Consistent and Truthful Interpretation with Fourier Analysis
Authors: Yifan Zhang
Haowei He
Yang Yuan
Venue: Submitted to ICLR 2023 (2023)
Keywords: AI Interpretability

Abstract: For many interdisciplinary fields, 
ML interpretations need to be consistent with \emph{what-if} scenarios related to the current case, i.e., if one factor changes, how does the model react?
Although the attribution methods are supported by the elegant axiomatic systems, they mainly focus on individual inputs, 
and are generally inconsistent. 
To support what-if scenarios, we introduce a new objective of consistency based on a notion called truthful interpretation. Towards this objective, 
we apply Fourier analysis of Boolean functions to get 
consistency guarantees. 
Experimental results show that 
for neighborhoods with various radii, 
our method achieves $2$x - $50$x lower inconsistency compared with the other methods.

Under review as a conference paper at ICLR 2023
CONSISTENT AND TRUTHFUL INTERPRETATION WITH FOURIER ANALYSIS
Anonymous authors Paper under double-blind review
ABSTRACT
For many interdisciplinary fields, ML interpretations need to be consistent with what-if scenarios related to the current case, i.c., one factor changes, how does the model react? Although the attribution methods are supported by the elegant axiomatic systems, they mainly focus on individual inputs and are generally inconsistent. In this paper, we show that such inconsistency not surprising by proving the impossible trinity theorem, stating that interpretability. consistency, and efficiency cannot hold simultaneously. When consistent interpretation required, we introduce a new notion called truthfulness, as relaxation of efficiency. Under the standard polynomial basis, we show that learning the Fourier spectrum is the unique way for designing consistent and truthful interpreting algorithms. Experimental results show that for neighborhoods with various radii, our method achieves 2x 50x lower interpretation enror compared with the other methods.
INTRODUCTION
Interpretability is central problem deep learning. During training, the neural network strives to minimize the training loss without other distracting objectives. However, to interpret the network, we have construct different mode!" which tends have simpler structures and fewer parameters, e.g., decision tree or a polynomial. l'heoretically, these restricted models cannot perfectly interpret deep networks due to their limited representation power. Therefore, the previous researchers had to introduce various relaxations. The most popular and elegant direction is the attribution methods with axiomatic systems (Sundararajan al. 2017 Lundberg & Lee 2017). which mainly focus on individual inputs.
The interpretations of the attribution methods do not automatically extend to the neighboring points. Take SHAP (Lundberg & Lee 2017) the motivating example, illustrated in Figure I on the task of sentiment analysis of movie reviews. In this example, the interpretations of the two slightly different sentences not consistent. It is not only because the weights of each word significantly different but also because, after removing word "very" of weight 19.9%, the network's 's output only drops by 97.8% 88.7% 9.1%. In other words, the interpretation does not explain the network's behavior even small neighborhood of the input.
Network Input
movic
good
prediction
baseline
Interprctation
11.9%
-7.6%
4.1%
19.9%
19.0%
97.8%
50.5%
Network Input
this
movic
is
good
prodiction
baseline
Interprctation
16.7%
-10.2%
5.5%
26.2%
88.7%
50.5%
Figure 1: Interpretations generated by SHAP on movie review.
Inconsistency is not vacuous concern. Imagine a doctor treating a diabetic with the help of an AI system. The patient has features A, B, and C, representing three positive signals from various tests.
For simplicity, below model denote the model that provides interpretation and network to denote the general black-box machine learning model that needs interpretation.

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AI recommends giving 4 units of insulin with the following explanation: A, B, and C have weights 1, 1, and 2, respectively, so 4 units in total. Then the doctor may ask Al: what if the patient only has A and B, but not C? One may expect the answer to be close to 2, A+B has weight of 2. However, the network is highly non-linear and may output other suggestions like units, explaining that both A and B have weight of 1.5. Such inconsistent behaviors will drastically reduce the doctor's s confidence in the interpretations, limiting the AI system's 's practical value.
Consistency (see Definition 3 is certainly not the only objective for interpretability. Being equally important, efficiency is a commonly used axiom in the attribution methods (Weber 1988 Friedman Moulin 1999, Sundararajan & Najmil 2020), also called local accuracy (Lundberg Lee 2017) )
or completeness (Sundararajan et al.) (017), stating that the model's 's output should be equal to the network's 's output for the given input (see Definition Naturally, one may ask the following question:
Q1: Can we always generate an interpreting model that is both efficient and consistent?
Unfortunately, this is generally impossible. We have proved the following theorem in Section
Theorem Impossible trinity, informal version). Interpretability, consistency, and efficiency cannot hold simulraneously.
A few examples following Theorem
(a) Attribution methods are interpretable and efficient, but not consistent.
(b) The original (deep) network consistent and efficient, but not interpretable.
(c) If one model is interpretable and consistent, it cannot be efficient.
However, consistency is necessary for many scenarios, so one may have the follow up question:
Q2: For consistent interpreting models, can they be approximately efficient?
The answer to this question depends on how "approximately efficient" is defined. We introduce a new notion called truthfulness, which can be seen as a natural relaxation of efficiency, i.e., partial efficiency. Indeed, we split the functional space of f into two subspaces, the readable part and the unreadable part (see Definition 5). We call g is truthful it can truthfully represent the readable part of f. Notice that the unreadable part is not simply a "fitting error". Instead, it truthfully represents the higher-order non-linearities in the network that our interpretation model g, even doing its best, cannot cover. In short, what tells is true, although it may not cover all the truth. Due to Theorem this is essentially the best that consistent algorithms can achieve.
Truthfulness is parameterized notion, which depends on the choice of the readable subspace. While theoretically there are infinitely many possible choices of subspaces, in this paper we follow the previous researchers on interpretability with non-lineame non-lineassties (Sundararajan et al. 2020: Masoomi et al. 2022 Tsang et al., 2020), and uses the basis that automatically induces interpretable terms like K; or TT which capture higher order correlations and are easy to understand. The resulting subspace has the standard polynomial basis (or equivalently, the Fourier basis). Following the notion of consistency and truthfulness, our last question is:
Q3: Can we design consistent and truthful interpreting models with polynomial basis?
It turns out that, when truthfulness is parameterized with the polynomial basis, designing consistent and truthful interpreting models is equivalent to learning the Fourier spectrum (see Lemma [ In other words, this is the unique way to generate truthful and consistent interpretations for high order correlations among input parameters.
In this paper, we focus on the case that f and are Boolean functions, i.e., the input variables are binary. This a commonly used assumption in the literature (LIME (Ribeiro et al.) 2016),
SHAP (Lundberg & Lee 2017). Shapley Taylor (Sundararajan et al. 2020), and other methods (Sun- dararajan & Najmi 2020; Zhang et al. 2021 Frye et al., [020 Covert et al. 2020 sai et al.) 2022)) although sometimes not explicitly stated. For readers not familiar with Boolean functions, we remark that Boolean functions have very strong representation power. For example, empirically most human
readable interpretations can be converted into (ensemble of) decision trees, and theoretically all
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(ensemble of) decision trees be converted into Boolean functions. The widely used algorithm XGBoost (Chen & Guestrin 2016) is based on an ensemble of decision trees.
