Analyzing the Effects of Classifier Lipschitzness on Explainers
Authors: Zulqarnain Khan
Aria Masoomi
Davin Hill
Jennifer Dy
Venue: ICLR 2022 Submitted (2022)
Keywords: Explainers
Explanation
Robustness
Astuteness
Lipschitz
Blackbox
Classifiers

Abstract: Machine learning methods are getting increasingly better at making predictions, but at the same time they are also becoming more complicated and less transparent. As a result, explanation methods are often relied on to provide interpretability to these complicated and often black-box prediction models. As crucial diagnostics tools, it is important that these explainer methods themselves are reliable. In this paper we focus on one particular aspect of reliability, namely that an explainer should give similar explanations for similar data inputs. We formalize this notion by introducing and defining explainer astuteness, analogous to astuteness of classifiers. Our formalism is inspired by the concept of probabilistic Lipschitzness, which captures the probability of local smoothness of a function. For a variety of explainers (e.g., SHAP, RISE, CXPlain, PredDiff), we provide lower bound guarantees on the astuteness of these explainers given the Lipschitzness of the prediction function. These theoretical results imply that locally smooth prediction functions lend themselves to locally robust explanations. We evaluate these results empirically on simulated as well as real datasets.

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ANALYZING THE EFFECTS OF CLASSIFIER LIPSCHITZNESS ON EXPLAINERS
Anonymous authors Paper under double-blind review
ABSTRACT
Machine learning methods are getting increasingly better at making predictions, but at the same time they are also becoming more complicated and less transpar- ent. As a result, explanation methods are often relied on to provide interpretability to these complicated and often black-box prediction models. As crucial diagnos- ties tools, is important that these explainer methods themselves are reliable. this paper we focus on one particular aspect of reliability, namely that explainer should give similar explanations for similar data inputs. We formalize this notion by introducing and defining explainer astuteness, analogous to astuteness of clas- sifiers. Our formalism is inspired by the concept of probabilistic Lipschitzness, which captures the probability of local smoothness of function. For variety of explainers (e.g., SHAP, RISE, CXPlain, PredDiff), we provide lower bound guarantees the astuteness of these explainers given the Lipschitzness of the prediction function. These theoretical results imply that locally smooth prediction functions lend themselves to locally robust explanations. We evaluate these results empirically on simulated as well as real datasets.
INTRODUCTION
Machine learning models have improved over time at prediction and classification, especially with the advances made in deep learning and availability of large amounts of data. These gains in pre- dictive power have often been achieved using increasingly complex and black-box models. This has led significant interest in, and a proliferation of, explanation models that provide explanations for the predictions made by these black-box models. We focus our work in analyzing the behaviour of these explanation models. In particular, we explore when are explainers reliable by analyzing the connection between the robustness of explainer methods and the smoothness of the black-box functions they are trying to explain. We propose and formally define explainer astuteness a prop- erty of explainers which captures the probability that given method provides similar explanations to similar data points. We then provide a theoretical way to connect this explainer astuteness to the probabilistic Lipschitzness of the black-box function that is being explained. Since probabilistic Lipschitzness is a measure of the probability that a function smooth in a local neighborhood, our results demonstrate how the smoothness of the black-box function itself impacts the astuteness of the explainer. This implies that enforcing smoothness on black-box functions lends them to more robust explanations.
Related Work. A wide variety of explainers have been proposed in the literature (Guidotti et al., 2018; Arrieta et al., 2020). Explainers can broadly be categorized as feature attribution or feature selection explainers. Feature attribution explainers provide continuous valued importance scores to each of the input features, while feature selection explainers provide binary decisions on whether a feature is important or not. Some popular feature attribution explainers can be viewed through the lens of shapley values such as SHAP (Lundberg Lee, 2017). LIME (Ribeiro et al., 2016) and LIFT (Shrikumar et al., 2016). Some models such as CXPlain (Schwab & Karlen, 2019), PredDiff (Zintgraf et al., 2017) and feature ablation explainers (Lei et al., 2018) calculate feature attributions by simulating individual feature removal, while other methods such as RISE (Petsiuk et al., 2018)
calculate the mean effect of a feature's 's presence to attribute importance to it. Feature selection methods, the other hand include individual selector approaches such as L2X (Chen et al., 2018) and IN INVASE (Yoon et al., 2018), and group-wise selection approaches such as gl (Masoomi et al., 2020). These models, while seemingly diverse have been shown to have striking underlying simi-

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larities, for example Lundberg & Lee (2017) unify six different explanation models under a single umbrella. Recently, Covert et al. (2020) went a step ahead and combined 25 existing methods under the overall class of mnnoval-based explanation models.
Similarly, there has been a recent increase in research focused on analyzing the behaviour of these explanation models themselves in ways similar to how classification models have been analyzed. Recent work has focused on dissecting various properties of explainers. Yin et al. (2021) propose stability and sensitivity (Yin et al., 2021) as measures of faithfulness of explainers the decision- making process of the black-box model. Li et al. (2020) explore connections between local ex- plainability and model generalization. Ghorbani et al. (2019) poke at the robustness of explainers to systemic and adversarial perturbations. Alvarez-Melis & Jaakkola (2018) empirically show that robustness, in the sense that explainers should provide similar explanations for similar inputs, is a desirable property and how forcing this property yields better explanations. Recently, Agarwal et al. (2021) explore the robustness of LIME (Ribeiro et al., 2016) and SmoothGrad (Smilkov al., 2017), and prove that for these two methods their robustness is related to the maximum value of the gradient of the predictor function. Our work is closely related to Alvarez-Melis & Jaakkola (2018) and Agarwal et al. (2021) on explainer robustness. However, as opposed only forcing explain- ers to be robust themselves (Alvarez-Melis & Jaakkola, 2018), our theoretical results suggest that ensuring robustness of explanations also depends the smoothness of the black-box function that is being explained. Our results are complementary to the results obtained by Agarwal et al. (2021) in that our theorems cover a wider variety of explainers (see contributions below). We further re- late robustness to probabilistic Lipschitzness of blackbox functions which is a quantity that can be empirically estimated.
