assumptions and the effectiveness of the proposed approach on real datasets, which I have added to the Question sections.

Questions

Q1. In Assumption 3.5, authors assume that the normalized score vector is sub-Gaussian with a parameter $C_1$. I want to know if this constant scales with respect to the dimension $D$ and model size $p$. If yes, what is the scaling? If no, can you clarify why it does not scale? Q2. The authors assume that the map $\phi$ to $P_{\phi}$ is continuous. My question is, in order to approximate the Fisher information $\hat{I_{m}}$, do we need to know what $P_{\phi}$ actually is? Q3. The authors back up their theoretical results with synthetic data experiments. Given the assumptions, it is not clear how applicable the proposed method is to real datasets, and how should the results be interpreted if these assumptions don't hold?"

Rating

6

Confidence

2

Soundness

3

Presentation

3

Contribution

3

Limitations

The authors clearly mention the assumptions before stating their theoretical guarantees. Also, the assumptions are explained clearly. I don't see any potential negative societal impact of their work.

Re.

Thanks for the response to my questions. A clarifying note on the Continuity assumption, as the authors mentioned, would be very useful. I still believe that the paper would benefit from some real data experiments or results. Overall, I will keep my score for the paper.

Response to rebuttal

Thank you very much for your responses! My main concern revolves around the time scale of the methods, which I believe is a key strength. However, this aspect hasn't been sufficiently highlighted in the paper. I hope the author can give this more attention in the revision and also follow through on addressing the larger problem as promised. Overall, I will maintain my score.