Given any Boolean function, may cxpand it on the Fourier basis and get its Fourier spectrum. It is well known that the Fourier spectrum's 's complexity naturally characterizes the complexity of Boolean functions (O'Donnell 2014), e.g.. small decision tree can be approximated with low degree sparse polynomial on the Fourier basis [Mansour 1994). Therefore, we restrict g functions on its Fourier spectrum. For learning truthful models, apply different methods from Boolean functional anslysis, Harmonica (Hazan et al. 2018) and Low-degree (Linial et al. 1993). Both algorithms have rigorous theoretical guarantees on recovery performance and sampling complexities. Afterwards, in Section 5 we will demonstrate that on datasets like SST-2 and IMDb, Harmonica can get 2x-50x lower interpretation error compared with other methods.
In summary, our contributions are:
We have proved the impossible trinity theorem for interpretability, which shows that interpretable algorithms cannot be consistent and efficient at the same time.
For interpretable algorithms that are consistent but not efficient, we have introduced a new notion called truthfulness, which can be seen partial efficiency. Due to the impossible trinity theorem, this the best that one can achieve when consistency is required.
For truthfulness with the polynomial basis, we have proved that the problem is equivalent to learning Boolean functions, and empirically demonstrated that the Harmonica algorithm can achieve much better interpretation errors and truthfulness, compared with the other methods.
2 RELATED WORK
Interpretability is critical topic in machine learning, and we refer the reader (Doran al. 2017 Lipton 2018) for insightful general discussions. Below we discuss different types of interpretable models.
Model-specific interpretable models Interpretable/white-bre models inherently ante-hoc and model-specific. One of the goals behind using interpretable models is to have inherent model interpretability. Current mainstream approaches include: Decision Trees (Wang et al. (2015a),
Balestriero (2017). Yang et al. (2018)), Decision Rules (Wang et al. (2015b). Su et al. (2015)
Decision Sets (Lakkaraju
(2019). Wang et al.al1220) (2017)) and Linear Model (Ustun & Rudin (2014).
Ustun et al. (204)).
Shapley value based explanations Shapley value (Shapley 1953) was first introduced in coop-
erative game theory, with several strong axiomatic theoretical properties (Weber 1988, Grabisch
& RRobeens 1999). ) Recently, it has been adopted for explanations of machine learning models
(Lundberg & Lee 2017 Strumbelj & Kononenko, 2014 Sundararajan & Najmi 2020 Wang et al.
2021a Zhang al. 202 Frye et al. 2020 Yuan et al. 2021) or feature importance (Covert et al.
2020
Based on the Shapley value, Owen (1922) proposed the Shapley interaction value study pairwise interactions between players. Grabisch & Roubens (1999) generalized it to study interactions of higher orders and provided an axiomatic foundation. Starting from that, many researchers worked on higher-order feature interactions from different perspectives (Sundararajan et al. 2020
Masoomi
ctt al. 2022 Tsang et al. 2020 Aas et al. 2021 Tsai et al. 2022).
However, computing Shapley value is NP-hard (Elkind et al. 2009 Van den Broeck al. 2022)
Recently many works on Shapley value focus on how to approximate Shapley value efficiently,
including sampling-basee based methods, regression-basee methods, or learning-based methods (Lundberg
Lee 2017 Chen et al.] 2019
Ancona et al. 2019 Covert & Lee, 2021 Jethani et al. 202
Hamilton al. 2022 Wang et al. 202
Gradient-based explanations Simonyan et al. (2014) proposed method approximate the network and propose the gradient as attribution linearly. Smoothgrad (Smilkov et al. (0177) generates explanations by introducing perturbation to the input and then observing the corresponding effect
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on the model's predictions. Integrated Gradients (Sundararajan et al. 2017) distributes the change in output with respect to a baseline input by integrating gradients between the two input states, and Janizek al. (202I) generalized this method to second order.
DeepLIFT (Shrikumar et al. 2017) and LRP (Montavon et al. 2017) backpropagated contribution layer-wise every feature of the input. Grad-CAM (Selvaraju al. 2017) leverage activation values of convolutional layers. CXPlain (Schwab & Karlen 2019) removes a single feature from a single input and measures the change in the loss function.
Consistent methods LIME (Ribeiro et al. 2016) is a classical method that samples the data points following predefined sampling distribution and computes function that empirically satisfies local fidelity. Chen et al. (2018) uses a two-layer additive risk model for interpreting the credit risk assessment, [Bastani et al. (2017) proposes an approach called model extraction via greedily learning a decision tree that approximates f. However, these methods are mainly designed with heuristics and do not have any theoretical guarantees.
Other methods Covert al. (2021) introduced general cemoval-based framework that unifies 26 interpretability algorithms, including classical methods like DeepLIFT (Shrikumar et al. 2017). LRP (Montavon al. 2017), etc.
3 OUR FRAMEWORK ON INTERPRETABILITY
We consider Hilbert space H. equipped with inner product (-), and induced & norm I| We denote the input space by X, the output space by y, which means H. C X' - V. We focus on Boolean functions in this paper, so X' 1-1, """3.
R. We use 1/1 instead of 0/1 to represent the binary variables because it easily fits into the Fourier basis. Due to space limit, we defer a brief introduction on Fourier analysis to Appendix A Fourier analysis of Boolean function is a fascinating field, and we refer the reader to Donnell (014) for a more comprehensive introduction.
We use G H to denote the set of interpretable functions, and F C H. to denote the set of machine learning models that need interpretation. In this paper, we focus on models that are not self- seff-interpretable, i.e.,
G.
Definition (Interpretable). A model is interpretable, if E EG.
Interpretable models are generated by interpretation algorithms.
Definition 2 (Interpretation Algorithm). An interpretation algorithm A takes J E H. x E X as inputs, and outputs A(f, , x) E for interpreting on x.
As we mentioned previously, for many interdisciplinary fields, the interpretation algorithm should be consistent.
Definition (Consistent), Given f E H. an interpretation algorithm A is consistent with respect to J. A(,,)) same for every x E
Efficiency is an important property of the attribution methods. Definition (Efficient). A model g E is efficient with respect to f F on x E X, if ( = f(x).
The following theorem states that one cannot expect to achieve the best of all three worlds.
Theorem (Impossible trinity). For any interpretation algorithmm and function sets G C FC H. there exists f E F such that with respect f, either A is not consistent, or A(f,x) is not efficient on x for some x E X.
2) e & for any z X. If for every I. E X. (()) f(z), we know g f g G. this is contradiction. Therefore, there exists I X' such that g(x) # f()).
Proof. Pick f F. A consistent with respect to
Theorem D says efficiency is too restrictive for consistent interpretations. However, being inefficient does not mean the interpretation is wrong, it can still be truthful. Recall a subspace V C. H. is closed if whenever (fn} C V converges some f E H. then f V. We have:
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Definition 5 (Truthful gap and truthful). Given closed subspace V C H. g S V and f F, the truthful gap of g to f for V is:
Tv(f.9) 1lf gl|
inf Il VEV
(1)
When Tv (f(.f)) O, we say g is truthful for subspace V with respect to f, and we know (see e.g. Lemma 4. I in Stein & Shakarchi 2009) Vu E V, (S 9, u) = 0.
Truthfulness means y fully captures the information in the subspace V of therefore it can be as a natural relaxation efficiency. To characterize the interpretation quality, we introduce the following notion.