Contribution:
We formalize and define explainer astuteness which captures the probability that given explainer provides similar explanations to similar points. This formalism allows us to the- oretically analyze robustness properties of explainers.
We provide theoretical results that connect astuteness of three classes of explainers; shap- ley value based (e.g. SHAP), explainers that simulate mean effect of features (e.g. RISE), and explainers that simulate individual feature removal (e.g. CXPlain), to the smoothness the black-box function. Our results suggest that enforcing Lipschitzness on black-box functions can result in explainers providing more astute explanations. Formally our theo- rems establish a lower bound on explainer astuteness that depends on the Lipschitness of the black-box function and square root of data dimensionality. Figure summarizes this main contribution of our work.
We demonstrate experimentally that this lower bound indeed holds in practice by compar- ing the astuteness predicted by our theorems to the observed astuteness on simulated and real datasets.
The remainder of this paper is organized as follows: we first provide background on the type of explanation methods we restrict ourselves to, i.c. removal of notation used in this paper in Section 2. We then provide formal definitions and theorems with proofs in Section Finally, we conclude the paper by providing experimental results in Section 4.
explainers, and the summary
2
BACKGROUND AND NOTATIONS
2.1 REMOVAL-BASEI FEATURE EXPLAINERS
As mentioned in Section 1 wide variety of explainer methods has been introduced. Owing this diversity, in this work, we concern ourselves with removal-bass feature attribution explainers as defined by Covert et al. (2020). Removal based feature attribution explainers are methods that define a feature's 's influence through the impact of removing it from a model and assign continuous valued scores to each feature signifying its importance. This includes popular approaches such as SHAP and SHAP variants including KernelSHAP, LIME, DeepLIFT (Lundberg & Lee, 2017), mean effect based methods such as RISE (Petsiuk et al., 2018), and individual effects based methods such as CXPlain (Schwab & Karlen, 2019), PredDiff (Zintgraf et al., 2017), permutation tests (Strobl al., 2008), and feature ablation explainers (Lei et al., 2018). All of these methods simulate feature re- moval either explicitly implicitly. For example, SHAP explicitly considers effect of using subsets
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Black-box
f()
Input space
Infe's pie)
Explainer
Φ(.) Aabbe
Figure 1: In this figure visualize the implication of theoretical results. For prediction function that locally probabilisticaly Lipschitz with a constant I,, the predictions for any two points 2, x' such that (x(x,x") S are within (x, x²) distance from each other with some probability - a. Given such prediction function, the explanation for the same data points are also expected to be within (z, z') of each other where 2 CLVd where C is constant, with the same probability a.
that include feature compared to the effect of removing that feature from the subset. RISE removes subsets of features while always keeping the feature that is being evaluated, and estimates the average effect of keeping that feature when other features are randomly removed. CXPlain ex- plicitly considers the impact of removing feature on the loss function used in training the predictor function.
2.2 NOTATION
Before formalizing explainer astuteness and our theoretical results, we explain the notations used in the rest of the paper. We denote d-dimensional input data as Rd from given dataset D. The black-box predictor function is denoted by f. where f(z) is the prediction given this function assumed have been trained on the given D. The explainer is represented by function where (xx) E I2 is the feature attribution vector representing attributions for all features in while (1(x) E IR is the attribution for the ith feature. To simulate the presence or absence features in given subset of features,
an indicator vector E {0, 1]d where z; 1 when the ith feature is present the subset. To indicate we are only using subsets where feature z; = 1, we use 244 and 215, Lastly, the p-norm induced distance
to indicate only using subsets where feature 2, 0,
between any two points v, x' denoted by dy (x, x²).
3 EXPLAINER ASTUTENESS
Our main interest is in defining metric that capture the difference in explanations provided by an explainer to points that are close to each other in the input space. The same question has been asked for classifiers and Bhattacharjee & Chaudhuri (2020) came up with the concept of Astuteness of classifiers, which captures the probability that similar points are assigned the same label by classifier. Formally they provide the following definition:
Definition Astuteness of Classifiers (Bhattacharjee & Chaudhuri, 2020): The astuteness of classifier over dataset DD denoted as (, D) is the probability that Vx. x' E D such that d(z, x') S the classifier will predict the same label.
Ar(f -
= =P. - s
The obvious difference trying to adapt this definition of astuteness to explainers that explana- tions for nearby points do not have be exactly the same. Keeping this in mind, propose and formalize explainer astuteness, as the probability that the explainer method assigns similar explana- tions to similar points. The formal definition follows:
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Definition 2. Explainer astuteness: The explainer astuteness of an explainer E over dataset DD, denoted as (, D) ) the probability that VI,x' 'D such that dz(x, x') I r the explainer E will provide explanations that at most X dz x²) away from each other, where X 2
A D) = Pp..:/-llpppp/ ((())) & x dd(() dp(2,2") srl
(2)
critical observation about definition 2 is that not only relates to the previously defined notion of classifier astuteness, but also connects to the concept of probubilistic Lipschitzness. Probabilistic Lipschitzness captures the probability of a function being locally smooth given a radius r. It is specially useful for capturing notion of smoothness of complicated neural network functions for which enforcing global and deterministic Lipschitzness is difficult. Mangal et al. (2020) formally defined probabilistic Lipchitzness as follows:
Definition 3. Probabilistic Lipschitz (Mangal al., 2020): Given 0 a L, r 2 0, a function f:X + R is probabilistically Lipschitz with a constant 2 Of
P Pa.r: s L d2(,,x x Sr) 2
(3)
Notice the similarity between definitions 2 and 3. This very correspondence forms the basis of our motivation to explore the connection between the Lipschitzness of the black-box prediction function, and the astuteness of the explainer, which we explore theoretically in this section and experimentally in 4.
