Paper:
The use of deep learning for a wide range of data problems has increased the need for understanding and diagnosing these models, and deep learning interpretation techniques have become an essential tool for data analysts. Although numerous model interpretation methods have been proposed in recent years, most of these procedures are based on heuristics with little or no theoretical guarantees. In this work, we propose a statistical framework for saliency estimation for black box computer vision models. We build a model-agnostic estimation procedure that is statistically consistent and passes the saliency checks of Adebayo et al. (2018b). Our method requires solving a linear program, whose solution can be efficiently computed in polynomial time. Through our theoretical analysis, we establish an upper bound on the number of model evaluations needed to recover the region of importance with high probability, and build a new perturbation scheme for estimation of local gradients that is shown to be more efficient than the commonly used random perturbation schemes. Validity of the new method is demonstrated through sensitivity analysis.

Review:
Summary
This paper proposes an attribution method, linearly estimated gradient (LEG) for deep networks in the image setting.
The paper also introduces a variant of the estimator called LEG-TV, which includes a TV penalty, and provides a 
theorem on the convergence rate of the estimator. The paper finds that the LEG attributions pass sanity
checks. 
My recommendation
Overall, I am recommending this paper as a weak accept. There are several points to address with
regards to the exposition and flow of the paper, which is my biggest issue with this paper. I believe
the authors can address this point and I am willing to raise my point on this basis. The paper also
provide some theoretical analysis of the proposed method, which is typically lacking for most
of the interpretation methods in this domain. 
Possible Improvements
- The LEG method is not sufficiently motivated. Here, I am specifically referring to the functional
form of the estimated itself in definition 1. See the question section  for some of the issues 
I raised there. 
- From figure 4, we see that the method passes the proposed sanity checks which seem like a 
key motivation for this work, however, the authors don't give an explanation for why this is the
case.
- The paper notes that LEG can be estimated using an LP; it would have been great for the authors
to completely spell this out in the appendix or somewhere in the text. What is the exact form of the
LP? What are the constraints? 
- As the authors know, the two evaluations presented in the paper: sanity checks, and the zeroing
out procedure (in figure 5) don't actually tell us which method is a good method, just rule out a method.
I would encourage the authors to design a toy task where the ground truth attributions are know, then
train a model to be 100 percent or so accurate on this task. You can then obtain LEG-TV estimates
from this model and compare to the ground-truth. 
- I found the proof of lemma 1 confusing, the authors say it follows trivially, but I don't see it. For example, 
there should be a factor of 2 somewhere after taking the derivative wrt to $vec(g)$, but I don't see it. It is
fine for the authors to spell out the derivation here if possible. 
-The paper ends quite abruptly with no conclusion or discussion. It would be great to include a wrap up
section that puts the contributions into context. 
- I get the sense that this method should be computationally intensive, though the paper says otherwise.
It is fine for a method to be computationally intensive, but can the authors speak to this issue?
Some Questions
Definition 1: I had a difficult time understanding this definition. What is $g$ here? I assume
it is the gradient based on the reference to the first order Taylor expansion. In addition, why
is the estimand squared? Further, What does it mean to take expectation wrt $F + x_0$. I was
particularly confused by the last point, because F is a continuous distribution, while presumably
$x_0$ is the point of interest. The paper notes in several places that it can sample from $F + x_0$,
is this equivalent to sampling from $F$ and adding point $x_0$?
What is LEG0 in figure 5?
Is $\kappa$ in your theorem 1, the condition number of the covariance matrix of the perturbation?
Conclusion
Overall, the paper provides a nice method along with analysis on convergence rates and other statistical
properties. Several of the key issues/questions I have about the paper are raised above. None of these
should be dealbreakers but would require the authors to flesh out more details and possibly justify
certain choices. In general, I think more effort should be put into the flow and writing of the paper.
Overall, this is an interesting contribution.
## After reading author responses
I believe the authors have clarified and improved the readability of the paper and clarified several of the questions
that I had. I am raising my score to an accept. While I believe this is a valuable contribution to the sea of attribution methods that have now been published, like the authors noted, it is still not clearly if attribution methods as a whole
are useful of decision making or understanding of a model by either a generic end user or the model developer. 
This is a huge problem in this area that deserves significant attention. This said, the goal of this current paper is to
take a step towards developing a principled method, so this is a step in that direction perhaps.