Definition 6 (Interpretation error). Given functions f.g X' + Y, the interpretation error between and with respect to measure ş is
P. (f.)) -(0)) 1/p
Notice that interpretation error is only loss function that measures the quality of the interpretation, instead of a metric in lp space. Therefore, fs can be a non-uniform weight distribution following the data distribution. For real-world applications, interpreting the model over the whole X' is unnecessary, so μ is usually defined as a uniform distribution on the neighborhood of input x (under a certain metric), in which case we denote the distribution Nz.
Other than loss functions defined over the whole function space, sometimes we also need the notion for pointwise interpretation error.
Definition 7 (Pointwise interpretation error). Given two functions f.g E H. an input x, the interpre- tation error of g to on x is u(x) = [xx) - g(x)]
Discussion on universal consistency. When talking about the notion of consistency, there are two entirely different settings, which we name as the "global consistency" and the "universal consistency". The global consistency is what we focus in this paper (Definition 3 i..., the interpretation only relates to the input features but not the others. This scenario belongs to the category of "removal-based explanations". and almost all the existing interpretable methods also belong to this category (26 of them are discussed in Covert et al. (202I)). On the other hand, the universal consistency means the interpretation may depend on the features that are different from the imput features. Think about interpreting the sentence "I am happy" with the interpretation that "this sentence does not include [very], so this guy is not super happy". Universal consistency is much more challenging than global consistency, and we suspect more powerful machinery like foundation models are needed, which we leave as future work.
4 LEARNING BOOLEAN FUNCTIONS
With the Fourier basis, we define our interpretable function set G as follows.
Definition (C-Readable function). Given a set of Fourier bases C, a function f is C-readable if it is supported C. That is, for any X & C. (f.x)) = 0. Denote the corresponding subspace as Vc-
The Readable notion is parameterized with C. because it may differ case by case. If we set C to be all the single variable bases, only linear functions are readable; we set to all the bases with the degree at most 2, functions with pairwise interactions are also readable. Moreover, if we further add one higher order term to C, e.g., X(a,, 2..... J. it means we can also reason about the factor T1XgX332 in the interpretation, which might be an important empirical factor that people can easily understand. Starting from the bases set C, we have the following formula for computing the truthful gap.
Lemma 1 (Truthful gap for Boolean functions). Given set of Fourier bases C, two functions f.g E 1- - R. the truthful gap of g to f for C is
(v.9)
Σ (f 9, 2 XsEC
(3)
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Proof. Denote the complement space as Vo. We may expand f,g, both bases, and get: fl - gll
inf |I vEV
v112
Σ (f - + (f.xs)² weVe
inf
Σ U2 + (tsxs))
SEC
SEC
SEC
SEC
(f-
inf
Σ (f v,x))
(f
veVe
SEC
SEC
SEC
Where the last equality holds because we can set the Fourier coefficients ès = fs for every S C, which further gives (f v, Xs) 0.
L
With the previous definitions, it becomes clear that finding truthful interpretation g is equivalent to accurately learning a Boolean function with respect to the readable bases set C. Intuitively, means want to find algorithms that can compute the coefficients for the bases in C. In other words, we want to find the importance of the bases like I1, T2 T22627, etc. Learning Boolean function is a classical problem in learning theory, and there many algorithms like KM algorithm (Kushilevitz & Mansour 1991). Low-degree algorithm (Linial et al. 1993) and Harmonica (Hazan et al. (0118).
We pick two algorithms for our task: Harmonica and Low-degree. Compared with Low-degree, Harmonica has much better sampling efficiency based on compressed sensing techniques. Specifically, for general real-valued decision trees which has s leaf nodes and are bounded by B, the sample complexity of Harmonica is Ô (B²s²/E log n). while Low-degree is Ô (B* /=2//_.... log ni) (Hazan et al. 2018). However, the Low-degree algorithm works for more general settings, while the theoretical guarantee Harmonica depends on the assumption that is approximately sparse in the Fourier space. Therefore, we include both algorithms for completeness. Empirically, Harmonica has a much better performance than Low-degree.
We defer the Algorithm description and theoretical guarantees to Appendix B and Appendix C and discussion on the comparison of our algorithms with the existing algorithms Appendix D
Remarks. The theoretical guarantee of Harmonica assumes the target function f is approximately sparse in the Fourier space, which means most of the energy of the function concentrated in the bases in C. This is not strong assumption, because if f is not approximately sparse, it means has energy in many different bases, more or specifically, the bases with higher orders. other words, f has large variance and is difficult to interpret. In this case, no existing algorithms will be able to give consistent and meaningful interpretations.
Likewise, although the Low-degree algorithm does not assume sparsity for it cannot learn all possible functions accurately as well. There are 2 different bases, and if we want to learn the coefficients for all of them, the cumulative error for is at the order of f2(2"c), which is exponentially large. This is not surprising due to the no free lunch theorem in the generalization theory, as we do not expect to be able to learn "any functions" without exponentially many samples.
EXPERIMENTS
5.1 ANALYSIS ON THE POLYNOMIAL FUNCTIONS
To investigate the performance of different interpretation methods, we manually investigate the output of different algorithms (LIME (Ribeiro et 2016), SHAP (Lundberg & Lee| (017). Shapley Interaction Index(wwen 1972). Shapley Taylor (Sundararajan et al., 2020) Hamilton et al. 2022),
Faith-SHAP (Tsai et al. 2022p, Harmonica and Low-degree) for lower order polynomial functions.
We observe that all algorithms can accurately learn the coefficients of the first order polynomial. For the second order polynomial function, only Shapley Taylor, Faith-SHAP, Harmonica and Low-degree can learn all the coefficients accurately. For the third order polynomial function, only SHAP Harmonica and Low-degree succeeded. Due to the space limit, we defer the details to Appendix
Faith-SHAP has delicate representation theorem, which assigns coefficients to different terms under the Möbius transform. Since the basis induced by the Möbius transform is not orthonormal, it is

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not clear to whether Faith-SHAP theoretically compute the accurate coefficients for higher order functions. However, the running time of Faith-SHAP has exponential dependency on n. so empirically weighted sampling subsets of features is needed Tsai et al. (2022). This might be the main reason that our algorithms outperform Faith-SHAP the experiments with real datasets.
5.2 EXPERIMENTAL SETUP
In the rest of this section, we conduct experiments evaluate the interpretation error Ip.Ns N. (f. g) and truthful gap Tv. gf of Harmonica and other baseline algorithms on NLP and vision tasks quantitatively. In our experiments, we choose 2nd order and 3rd order Harmonica algorithms, which correspond to setting C' to be all terms with the order at most and 3. The baseline algorithms chosen for comparison include LIME, Integrated Gradients (Sundararajan et al., (017). SHAP, Integrated Hessians (Janizek et al. 2021), Shapley Taylor interaction index, and Faith-Shap where the first three are first-order algorithms, and the last three are second-order algorithms.
The two language tasks we pick are the SST-2 (Socher et al. 2013) dataset for sentiment analysis and the IMDb (Maas et al.) (011)) dataset for movic review classification. And the vision task the ImageNet (Krizhevsky et al. 2012) for image classification.
5.3 SENTIMENT ANALYSIS
We start with a binary sentiment classification task on SST-2 dataset. We try to interpret a convolu- tional neural network (see details in Appendix [P) trained with Adam (Kingma & Ba 2015) optimizer for 10 epochs. The model has test accuracy of 80.6%. For given input sentence with length 4, we define the induced neighborhood N₂ by introducing a masking operation on this sentence. The radius 0 < S is defined as the maximum number of masked words.