3.1 THEORETICAL BOUNDS OF ASTUTENESS
A cursory comparison between equation 2 and equation 3 hints at the two concepts being related to each other. In fact, explainer astuteness can be viewed as probabilistic Lipschitzness of the explainer when it is viewed tion to explore is how the astuteness of explainers is connected to the Lipschitzness of the blackbox model they are trying to explain. We introduce and prove the following theorems which provide theoretical bounds that connect the Lipschitz constant L of the blackbox model to the astuteness of various explanation models including SHAP (Lundberg & Lee, 2017), RISE (Petsiuk et al., 2018), and methods that simulate individual feature removal such as CXPlain (Schwab & Karlen, 2019).
function with Lipschitz constant À. However, much more interesting ques-
3.1.1 ASTUTENESS OF SHAP
SHAP (Lundberg & Lee, 2017) is one of the most popular feature attribution based explanation models in use today. Lundberg & Lee (2017) unify existing explanation approaches within the SHAP framework. Each of these explanation approaches (including LIME, DeepL.FT, and ker- nelSHAP) can be viewed as approximations of SHAP, since SHAP in its theoretical form difficult to calculate. However, in this section we use the theoretical definition of SHAP to establish bounds on astuteness.
For given data point X and prediction function f the feature attribution provided by SHAP for the jth feature is given by:
(x(2) Σ |2-i!!(d
12- -2--1dd-1--11111 1)! [f(x 241) f(x 2-1)
(4)
d!
Theorem 1. (Astuteness of SHAP) For a given r 2 0 and 0 S a S 1, and a trained predictive function f that is probilistic Lipschitz with a constant T. with probability at least 1 at radius r. Then for SHAP explainers we have astuteness Ar,
a for À 2 2/dL
Proof. Given input and another input x' s.t. d(x, x') < And letting |==d C..
Using equation 4 we can write,
(1(01(z) E.
f(x z -(0 C1 (21)))2))))))) [ a oz))))
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Using the fact that dz(2, y) || ullp where |1.1|p is the p-norm, the right hand side gives us, dp(o:(z), (x())) = E,,,,, C2 (= f(x32_1) - -.., Czlf(2' G2+1) - f(z' : 2z.)lllp
Combining the two sums and re-arranging,
(((((1))
f(z' @ oz1)]lla
(5)
=11E
f(x' 3 244) + f f(x 3 2.)lllp
Using triangular inequality on the R.H.S twice,
(,(0(()). - Σ 0z) - f(x' o =41)llb + Σ f(t =
2-4
2-1
Σ c.11((00+1) f(z' @ 2+))1lp + Σ c oz-1) - f 3 211)
We can replace each value inside the sums in equation 6 with the maximum value across either sums. Doing so would still preserve the inequality equation 6, as the sum of values always less than the maximum among those summed n times. Without loss of generality let us assume this maximum 1f の 2 f(x' @ 24,) for some particular 2". This gives us:
(1(0(()) < 11/ (x 1l1
ΣG (7)
However, E. C2 = Σ. |=
1, which gives us,
d!
(,(1(1))
2 11 @ =1. - s(x' o =iI)l,,-----1 = (d/x 2 21,), 0211)
(8)
Using the fact that f is probabilistic Lipschitz with given some constant L 2 0, (x(xr=²) S r and dp(2 24. x' x) s dx(x, x²). From the definition of probablistic Lipschitz we get:
( (addaa)) 1-a *12d, (1)) o < 2L dx(z, 21
o
Since equation 8 establishes that (x()) s 2dp(f(x 3 f(x o o(t)),, the below inequality can be now established:
(1)) $ 2L x do(x, 21-a
Note that equation 9 is true for each feature E (1....,d) d}. To conclude our proof, we note that dp(((x), o(x')) < / v' * max, (z), (x'))) Utilizing this in equation 9, assuming the is the one corresponding to the maximum difference, gives us:
(11(6()).((1))) $ 22 d,(x,,²))
(10)
Since Pidp(ó(x), 6(x')) < 2 2/dL * (z, in equation 10 defines Ax,r for A 2 2 2/L this concludes the proof.
'd,(z.)) V/E. ||ا
vill E max: ||4 vdlp Vdmax (1,,))))
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Corollary 1.1.
the fiunctiioo f locally L L-lipschit radius then shapley explainers a-astute for radius for X > 2VVdL
Proof. Note that definition reduces to the definition of deterministic lipschitz a = O. Which means equation 10 will be true with probability 1. Which concludes the proof.
3.1.2 ASTUTENESS OF RISE
RISE determines feature explanation for the ith feature by sampling subsets features and then calculating the mean value of the prediction function when feature is included in the subset. RISE feature attribution for given point and feature for prediction function f can be written as:
(x(x) 11
(11)
The following theorem establishes the bound on X for explainer astuteness of RISE in relation to the Lipschitzness of blackbox prediction function.
and 0 I S 1, and trained predictive function f that is locally probabilistic lipschitz with a constant L with radius r and probability at
Theorem 2. (Astuteness of RISE) For given
least 1 - a. Then for RISE explainer we have the astuteness A,, 2 1 - a, for X 2 V/L..
Proof. (Sketch, full proof in Appendix A) Given input & and another input x' s.t. d(x, S r, using equation 11 we can write
@ dp(0((z). o((=)) = (f
2)))
= [| - 0z)]],
(12)
[[Exala= - f @ 2)]llp
Using Jensen's 's inequality on R.H.S followed by the fact that Eif < max f
())) 5 m の2))
(13)
Using the fact that f is probabilistic Lipschitz gives and using dp(0(z), < Vd max, (1) gives us,
Pld,(o()) VaL * dx(x,x')] 2
(14)
Since P(dp(ó(x), o(x') s VdL (z x')] defines Ax, for X 2 VdL, this concludes the proof.
Corollary 2.1. If the prediction function is locally L L-pisschitz at radius 2 0, then RISE explanations are x-astute for radius and X 2 VdL
Proof. Same as proof for Corollary 1.1.
3.1.3 ASTUTENESS OF "REMOVE INDIVIDUAL." EXPLAINERS
Within the framework feature removal explainers, a sub-category is the explainers that work by removing single feature from the set of all features and calculating feature attributions based on change in prediction that result from removing that feature. This category includes Occlusion, CXPlain (Schwab & Karlen, 2019), PredDiff (Zintgraf et al., 2017) Permutation tests (Strobl al., 2008), and feature ablation explainers (Lei et al., 2018).
Remove individual explainers determine feature explanations for the zth feature by calculating the difference prediction with and without that feature included for given point x. Let z-, E 0, 1d represent binary vector with 2, 0, then the explanation for feature can be written
((x;) f(x: (x)))
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Theorem 3. (Astuteness of Remove individual explainers) Consider given 2 0 and 0 I and a trained predictive function f that is locally probabilistic Lipschitz with a constant with radius measured using do(..-) induced by the p-norm and probability at least - a. Then for Remove individual explainers, we have the astuteness A,, 2 a, for À 2 2/L where d the dimensionality of the data.
Proof. (Sketch, full proof in Appendix A) By considering another point x' such that dp(x, x') S r and equation we get,
S(x' (('00=))) @
(16)
then following the exact same steps as the proof for Theorem 1 i.c. writing the right hand side in terms of p-norm, utilizing triangular inequality, and the definition of probabilistic Lipschitzness leads us to the desired result.
Corollary 3.1. If the prediction function locally L- L-piscchitz at radius 2 0, then remove individual explanations are x-astute for radius and X 2 2 2VLL.
Proof. Same proof for Corollary 2.1.
3.2 IMPLICATIONS
The above theoretical results all provide the same critical implication, that is, explainer astutness is lower bounded by the Lipschitzness of the prediction function. This means that black-box classifiers that are locally smooth (have a small L at a given radius r) lend themselves to probabilistically more robust explanations. While there has already been work enforcing Lipschitzness on neural networks through regularization (Gouk et al., 2021), it has primarily been with the goal of improving performance. Our results provide motivation for the same except by showing that doing so will result in higher lower bound on the astuteness of explanations on such Lipschitz controlled classifiers.
4 EXPERIMENTS
To demonstrate the validity of our experimental results, we perform a series of experiments on datasets. We train four different classifiers each of these datasets, and then explain the decisions of these classifiers using three explainer methods. The details are as follows:
Datasets. We use five simulated datasets based on datasets introduced by Chen al. (2018). The first three, labelled XOR, Orange Skin(OS), and Nonlinear Additive(NA) are generated such that the ground truth predictions depend on the same features for all points. The last two labelled Switch and Switch++ are constructed such that the ground truth predictions depend on different features in different regions of the input space. More details about these datasets are provided in Appendix B.
In addition to the above simulated datasets, we also provide evaluations on two real datasets from the UCI Machine learning repository (Asuncion & Newman, 2007) for binary classification. Rice (Cinar & Koklu, 2019) consists of 3810 samples of rice grains of two different varieties (Cammeo and Osmancik). 7 morphological features are provided for each sample. Telescope (Ferene et al., 2005) consists of 19000+ Monte-Carlo generated samples to simulate registration of high energy gamma particles a ground-based atmospheric Cherenkov gamma telescope using the imaging technique. Each sample is labelled as either background gamma signal and consists of 10 features.
Classifiers. For each dataset we train the following four classifiers; 2layer: A two-layer MLP with ReLU activations. For simulated datasets each layer has 200 neurons, while for the 2 real datasets we use 32 neurons in each layer. 4layer: A four-layer MLP with ReLU activations, with the same number of neurons per layer as 2layer. linear: A linear classifier constructed by removing the non- linear activations from the 2layer classifier. svm: A support vector machine with Gaussian kernel.
The idea here is that each of these classifiers will have different probabilistically Lipschitz behavior, and that can be used to lower bound the explainer astuteness when explaining each these classifiers according to theoretical results.
Explainers. We evaluate 3 explainers here that are representative of the theorems provided in SHAP (Lundberg & Lee, 2017) serves Representative of Theorem 1. We use the gradient based
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approximation for the neural-network classifiers and the kernel shap approximation for SVM. Both are included in the implementation provided by the authors". RISE (Petsiuk et al., 201 8018) servves as representative method for Theorem 2. The official implementation provided by the authors is primarily for image datasets S. We adapt this for tabular datasets. CXPlain (Schwah & Karlen, 2019) serves as representative method for Theorem 3. We use the official implementation provided by the authors 4 "Secctinn 3:
4.1 ESTIMATING PROBABILISTIC LIPSCHITZNESS AND LOWER BOUND FOR ASTUTENESS
To demonstrate the connection between explainer astuteness and probabilistic lipchitzness as alluded to by our theory we need to estimate probabilistic Lipschitzness for classifiers. In our experiments we achieve this by by empirically estimating the 2z,,'' D (equation 3) for a range of values of L E {0,1} incremented at 0.1. We do this for each classifier and for each dataset D and set r as median of pairwise distance for all training points. According to equation this gives us an upperbound on - a i.c. we can say that for a given L, the classifier is Lipschitz with probability least 1 a.