Results on interpretation error Figure Figure[shows shows the interpretation error evaluated under different neighborhoods with radius ranging from to Here co represents the maximum sentence length, which may vary for different data points. The interpretation error is evaluated under L L', and L norms. Here 2 and L are defined according to Eqn. with p 2 and p 1, respectively. And L denotes f 1 {f) - (()) | 2 0.1) dd1)).((x) The detailed numerical results are presented in Table 4 in Appendix G We can see that Harmonica consistently outperforms all the other baselines all radii.
10a
Harmenicc 22d andarr
Interpr Interprtatton cerrrtattinn crror
Like
Interpretaion
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UNC
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Mtiply Tayoe teee (inderdal
Inaignied Svar ey Taylo Inlos
10-1
reageasic Geacians
Shockss Tar 2edionend
10-1
16 Subspace radius
Subspace radius
Subspace radius
(a) T,² norm
(b) L norm
(c) I. norm
Figure 2: Visualization of interpretation error N. (f.g) evaluated on SST-2 dataset.
Estimating truthful gap For convenience, we define the set of bases dd up to degree d as Cd = SS S [n], & d}. We evaluate the truthful gap on the set of bases C3 C2, and C By
definition in Eqn. 3). we have
Σ Σxs(x) x ))
Tv (f.9) Σ (f-sss))
E 1~1-111-
(f(x) ((x))
g(x))
XsEC
XsEC
Then we perform sampling- based estimation of the truthful gap. Worth mentioning that since the size of set Od satisfies |Cd|= Σi o (?) and the max length all sentences n = 50, Exe Xs (x(x), as the summation function of orthonormal basis, is easy to compute on every sample x E {-1, 1] (for n" very large, we will perform another sampling step this function).
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Results truthful gap Figure shows the truthful gap evaluated on SST-2 dataset. We can see that Harmonica achieves the best performance for C² and C For the simple linear case C*,, Hanmonica is almost as good as LIME.
1.2
Truor goo u
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(a) bases C
(b) bases C2
(c) bases C*
Figure 3: Visualization of truthful gap Tc(f.g) evaluated SST-2 dataset.
5.4 MOVIE REVIEW CLASSIFICATION
The IMDb dataset contains long paragraphs, and each paragraph has many sentences. For the readability of results, we treat sentences as units instead of words - masking several words sentence may make the whole paragraph hard to understand and even meaningless, while masking critical sentence has meaningful semantic effects. Therefore, the radius is defined as the maximum number of masked sentences. By default, we use periods, colons, and exclamations to separate sentences. The target network to be interpreted is convolutional neural network (see details in Appendix trained over this dataset with an accuracy of 85.6%.
Results on interpretation error Figure 4 shows the interpretation error evaluated on IMDb dataset under the same settings of SST-2 dataset, i.e., sampling-han based estimation, with slight modification of this procedure that on IMDb dataset, masking operation is performed on sentences in one input paragraph instead (we also change the definition of radii accordingly). We can see that Harmonica consistently outperforms all the other baselines all radii. The detailed numerical results are presented Table i in Appendix
Intorpretation arror
10-1
Grddens
Interpretation
UMC
Ordorter
10-1
Staple Tayor Few (isterae)
reogratee Dtadioms
Svar Tayler ender
Sheclss Taa 2edionent
Subspace radius
radius
Subspace radius
(a) T norm
(b) L norm
(c) I norm
Figure 4: Visualization of interpretation error N. (f.g) evaluated on IMDb dataset.
Results on truthful gap Figure S truthful gap evaluated on IMDb dataset under the same settings of SST-2 dataset. We can see that Harmonica consistently outperforms all the other baselines.
5.5 IMAGE CLASSIFICATION
use ImageNet (Krizhevsky et al. 2012) dataset as the vision interpretation dataset. For this task, we aim to provide class-speciff interpretation, which means that only the class with the maximum predicted probability is taken into consideration for each sample. Following LIME (Ribeiro et al. 2016), we split each 224 224 image into 16 superpixels to balance the human readability and computational efficiency. The official pre-trained ResNet-100 101 (He et al. 2016) model from Py PoTch used.
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14 22-
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(a) bases O
(b) bases C2
(c) bases C
Figure 5: Visualization of truthful gap Tc(fsg) evaluated on IMDb dataset.
Results on interpretation error 6 shows the interpretation error evaluated on 1000 random images from ImageNet while the masking operation is performed on 16 superpixels in one input image instead (we also change the definition of radii accordingly). We can see that when the neighborhood's radius is greater than 1, Harmonica outperforms all the other baselines. The detailed numerical results are presented Table in Appendix
10
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10-1
10
Subspace radius
Subspace radius
Subspace radius
(a) L norm
(b) L' norm
(c) L norm
Figure 6: Visualization of interpretation error A, (f.g) evaluated on ImageNet dataset.
Results on truthful gap Figure 7 shows the truthful gap evaluated on ImageNet dataset. We can see that Harmonica outperforms all the other baselines consistently.
10
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Figure 7: Visualization of truthful gap Te evaluated on ImageNet dataset.
5.6 ADDITIONAL EXPERIMENTS
We further explore the sample complexity of Harmonica and Low-degree algorithms in Appendix H which show that Harmonica achieves better performance with the same sample size.

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A PRELIMINARIES ON FOURIER ANALYSIS
We first introduce the Fourier basis: Definition 9 (Fourier basis). For any subset of variables S _ [n]. we define the corresponding Fourier basis
xs(r) lkesst --1,1,
The Fourier basis complete orthonormal basis for Boolean functions, under the uniform dis- tribution {-1,1}". We remark that this uniform distribution is used for theoretical analysis and algorithm design, and different from the measure for interpretation quality assessment in Definition 6 We define the inner product follows.
Definition 10 (Inner product). Given two functions f.g E 1-1, - R, their inner product is:
(f.)) = 2
Σ DE -1(11
f(z)g(r) = E
2~(-1.11" [f(x)g(z)
Then we can compute the Fourier spectrum any Boolean functions based on Fourier basis.
Definition 11 (Fourier expansion). For any Boolean function f E 131 R. we can expand as
[(x)
Σ fsxs(x),
Sclm
where fs Xs) is the Fourier coefficient All the Fourier coefficients together called the Fourier spectrum of f.
The inner product defines one kind of similarity between two functions and invariant under different basis. Specifically, we have the following Theorem.
Definition 12 (Plancherel's Theorem). Given two functions f.g E 1-1,1)" + R,
(f.g) = Σ
fs0s
SC(n]
When setting f g. get the Parseval's identity: Elfa] =1ff=
B HARMONICA ALGORITHM
Algorithm 1 Harmonica
1. Given uniformly randomly sampled .., .r, evaluate them on f: (f(z1), ..... ((Tr)
2. Solve the following regularized regression problem.
T Σ
argmin aERICI
Σ SXXEC
asxs(x;) Masss) + Alaalli
(4)
3. Output the polynomial g(x) Esxxec asxs(x).
To present the theoretical guarantees of the Harmonica algorithm, introduce the following definition, which is slightly different from its original version in Hazan et al. (2018).
Definition 13 (Approximately sparse function). We say a function f E (-1,1]" - is (e, C)-
bounded, if [B - Exsec f(S)xs)² < and Es [f(S) S.