We can use the estimates for probabilistic Lipschitzness to predict astuteness using our theorems. We do this by noting that our theorems imply that for X 2 CLVd explainer astuteness is at least 1 This means we can simply multiply the range of Lipschitz constant I. with C'dd and for À greater or equal to that value we can guarantee that explainer astuteness should be lower bounded by 1 a. This is how we arrive at the dashed lines in Figure 2.
4.2 RESULTS
The goal of our experiments is to empirically show that we can use the probabilistic Lipschitzness of classifiers to predict the lower bound for explainer astuteness. If we can plot astuteness lower bound predicted by our theory against observed astuteness for different values of epsilon and the observed astuteness curves are indeed shown to be lower bounded by the predicted astuteness curves, which are the desired result.
To this end, for each of the seven datasets we first train the four classifiers. We then measure the probabilistic Lipschitzness of each of these classifiers for each dataset. Afterwards we explain the predictions of these classifiers using each of the three explainer methods listed above. This results in subplots for each dataset, one for each explainer, as shown in Figure 2, for Nonlinear Additive, Switch++, Rice and Telescope datasets. Plots for all datasets can be seen in Figure 3.
Each subplot in Figure presents two types of curves, the dashed curves represent the predicted lower bound on explainer astuteness given classifier, described in Section 4.1. The solid curves are the actual estimations of explainer astuteness using Definition 2. According to our theoretical results, given À the estimated explainer astuteness (solid curves) should stay above the predicted astuteness (dashed curves) based on the Lipschitzness of classifiers. As Figure 2 demonstrates, this indeed the case. Notice that some deviations from this can be expected since the theoretical results assume the ideal scenario for each of the methods and assume that the probabilities can be computed exactly. In practice e.g. while our theorems use the theoretical definition of SHAP, the implemen- tation uses Gradient Explainer approximation provided by Lundberg & Lee (2017). Similarly, all probabilities are calculated using empirical estimates using the available training points. Table presents the results in tabular form for all datasets. It shows the difference between the AUC under the estimated astuteness curves (AUC) and the AUC under the predicted lower bound (AUCIb). This number captures the average tightness of the lowerbound over range of À values. This number should be non-negative, which we observe it is, except in one case where it's a small negative value due to estimation errors. We also observe that given at least some of the values in this table not close to 0 our proposed lower bound can be improved upon in the future to make it tighter.
4.3 CONCLUSION AND FUTURE WORK
In this paper we formally defined explainer astuteness which captures the probability that given explainer will assign similar explanations to similar points. We theoretically prove that this explainer astuteness proportional to the probabilistic Lipschitzness of the black-box function that is being
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explained. As probabilistic Lipschitzness captures local smoothness properties of function, this result suggests that enforcing smoothness on black-box models can lend these models to more robust explanations. By way of future work observe that experimental results suggest that our predicted lower bound can be tightened further. We intend to pursue this in future research.
SHAP
RISE
CXPLAIN
Noniinear Additive
Astutene5s
Jayer
Rayer
Insar
SVm
0.0
632
0.
316
0.0
632
Astuteness
Zayer dayer
Switch++
0.6
0.4
Inear
1.0
Astutene55
0.8
Rice
Zayer
4ayer
Inear
smm
0.0
632
00
316
0.0
632
Astuteness
telescope
Zoyer
0.
À
À
X
Figure 2: This figure experimentally shows the implication of our theoretical results. Given the datasets and the four classifiers trained each of these datasets, we observe that the explainer astuteness for SHAP, RISE and CXPLAIN is lower bounded by the astuteness predicted by our theoretical results given value of À along the x-axis. The predicted lower bound is depicted by dashed lines, while solid lines depict the actual estimate of explainer astuteness. See Figure 3 for plots for all seven datasets.
Table 1: AUC AUC (4). The observed AUC is lower bounded by the predicted AUC. The difference between the two is always 2 0. This represents the average tightness of the lower bound.
2layer
4layer
lincar
Datasets
RISE
CXPlain
RISE
CXPlain
SHAP
RISE
CXPlain
SHAP
XOR
.09
.050
-013
.049
.050
.033
.049
.050
.031
.642
.578
589
OS
.585
.477
.551
.489
.415
.426
.043
.017
.043
.761
.628
.732
NA
.359
.289
.318
.285
.216
.244
.452
.391
.474
.742
.653
.708
Switch
.053
.053
.003
.086
.083
.039
.043
008
.034
.577
.472
.544
Switch++
.618
-557
.900
.455
342
.398
.011
025
.433
377
399
Rice
.159
004
.171
.138
.031
.138
166
.102
.162
265
.192
224
Telescope
.324
.213
.317
.24
.223
.149

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REFERENCES
Sushant Agarwal, Shahin Jabbari, Chirag Agarwal, Sohini Upadhyay, Zhiwei Steven Wu, and Himabindu Lakkaraju. Towards the unification and robustness of perturbation and gradient based explanations. arXiv preprint 1rXiv:2102.00 10618, 2021.
David Alvarez-Melis and Tommi S Jaakkola. On the robustness of interpretability methods. arXiv preprint 1r0i wrX0:1800600849, 2018.
Alejandro Barredo Arrieta, Natalia Díaz- Rodz-oorrg Javier Del Ser, Adrien Bennetot, Siham Tabik, Alberto Barbado, Salvador García, Sergio Gil-López, Daniel Molina, Richard Benjamins, et al. Explainable artificial intelligence (xai): Concepts, taxonomies, opportunities and challenges to- ward responsible ai. Information Fusion, 58:82-115. 2020.
Arthur Asuncion and David Newman. Uci machine learning repository, 2007.
Robi Bhattacharjee and Kamalika Chaudhuri. When non-parametric methods robust? In Inter- national Conference on Machine Learning, pp. 832-841. PMLR, 2020.
Jianbo Chen, Le Song, Martin Wainwright, and Michael Jordan. Learning to explain: An ceformation-theorette perspective model interpretation. In International Conference Ma- chine Learning, pp. 883-892. PMLR, 2018.
Iika Cinar and Murat Koklu. Classification of rice varieties using artificial intelligence methods. In- lernational Journal of Intelligent Systems and Applications in Engineering, 183):188-194, 2019.
Ian Covert, Scott Lundberg, and Su-In Lee. Explaining by removing: A unified framework for model explanation. arXiv preprint arXiv:2011. 2020.
Daniel Ferenc, MAGIC collaboration, et al. The magic gamma-ray observatory. Nuclear Instru- ments and Methods Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 553(1-2):274 281, 2005.
Amirata Ghorbani, Abubakar Abid, and James 7ou. Interpretation of neural networks is fragile. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 33, pp. 3681-3688, 2019.
Henry Gouk, Eibe Frank, Bernhard Pfahringer, and Michael Cree. Regularisation of neural net- works by enforcing lipschitz continuity. Machine Learning, 110(2):393-416, 2021.
Riccardo Guidotti, Anna Monreale, Salvatore Ruggieri, Franco Turini, Fosca Giannotti, and Dino Pedreschi. A survey of methods for explaining black box models. ACM computing surveys (CSUR), 51(5): 42, 2018.
Jing Lei, Max G'Slll, Alessandro Rinaldo, Ryan J Tibshirani, and Larry Wasserman. Distribution- free predictive inference for regression. Journal of the American Statistical Association, 113 (523):1094-1111 2018
Jeffrey Li, Vaishnavh Nagarajan, Gregory Plumb, and Ameet Talwalkar, learning theoretic per-
spective on local explainability. arXiv preprint 0rXiv:2011.01205. 2020.
Scott M Lundberg and Su-In Lee. A unified approach to interpreting model predictions. In Proceed- ings of the 31st international conference on neural information processing systems, pp. 4768- 6777,2017
Ravi Mangal, Kartik Sarangmath, Aditya V Nori, and Alessandro Orso. Probabilistic lipschitz analysis of neural networks. International Statie Analysis Symposium, pp. 274-309. Springer, 2020.
Aria Masoomi, Chieh Wu, Tingting Zhao, Zifeng Wang, Peter Castaldi, and Jennifer Dy. Instance- wise feature grouping. Advances in Neural Information Processing Systems, 33, 2020.
Vitali Petsiuk, Abir Das, and Kate Saenko. Rise: Randomized input sampling for explanation of black-box models. arXiv preprint arXiv: 100.07200 2018.
small negative numbers possible due estimation error..
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Under review as a conference paper at ICLR 2022
Marco Tulio Ribeiro, Sameer Singh, and Carlos Guestrin. "why should i trust you?" explaining the predictions of any classifier. In Proceedings of the 22nd ACM SIGKDD international conference on knowledge discovery and data mining, pp. 1 1135-11444 2016.
Patrick Schwab and Walter Karlen. Cxplain: Causal explanations for model interpretation under uncertainty. arXiv preprini arXiv:1910. /2336, 2019.
Avanti Shrikumar, Peyton Greenside, Anna Shcherbina, and Anshul Kundaje. Not just black box: Learning important features through propagating activation differences. arXiv preprint arXiv:1605.01713, 2016.
Daniel Smilkov, Nikhil Thorat, Been Kim, Fernanda Viégas, and Martin Wattenherg. Smoothgrad: removing noise by adding noise. arXiv preprint arXiv: 17Xiv:/706.03255,, 2017.
Carolin Strobl, Anne-Laure Boulesteix, Thomas Kneib, Thomas Augustin, and Achim Zeileis. Con- ditional variable importance for random forests. BMC bioinformatics, 9(1):1-11, 2008.
Fan Yin, Zhouxing Shi, Cho-Jui Hsieh, and Kai- Wei Chang. On the faithfulness measurements for model interpretations. arXiv preprint a 202.00444..222 2021.
Jinsung Yoon, James Jordon, and Mihaela van der Schaar. Invase: Instance-wise variable selection
using neural networks. In International Conference on Learning Representations, 2018.
Luisa M Zintgraf, Taco S Cohen, Tameem Adel, and Max Welling. Visualizing deep neural network decisions: Prediction difference analysis. arXiv preprint 1rXiv:1702.04595, 2017.

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Under review as a conference paper at ICLR 2022
A
DETAILED PROOFS
We include the detailed proofs for Theorems and here.
Proof. (For Theorem 2)
Given input æ and another input x' s.t. d(z, x²) S using equation 11 we can write
(((=') =
E,(1/2 =
(17)
| f(z' oz)]1p
Using Jensen's inequality on R.H.S.
d,(0((t) ( f(r o =)]b
(18)
Using the fact that B[f S maxf.
d,(di(z), < max - oz]| max (,([(( (()
(19)
Using the fact that f is probabilistic Lipschitz with some constant 2 0, and dp(x @ 2, C2) 2'),,,), Vz. Then using the definition of probabilistic Lipschitz we get,
(f(= @ < + d(x 2 -
(20)
Using this in equation 19 gives us
Pldp(os(z). oz(2)) S L * d(x, 22)]210
1111-4 1-a
(21)
Note that equation 21 is true for each feature i fl...., d). To conclude the proof note that
do("))),d(1') & vd max; d)(d)(()), (x())) Utilizing this with equation 21 leads us to
Pld,(o(s), o(x') s VdL * dz x')l 2 a
(22)
Since P(dp(o(c), & vaL dx( defines As, for X 2 VdL, this concludes the proof.