Here is (c, -ouuuddd means it is almost readable and has bounded & norm. Our algorithm is slightly different from the original algorithm proposed by Hazan et al. (2018). but similar theoretical guarantees still hold, stated below.
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Theorem 2. Given f E {-1,1}} - (c/4, -)-bounded function, Algorithm Z Finds function g with interpretation emoor at most e in time O( + [CI/e) (C]) and sample complexity T D0s2/e log CC])
Our proof is similar to the one in the original paper Hazan et al. (2018), with changes in the readable notion, which now more flexible than being low order. First recall the classical Chebyshev inequality.
Theorem (Multidimensionaa) Chebyshev inequality). Let X be dimensional random vector, with expected value μt = [[ and covariance matrix V = IE [[X - p) - =. If V positive definite matrix, for any real number 8 0:
11) p P(V (( 11)=-1)00-
m
S
8
Let f be an (e/4, C')-buunded function written in the orthonormal basis as Es (S) We can equivalently write as h 9. where h is supported C' that only only includes coefficients of magnitude at least e/ /4s and the constant term the polynomial expansion of f.
Proof of Theorem 2
Since -(f)= fs S. s, we know h is 4s /E + 1 sparse. The function is thus the sum of the [s|==
remaining x(5)xs terms not included in h. Denote the of bases that appear in C' but not in g as R, so we know the coefficient of f on the bases in R is at most e e/4s.
Draw m (to be chosen later) random labeled examples {z ...., (=th,
y²') and enumerate all N = | basis functions Xs E C as fxi,-..,X0 } Form matrix A such that Aij = Xi (=*) and consider the problem of recovering 4s² / + sparse x given Ar + where r is the vector of coefficients of h, the th entry of y equals y², and e, = g (z*).
....,
We will prove that with constant probability over the choice m random examples, |/e/l2 S VEm. Applying Theorem 5 in Hazan al. (0018) by setting 1) V and observing that /641 (I)) (see definition in the theorem), we will recover x' such that || - x' 2'1 < ege for some constant c2.
As such, for the function f f= Em x/x: we will have E |/ ~1/2 < dze by Parseval's identity.
Note, however, that we may rescale by constant factor 1/ (2c3) to obtain error s/2 and only incur an additional constant (multiplicative) factor in the sample complexity bound. By the definition of g. we have
||
Σ SX88C
N(S)+ + i(S)2
(5)
SER
where each f(S) for S E R is of magnitude at most E/4s. By Fact 4 in Hazan et al. (018 and Parseval's identity we have Σn (1R)2 I e/4. Since f is (s/4, C')-oonentraated we have Esxsec f(S)2 S E/4. Thus, ||9|12 is most e/2. Therefore, by triangle inequality E ||4 1 A|P] + [[9|| S s. It remains to bound ||e||2. Note that since the examples are chosen independently, the entries
E [I 115 <
(z*) are independent random variables. Since g is a linear combination of orthonormal monomials (not including the constant term), we have [z-D[g(=)] = O. Here we can apply linearity of variance (the covariance of and Xj is zero for all i # and calculate the variance
Var (g (=²))
Σ J(S) + Ei(sy
Sx8gC
SER
With the same calculation as we know Var (g (z1)) is at most e/2. Now consider the covariance matrix V of the vector e which equals E [ee J] (recall every entry of e has mean 0). Then is a diagonal matrix (covariance between two independent samples is zero), and every diagonal entry is at
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most e/2. Applying Theorem 3
have
[El22 / (2
1 Hence with probability at least 1/2, we have that ||el|a Vem. From Theorem 5 in Hazan et al. (2018). we may choose m
Setting 8 = 22m, we conclude that P (llell2 > /m/ S
Ó (s2/e- logn"). This completes the proof. Note that the probability 1/2 above can be boosted to any constant probability with constant factor loss in sample complexity.
For the running time complexity, we refer to Allen-Zhu & Yuan (0T6) for optimizing linear regression with & regularization. The running time is (( log - L/e) CC). where is the smoothness of each summand in the objective. Since each Xs takes value in {-1, 1}. the smoothness is bounded by the number of entries in each summand, which is |C|. Therefore, the running time is bounded by O(TT log + [CI/c) (CI).
C LOW-DEGREE ALGORITHM
The low-degree algorithm is based on the concentration inequality, and it estimates the coefficient of each axis individually.
Algorithm 2 Low-degree
1. Given uniformly randomly sampled æ1, ... sT, evaluate them on f: f(z,
2. For any Xs C, let ås Σls ((x))xx(a;))
3. Output the polynomial g(z) Es. (sxs(x).
Theorem 4 Linial et al. (993)). Civen any e, >> O, assuming that function f is bounded by B,
when T
28 log 20
have
Pr
Vxs E C, I9s ssl S 2
Theorem proved using the Hoeffding bound, and included the proof here for completeness.
Proof. Since we are given samples to estimate f(S) for every S, we can directly apply the Hoeffding bound (notice that the function is bounded by B):
2T
Te²
Pr (las f(()) 2) |C| < exxp
2 exp
4B2
2B3
Notice that T 2 proved.
2Ba 22
2|0| log 8 we know the right hand size is bounded by C so Theorem is
D DISCUSSION ON THE EXISTING ALGORITHMS
In this section, we compare our approach with the existing techniques from the perspectives of interpretation error and truthfulness.
LIME (Ribeiro al., 2016) Given an input æ, Lime samples the neighborhood points based sampling distribution II, and optimizes the following program:
min
9. L(f.g, 9Eg
52g)
where L is the loss function describing the distance between and the sampled data points, G is the set of readable functions (e.g. the set of linear functions). 2(-) is a function that characterizes the complexity of g. In other words, LIME tries minimize the fitting error and simultaneously
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minimizes the complexity of g (which is usually the sparsity of the linear function). By minimizing L, LIME also works towards minimizing the interpretation error, but their approach is purely heuristic, without any theoretical guarantees. Although their readable function set can easily generalize to the set with higher order terms, the sampling distribution I is not uniform, so it difficult to incorporate the orthonormal basis into their framework, In other words, the model they compute is not truthful.
Attribution methods As we discussed in the introduction, attribution methods mainly focus on individual inputs, instead of the neighboring points. Therefore, it is difficult for the attribution methods to achieve low inconsistency. especially for first-order methods like SHAP (Lundberg & Lee 2017) and IG (Sundararajan et al. 2017).
For higher-order attribution methods, consistency can potentially be improved due to their enhanced representation power. The classical Shapley interaction index has the problem of not precisely fitting the underlying function, as observed by (Sundararajan et al. (2020), who proposed Shapley Taylor interaction index (Sundararajan al.) (2200) with better empirical performance. Shapley Taylor interaction index satisfies the generalized efficiency axiom, which says for all f E 1-1,1}" - R.
Σ
It(() f(m)) - f(0)
SE
We should remark that both the Shapley interaction index and Shapley Taylor interaction index were not originally designed for consistent interpretations, so they did not specify how to generalize the interpretation for the neighboring points. To this end, we make global extension to the Shapley value based interpretation, that is, using Shapley interaction indices Shapley Taylor interaction indices as the coefficients corresponding terms of the polynomial surrogate function.
g (21, F2, ... F.) f(0) +
I(f,S) (C
Σ
However, these higher-order Shapley value based methods all focus on the original Shapley value framework, so their interpretations are not truthful, i.c., not getting the exact coefficient of f even on will show in experiments, higher-order methods still incur
the "simple bases". Moreover, high interpretation errors compared with our methods.