Proof. (For Theorem 3)
By considering another point x' such that dz, x²) and equation 15 we get,
(.(()))) = (/ f(x' o
f(xoz))
(23)
using the fact that dz v) = ||x ||ı= where ||-11p is the p-norm, the RHS gives us,
(1). 11f(x) ((o22.)-((a)) + f(xff=(''' 1(2100000) 2z using triangular inequality,
(24)
dr(),(z), < [lf(z)) - f(z')lp + 0 - - 0
(25)
w.l.o.g assuming the first term the right is bigger than the second term
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Under review as a conference paper at ICLR 2022
d)(0(()). & = 24,
(26)
using the fact that f is probabilistic Lipschitz get us,
Pl44()(()) - 2Ld x') 21))
(27)
to conclude the proof note that dp(0(z), o(z)) - dd max, (p( oz(x))) which gives us,
Pidrlo(z).o(z')) s 2 d1 r')l ) 11-0 -
(28)
B DATASET DETAILS
XOR:
A dataset with classes. The input data is generated from 10-dimensional standard Gaussian distribution. The class probabilities generated proportional to exp [x X2)) That is only the first two features are important in generating predictions for all data points.
Orange-skin: The input data again generated from 10-dimensional standard Gaussian distribution. The ground truth class probabilities are proportional to exp In this case the first 4 features important globally for all data points.
4).
Nonlinear-additive Similar to Orange-skin dataset except the ground trugh class proba- bilities are proportional to exp sin 2X, + 2| X3 + exp(-Xa} and therefore each of the 4 important features for prediction nonlinearly related to the prediction itself.
Switch: This simulated dataset is specifically for instancewise feature explanations. For the input data feature X; is generated by mixture of Gaussian distributions centered at +3. X₁ is generated from the Gaussian distribution centered at +3, X2 to Xs are used to generate the prediction probabilities according to the Orange skin model. Otherwise Xe to X9 are used to generate the prediction probabilities according to the Nonlinear-additiee model.
Switch++: This is a modified version of the Switch dataset where instead of two classes there are classes with varying weights.
C ADDITIONAL RESULTS
Table shows the normalized AUC for the estimated explainer astuteness the predicted AUC based on the predicted lower bound curve. As expected the predicted AUC lower bounds the esti- mated AUC.
Figure 3 shows the same plots shown in Figure 2 but includes all datasets.
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Under review as a conference paper at ICLR 2022
Table 2: Observed AUC and (Predicted AUC). The observed AUC is lower bounded by the
dicted AUC and so the observed AUC should always be higher than the predicted AUC. The AUC values are normalized between O and 1.
2layer
4layer
linear
Datasets
SHAP
RISE
CXP
(LBB)
SHHP
RISE
CXP
(LBB
SHAP
RISE
CXP
(LB)
SHAP
RISE
CXP
(LB)
(.550)
(.950)
(.950)
(.333)
OS
954
847
920
(.369)
966
896
9906
(-480)
.994
.967
.994
(,950)
.945
8813
.917
(.184)
NA
978
.909
.936
(618)
.981
.926
.940
(.6%6
.972
.912
.994
(.520)
.971
.937
(.229)
Switch
.98
.996
948
(945)
.99
988
.448
(.905)
.994
.978
.988
(.950)
.969
.885
.936
(.412)
Switche+
969
.908
.441
(.350)
.967
.894
950
(.552)
.992
.974
.985
(-950)
.982
926
.947
(.548)
Rice
962
.866
974
(8003
.932
824
932
(.793)
.968
.962
(.800)
(,715)
Telescope
962
.863
954
(.637)
955
8663
.44
(.610)
.90
9006
966
(.756)
.969
9009
.72
(.467)
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Under review as a conference paper at ICLR 2022
SHAP
RISE
CXPLAIN
10
Nonlinear Additive
Zayer Bayer
0.
6.32
00
316
00
632
Zayer
Aayer
XOR
0.0
0.
00
316
632
Orange Skin
Zayer
8.6
0.4
0.
0.0
00
316
632
1.0
0.8
0.6
Jayer
Aayer
Switch
0.4
Inear
02
00
10
0.8
SW switch++
0.6
Rayer
0.4
Inear
0.
0.
1.
0.8
0.6
Rice
0.4
Bayer
4ayer
02
2mm
0.
1.0
0.8
Telescope
Zayer
0.6
4ayer
0.4
Inear
02
À
Figure 3: This figure experimentally shows the implication of our theoretical results. Given the seven datasets and the four classifiers trained on these each of these datasets, we observe that the explainer astuteness for SHAP, RISE and CXPLAIN is lower bounded by the astuteness predicted by our theoretical results given value of The predicted lower bound is depicted by dashed lines, while solid lines depict the actual estimate of explainer astuteness.