When applying Shapley value techniques for visual search, Hamilton et al. (2022 proposed an interesting and novel sampling + Lasso regression algorithm for efficiently computing higher order Shapley Taylor index in their experiments. However, their methods bility distribution generated from permutation numbers, which is far from the uniform distribution. Additionally, their algorithm is based on the Shapley Taylor index, so their method truthful as well.
based on sampling proba-
E TEST WITH LOWER ORDER POLYNOMIAL FUNCTIONS
E.
FIRST ORDER POLYNOMIAL FUNCTION
To investigate the performance of different interpretation methods, let us take closer look at a 1st order polynomial function:
1 2 3 42
f1 (11,
For this simple function, can manually compute the outcome of each algorithm, as illustrated in T If the algorithm's 's output is correct, i.e., equal to the output of f1, we write a check mark. Otherwise, write down the actual output of the given interpretation algorithm.
As we can see, all methods are consistent and efficient for all cases. In fact, all variants of Shapley indices degraded to 1st order Shapley values.
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Alporithms
Ground Truih
0.417
1.M
HILLT
SHAP
Shaples Interaction Index
Shopley Taylor Index onder)
Falth-Shap (is)
Low-deegree (Ist enter)
Harmonica (ia onler)
Table I: Interpretations by LIME, SHAP, Shapley Interaction Index, Shapley Taylor Index, Faith- Shap, Low-degree, and Harmonica on the 1st order polynomial function f1.
E.2 SECOND ORDER POLYNOMIAL. FUNCTION
In addition, let us take closer look at a 2nd order polynomial function:
1 -0 7 2212 + TT1ss 72223
fe (21,
For this simple function, we can manually compute the outcome of each algorithm, as illustrated in Table If the algorithm's 's output is correct, i.e., equal to the output of /2, write check mark. Otherwise, we write down the actual output of the given interpretation algorithm.
As see, 2nd order interpretation algorithms, including Shapley Taylor index, Faithful Shapley, Low-degree, and Harmonica are consistent and efficient for all cases. Other methods can only fit few inputs, the 2nd order Shapley interaction index misses all the cases because it is not efficient, and LIME misses all the cases because f2 is not linear function.
Algorilams
L.
Geannl Truth
-0.998
-0.140
-0574
-0.000
+0.474
+1.503
-0307
+0990
LIME
1.4.7
+0.05
0.53
10383
T1081
0153
Shappey
Shapley Taylor Indes (aal ondes)
Shupley
Low-deyrve (2d coler)
Table Interpretations by LIME, SHAP, Shapley Interaction Index, Shapley Taylor Index, Faith- Shap, Low-degree, and Harmonica the 2nd order polynomial function f2-
E.3
THIRD ORDER POLYNOMIAL FUNCTION
Finally, we investigate the following 3rd order polynomial, and present the result Table 3
1 4 T3
Ja (11, 22, 13) = 2 T1
12 +
1133 - =2 + 51222
As we can see, Fhith-Shap, Low-degree, and Harmonica are consistent and efficient for all cases. Other methods can only fit a few inputs, the 3rd order Shapley interaction index misses all the cases because is not efficient, and LIME misses all the cases because fa not linear function.
Algorsthos
[-,-1,-1
[-1-1,+1]
[-,
Truth
SHAP
-0,65
402222
-1.992
-0,85
+1.085
41.252
-022
Shupley Interation Index Ond order)
-0.385
40.251
-0935
-0.96
+0.791
+1.300
-6.10
+1.88
Shupley Taylor Inles (Sd orde)
-0.70
-0642
Faith-Shap Gnl order)
Low-degree ordo)
Harmanica Ond oeler)
Table 3: Interpretations by LIME, SHAP, Shapley Interaction Index, Shapley Taylor Index, Faith- Shap, Low-degree, and Harmonica on the 3rd order polynomial function fa-
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F EXPERIMENT SETTINGS
In this section, we will describe the detailed experimental settings, For the two language tasks, i.c., SST-2 and IMDb, we use the same CNN neural network. The word embedding layer is pre-trained by GloVe (Pennington et al. 2014) and the maximum word number is set to 25, 000. Besides the embedding layer, the network consists of several convolutional kernels with different kernel sizes (3, 4, and 5). After that, we use several fully connected layers, non-linear layers, and pooling layers to process the features. Sigmoid function attached to the tail the network to ensure that the output can be seen as probability distribution. Our networks
trained with Adam (Kingma Ba 2015) optimizer with a learning rate of 0.01 for 5 epochs. For the vision task, we choose the official ResNet (He et al. 2016) architecture, which is available on PyTorch and we do not discuss the architecture details here.
Notice that algorithm is post-hoc model-agnostie interpretation algorithm, only need to the original neural network f to infer on given input as an oracle. This means that one can easily change the network architecture without any additional changes.
All the experiments are run on a server with Nvidia 2080 Ti GPUs. More information about the runtime python environment and implementation details can be found in code.
G DETAILED NUMERICAL RESULTS
In this section, we provide numerical results for Figure
and
in Table and respectively.
radius
L
L'
L"
radius
L norm
L'
E
morm
norm
L.
0.0536
0.0460
0.1154
0.0434
0.0363
0.0995
0.0981
0,0870
0.3572
2
0.0576
0.0463
0.1219
2
0.0484
0.0376
0.1046
0.1010
0.0861
0,3458
0.0639
0.0483
0.1350
0,0561
0.0405
0.1157
0.1043
0.0863
0.3421
8
0.0768
0.0549
0.1696
8
0.0700
0.0474
0.1456
8
0.1135
0.0919
0.3659
16
0.0945
0.2173
16
0,0876
0.0575
0.1900
16
0.1285
0.1015
0,4034
32
0.0990
0.0677
0.2275
32
0.0921
0.0600
0.2003
0.1323
0.1038
0.4111
26
0.0991
0.0677
0.2279
0.0922
0.0601
0.2005
0.1322
0.1037
0,4112
Harmonica 2
Harmonica
LIME
radius
L norm
L norm
L norm
radius
L nomm
L. norm
E norm
radius
L norm
L norm
L norm
0,0792
0.0568
0.1819
0.1133
0.0681
0.2010
010865
0,3477
0.1215
0.0836
0.3167
0.3361
0.1232
0.1470
0.1186
U.4847
0.2081
0.1563
0.5066
0.1753
0.1183
0.3406
8
0.1865
0.1554
0.6252
8
0,2598
0.2075
0.6476
0.2343
0.1691
0.4859
0.6891
32
0.2137
0.1814
U.6958
32
0.2872
0.2357
0.7099
32
0.2709
0.2028
0.5823
0.2137
0.1814
0.6961
00
0,2874
0.2359
0.7105
0.2711
0.2029
0.5823
SHAP
Integrated Gradients
Integrated Hessians
radius
L norm
1. norm
1." norm
radius
1. norm
L. norm
1 norm
0.0623
0.0472
0.1090
0.0602
0.0451
0.1034
0.0853
0.0649
0.1971
0.0813
0.0614
0.1795
4
0.1144
0.0874
0.3069
0.1091
0.0824
0.2794
0.3707
16
0.1612
0.1217
0.4404
16
0.1586
0.1190
0.4271
32
0.1642
0.1241
0.4486
32
0.1616
0.1213
0.4356
CC
0.1642
0.1240
0.1616
0,1213
0.4357
Shapley Taylor Index
Faith- Shith-Shap?