Reviews:
[{"review_id": "s210Yt-ADx4", "reviewer": "Reviewer_Np65", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "Strengths:\n+ very important problem to study. The robustness of explanation problems has been studied by prior work, but there is a lack of theoretical analysis from the perspective of function smoothness.\n+ the proofs are detailed and correct.\n+ covers some of the representative explanation methods, such as SHAP.\n+ data and codes are released.\n\nWeaknesses:\n- The proof does not bring much new insight that are relevant to explanation robustness. There two sub-problems that are important but not addressed: 1) what's the worst-case change in the explanations (that would require a upper-bound over \\epsilon, rather than a lower-bound); 2) the individual explanation element are treated independently but in practice, the features are ranked and only a small number of them are selected as an explanations, and a small change in explanation vector may lead to rather different top-k salient features.\n\n- experimental sections are not comprehensive and vague. For example, it is not clear how to estimate the Lipschitz constant of a neural network. It is also not clear why the predicted astuteness should lower-bound the estimated one. In fact, the definition of the predicted/estimated astuteness is not given. How the experiment in Figure 2 support the theory is not explained clearly. Lastly, from Figure 2, the bounds seem quite loose: will they be useful in practice, and how they can be used?", "summary_of_the_paper": "This paper studies the problem of robustness of local explanations of a classifier at some input x. Explanations can be generated by several methods: Shapley-value-based, mean-effect-based, and removal-based. Since the explanation methods use the output of the classifier f, the change in the explanations can be related to the change in the output of f, and the latter is quantified by the Lipschitz constant of f.\nLower bounds of \\epsilon that bounds the change in the explanation in \"explanation astuteness\" is proved for three example explanation methods. This gives the \"best-case\" robustness in the explanations given perturbation in the input x. Experiments on 7 datasets evaluated the tightness of a low-bound of \"explanation astuteness\", which is the probability that two close inputs x and x' have their explanations \\phi(x) and \\phi(x') differ by no more than some upper-bound related to d(x,x').", "summary_of_the_review": "Problem studied are good, but theory and experiments have room for improvement. ", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "2: Several of the paper’s claims are incorrect or not well-supported.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "lwJSutva-yI", "reviewer": "Reviewer_ARPU", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "The definition of astuteness introduced as a criterion for assessing the reliability of explanations seems to be a natural and useful definition. It is also interesting that lower bounds of astuteness can be derived for well-known explanatory methods. On the other hand, the derivation is elemental and does not lead to surprising results or useful results that can be used to improve the reliability of explanations. The result that the robustness of the explainer is bounded by the Lipschitz constant of the prediction function also seems obvious, since useful explanatory methods are naturally required to be robust against perturbation.", "summary_of_the_paper": "In this study, a metric called Astuteness is introduced as a criterion to evaluate the reliability of explanations. Intuitively, if the distance between explanations given for two samples that are within a certain distance is also within a certain distance, then the explanation has astuteness. This definition can be interpreted as defining the robustness of an explanation. More specifically, it relates the distance in the sample space to the distance in the explanation space by probabilistic Lipschitzness. In this study, the authors show that for some certain classes of explanatory methods, such as SHAP and CXplain, the probabilistic lower bound of the astuteness of an explanation depends on the square root of the number of feature dimensions D and the Lipschitz constant L. The reliability of this probabilistic lower bound is also evaluated experimentally with several datasets.", "summary_of_the_review": "Although the proposed definition is natural and useful as a criterion for evaluating the reliability of explanations, it cannot be said that the analysis using the criterion leads to useful results for improving explanatory methods, and it seems that more detailed analysis and deeper insight into the relationship between explanatory methods and the proposed criterion are needed for publication.", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "2a6lnrZDKdA", "reviewer": "Reviewer_ftCD", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "The paper essentially talks about the robustness of an explanation method that has been extensively studied in the XAI literature [1,2. 3]. Given below is a list of the papers' strengths and weaknesses:\n\nStrengths:\n1. Given the Lipschitzness of the prediction function, the paper provides lower bound guarantees on the astuteness of different explanation methods.\n2. It analyzes three types of explanation methods: i) Shapley value-based, ii) explainers that simulate the mean effect of features, and iii) explainers that simulate individual feature removal.\n3. Empirical evaluations using real and synthetic datasets demonstrate that the lower bound holds in practice.\n\nWeaknesses:\n1. Alvarez-Mellis et al. [1] defined that an explanation satisfies robustness if similar inputs do not result in substantially different explanations. The explainer astuteness (defined by the authors) is similar to the above definition of robustness. However, both robustness and astuteness do not consider the model behavior while defining similarity. The underlying assumption is that the model prediction remains the same for similar inputs, but we should also consider that the model can output the same predictions using different logic paths. It would be great if the authors can comment on this, or will the astuteness formulation change on considering this.\n2. The paper is hastily written with errors in proofs: i) In the proof of Theorem 1, the last two $f(\\cdot)$ terms of the first two equations should be $f(x' \\odot  z_{-i})$. ii) In Equation 5, the sign associated with $f(x' \\odot  z_{-i})$ should be $+$, and iii) typos in Equation 12.\n3. The assumption made in Equation 6 for taking the maximum is unclear. It would be helpful if the authors can expand that.\n4. In the experiments, the authors mention that \"some deviations can be expected since the theoretical results assume the ideal scenario for each of the methods and assume that the probabilities can be computed exactly, which is not the case in practice\" -- what do the authors mean by the ideal scenario? and if the calculated probabilities are not exact, then how reliable are the bounds?\n\n[1]  Alvarez-Melis, David, and Tommi S. Jaakkola. \"On the robustness of interpretability methods.\" ICML WHI. 2018.\n\n[2] David Alvarez-Melis, Tommi S. Jaakkola, \"Towards Robust Interpretability with Self-Explaining Neural Networks\". In NeurIPS, 2018.\n\n[3] Lakkaraju, Himabindu, Nino Arsov, and Osbert Bastani. \"Robust and stable black box explanations.\" In ICML, 2018.", "summary_of_the_paper": "The paper focuses on the reliability of explainers, where an explainer should give similar explanations for similar data inputs. To this end, the authors introduce and define \"explainer astuteness\", which leverages the concept of \"probabilistic Lipschitzness\" (the probability of local smoothness of a predictor/classifier function). The paper explores the reliability of different types of explainers by analyzing the connection between the robustness of explanation methods and the smoothness of the black-box predictor function they are trying to explain. Finally, the authors provide a theoretical way to connect the explainer astuteness to the probabilistic Lipschitzness of the black-box function and explore how the smoothness of the black-box function impacts an explainers' astuteness.", "summary_of_the_review": "The paper presents theoretical guarantees for different classes of explanation methods. However, the notion of stability/robustness/astuteness has been extensively studied in the existing literature and the novelty of the work is thus a bit unclear.", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}]

Author rebuttal:

Meta-review: The paper explores "Astuteness of explainer", to measure reliability of the explanations. There were concerns about the overlap of the proposed work with existing literature.  It was felt that both theory and experiments need more development
Decision: Reject