Table 4: The interpretation error of Harmonica and other baseline algorithms evaluated on the SST-2 dataset for different neighborhoods with radius ranging from to co under L L and L norm.
H DISCUSSION ON THE LOW-DEGREE ALGORITHM
From Theorem 2 and Theorem we know that the sample complexity of the Harmonica algo- rithm (Ō(!)) )) is much more efficient than the Low-degree algorithm (Ö(+))) Figure shows that when evaluating the interpretation
SST-2 dataset, with the same sample size, the Harmonica
19

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Under review as a conference paper at ICLR 2023
radius
L porm
L norm
Z" norm
radius
L. norm
L porm
L norm
radius
1. norm
1.1
F.
0.0224
0.0192
0.0040
0.0175
0.0149
0,0027
0.0624
0.0525
0.1175
0.0234
0.0193
0.0157
00346
4
0.0300
0.0228
0.0140
0.0264
0.0195
0.0098
0,0740
0.0583
0.1606
8
0.0441
0.0298
0.0433
8
0.0407
0.0265
0.0367
8
0.084
0.0640
0.1903
16
0.0506
0.0573
16
0.0472
0.0296
0.0505
32
0.0514
0.0354
0,0595
32
0,0481
0.0301
0.0527
32
0.0892
0.0667
0.2033
2CC
0.0514
0.0334
0.0596
00
0.0481
0.0301
0.0528
0,0892
0.0667
0,2033
Harmonica 2
Harmonica
LIME
radhus
L norm
7 norm
L porm
radius
L norm
L norm
L nomm
radius
L porm
L norm
L norm
0.0849
0.0650
0.2266
0.1228
0.0810
0.2702
0.1178
0.0635
0.1929
2
0.1123
0.0903
0.3589
0.1576
0.1169
0.4102
0.1567
0.0997
0.3125
4
0.1406
0.1162
0.4891
4
0.1882
0.1494
0.5333
0.1969
0.1415
0.4501
8
0.1398
0.1341
0.5610
0,2031
0.1657
0.5885
0.2227
0.1702
0,5423
16
0.1662
0,1402
0.5817
16
0.2067
0.1697
0.6004
16
0.2308
0.1795
0,5702
32
0.1671
0.1411
0.5842
32
0.2071
0.1702
0.6014
32
0.2319
0.1807
0.5739
2C
0.1671
0.1411
0.5843
80
0.20V1
0.1702
0.6015
30
0.2319
0.1807
0,5738
SHAP
Integrated
Gradients
Integrated Hessians
radius
L norm
L norm
Z" norm
radius
L norm
L norm
I norm
0.0365
0,0337
0.0590
0.0470
0.0999
0.0469
0.0378
0.0466
0.0684
0.0538
0.1339
0.0547
0.0440
0.0657
0.0698
0.0842
0.1300
0.0620
0.0494
0.0808
16
0.0719
0.0558
0.1287
16
0.0684
0.0543
0.0851
0.0741
0.0575
0.1294
0.0859
0.0743
0.0577
0.1294
0.0744
0.0590
0.0859
Shapley Taylor Index
Faith-Shap
Table The interpretation error of Harmonica and other baseline algorithms evaluated on the IMDb dataset for different neighborhoods with radius ranging from to co under L,2, I, and I norm.
radius
L
L.
7."
radius
1."
E
radius
1 norm
I,a norm
1.9 norm
0.1290
0.1108
0.4248
0.1048
0.0880
0.3202
0,2422
0.2208
0.7274
0.1308
0.1073
0.0887
0.3260
0.2347
0.2036
4
0.1373
0.1094
0.4332
0.1220
0.0953
0.3723
0.2346
0.1918
0.6540
8
0.1584
0.1264
0.3156
8
0.1443
0.1139
0.4698
8
0,2897
0,2304
0.6893
XC
0.1693
0.1342
0.1230
00
0.3261
0.2579
0.7196
Harmonica
Harmonica
LIME
radius
L norm
L norm
L norm
radius
L norm
L norm
1 norn
radius
L norm
nora
L noma
0.1197
0.0867
0.2892
0,2322
0.1650
0.4871
1,3681
0.9504
0.8443
2
0.1658
0.1261
0.4566
2
0.3141
0.2375
0.6406
2
1,8603
1,4006
0.9025
4
0.2306
0.1862
0.6483
0,4200
0.3346
0.7722
2.3168
2.0427
0.9394
0.2943
0.2409
0.7318
0,5010
0.4094
0.8300
3.1095
2,6844
0,9543
0.3115
0.2523
0.7362
0,5113
0.4177
0.8340
3.2139
2,8140
0.9562
SHAP
Integrated Gradients
Integrated Hessians
radius
1.
1.
1. norm
radios
1." morm
1. norn
1."
0.1337
0.1039
0,3939
0. 0.18
0.0948
0.3443
0.4338
4
0.2028
0.1600
0.5830
0.1718
0.1351
0.5126
8
0.2226
0.1771
0.6365
0.1960
0.1554
0.5872
0.2274
0.1804
0.6418
0.2099
0.1654
0.6071
Shapley Taylor Index
Faith-Shap²
Table 6: The interpretation error of Harmonica and other baseline algorithms evaluated on the ImageNet dataset for different neighborhoods with radius ranging from 1 to under L2, L and 6 norm.
algorithm outperforms the Low-degree algorithm by a large margin. We further increase the sample size for the Low-degree algorithm and see that its interpretation error gradually approaches that of Harmonica. However, even with 5x sample size, the Low-degree algorithm still gives a larger interpretation error compared with Harmonica.
20

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Under review as a conference paper at ICLR 2023
2000 wimk
10%
Low drome (2 200 same
Intorp cwrtonattonooc citer
Lond
Low-liphe 2000
Intorpre ctalion
2011
Subssaac
(a) 2 norm
(b) L' norm
(c) L norm
Figure 8: Visualization of interpretation error Ip (f. g) evaluated on SST-2 dataset, while Harmon-
ica and Low-degree algorithms using different sample size varying from 2000 to 10000.
21


Reviews:
[{"review_id": "O6kEUV3sqA", "reviewer": "Reviewer_hMRo", "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": "", "summary_of_the_paper": "This paper proposes an explanation method to provide consistent and truthful explanations of black-box models. Experimental results verify the effectiveness of the proposed method.", "summary_of_the_review": "There exists a significant theoretical defect of the proposed method and the authors do not provide explanation results of the proposed method. Besides, experimental settings are unclear.", "strength_and_weaknesses": "[Strengths]\n1. This paper focused on an important topic, i.e., learning consistent and truthful explanations.\n2. This paper is well written. It is easy to follow the authors’ ideas.\n\n[Weaknesses]\n\n1. There exists a significant theoretical defect in the algorithm. Algorithm 1 and Algorithm 2 aim to learn the polynomial $g(x)=\\sum_{S,\\chi_S\\in C} \\alpha_S \\chi_S(x)$ as the explanation model. However, such an explanation model cannot provide universal explanations because it does not consider the dimension alignment issue. For example, given two sentences, “How are you.” and “Hello, how are you.”, $\\chi_{\\\\{x_1, x_2, x_3\\\\}}$ and $\\chi_{\\\\{x_2, x_3, x_4\\\\}}$ both correspond to the sub-sentence “how are you” of these two sentences. Therefore, the network is supposed to predict the same score for the input of these two sub-sentences, but the explanation model provides different explanations, i.e., $\\alpha_1\\chi_{\\\\{x_1, x_2, x_3\\\\}}$ and $\\alpha_2\\chi_{\\\\{x_2, x_3, x_4\\\\}}$. Such explanations do not make any sense. It seems that such explanation models are only suitable for data where each dimension has a fixed meaning. This is our main concern.\n\n2. The authors do not provide explanation results of the proposed method. Therefore, it is difficult for us to make any comment on the interpretability of the proposed method.\n\n3. The metric of consistency is unable to strictly measure the consistency between normal samples and masked samples. It is because the masked sample is an out-of-distribution (OOD) sample. The explainable model is supposed to provide different explanations for normal samples and OOD samples. For example, a normal sentence contains ten words, and the masked sentence only contains two words, whose explanations are almost absolutely inconsistent. In this way, even the supposed optimal explanation is unlikely to get the inconsistency of zero, i.e., $I_D(f,g)\\ne 0$. The authors are supposed to prove that your definition of inconsistency can correctly reflect the absolute inconsistency of the explanation method.\n\n4. The authors’ three claims about “interpretability, consistency, and efficiency” lack clear definition and proof, i.e., “(a) Attribution methods are interpretable and efficient, but not consistent. (b) The original network is consistent and efficient but not interpretable. (c) If one model is interpretable and consistent, it cannot be efficient.” I find that it is very difficult for me to provide any comments on these unprecise claims. The authors are supposed to provide a clear definition of “interpretability, consistency, and efficiency,” and provide proofs of these three claims.\n\n5. Concern about the correctness of the input space of functions $f$ and $g$. In some Definitions, e.g., Definition 3.2 and Definition 3.3, $f,g\\in\\\\{-1,1\\\\}^n$, but in other Definitions, e.g., Definition 3.4 and Definition 3.6, $f,g\\in\\\\{0,1\\\\}^n$. If they are all correct, then the authors are supposed to explain the reason why the input space is different.\n\n6. Definition 3.8 gives the lower bound of the fitting error of the explanation model. Why do you focus on the lower bound of the fitting error, instead of the upper bound?\n\n7. The experimental comparisons of the proposed explanation method and other explanation methods in Section 5.1 are unfair. It is because the majority function contains the product relation $x_1x_2x_3$, which naturally prefers the explanation method proposed in this paper, where $\\chi_S$ is also defined on the product relation $\\prod_{i\\in S} x_i$. The authors are supposed to conduct experimental comparisons on other functions that do not depend on the product relation of input variables.\n\n8. Ask for more experiments. (1) The authors are supposed to compare the inconsistency between the proposed method and the Faith-SHAP [cite1], which is also an efficient explanation method. (2) The authors are supposed to conduct experimental comparisons with more models, such as LSTMs or Transformers. (3) The authors are supposed to conduct experiments to verify the effectiveness of Algorithm 2.\n\n[cite1] Tsai et al. “FAITH-SHAP: THE FAITHFUL SHAPLEY INTERACTION INDEX” in arXiv:2203.00870 \n\n9. In Figure 3, results show that the proposed method did not exhibit significantly lower readable inconsistency than other methods. This makes me question the effectiveness of the proposed method.\n\nMinor.\n- Which convolutional neural network is the authors use for experiments in Section 6.1?\n\n- Please provide the actual radius in Figures 2, 3, and 4, instead of $\\infty$\n\n- In Definition 3.1, I think it should be $\\chi_S(x)$ instead of $\\chi_S$ in the left-hand side of the equation.\n", "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": "JshwyitRvM", "reviewer": "Reviewer_BSNn", "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": "", "summary_of_the_paper": "Focus on the consistent of ML interpretations, this paper introduces a new objective of consistency based on a notion called truthful interpretation by applying Fourier analysis of Boolean functions. Experimental results show that the method achieves higher consistency compared with other methods.", "summary_of_the_review": "Contributions, especailly technique improvement need to be cleary addressed.", "strength_and_weaknesses": "\nStrong Points:\n[1]This paper produced a novel method of Fourier analysis of Boolean function to get consistency guarantees.\n\n\nWeak Points:\n[1]It is suggested that the author need to highlight the contributions of this paper, especially the part of technique improvements.\n[2]This paper lacks an understanding of where the paper's going, the authors, for example, fail to focus on the consistency of interpretability and the improvements of truth interpretation.\n[3]As above, it needs to review the works of about ML inconsistent rather than the different types of interpretable models in Related Work.\n[4]For any model  and  especially with different structure, the weight distribution which is defined to evaluate the inconsistency by Equation (1) is lack of some evidence. It might be needed to provide some demonstrations.\n[5]It is better here to supplement more clear and plain explanations about algorithm 1 and 2.\n[6]How to evaluate that a function can faithfully interprets the readable part of the network or not in section 3.2?\n[7]Although the experiment results based on SST-2 and IMDb datasets is impressive compared with other baselines, it is needed more clear and sufficient analysis about that. And the experimental hardware environment is not reported.\n[8]Many of the reference should be further improved. For example, few references miss page, and the format of references is not uniform and either AAAI or its full name. In addition, the authors may try to discuss the existing work in published papers (rather than a number of preprinted references from arxiv).", "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": "1dsHy02plb", "reviewer": "Reviewer_dqjk", "summary": "", "questions": "", "limitations": "", "rating": 3, "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": "", "summary_of_the_paper": "The paper investigates the tension between consistency and simplicity in interpretability. The goal is to find a model from a restricted function class (e.g. of simple functions) that matches the network on as much of the input space as possible. The paper focuses on Boolean functions, takes the function space to be functions that are space in the Fourier basis, and investigates approaches to find the sparse function that best approximates a given function. They investigate the performance of (slightly modified versions of) the Harmonica algorithm (Hazan et al., 2017) and Low-degree (Linial et al., 1993). The paper then evaluates these algorithms and compares them to existing approaches—including LIME and SHAP—on datasets for sentiment analysis and movie review classification. ", "summary_of_the_review": "The paper investigates the tensions between consistency and simplicity in the context of interpretability, with a focus on interpreting Boolean functions with functions that are sparse in Fourier basis: however, the setup is quite restrictive, and furthermore, both the algorithmic and empirical contributions are limited. ", "strength_and_weaknesses": "Strengths\n- The paper illustrates that existing tools from learning sparse approximations to Boolean functions could be related to interpretability. \n\nWeaknesses\n- The analysis in the paper is entirely restricted to Boolean functions, and the restricted function class is assumed to be sparse functions in the Fourier basis. The algorithms studied in this paper rely heavily on these assumptions. \n- The algorithms investigated by the paper are standard in the learning Boolean functions literature (as the paper does acknowledge). For example, Section 4 is entirely devoted to analyzing existing algorithms (with minor modifications). \n- The empirical evaluation is restricted to small-scale setups: a convolutional network achieving ~80% accuracy on the SST-2 dataset and a convolutional network achieving trained on the IMDb dataset. It would be useful to more extensively empirical evaluate their methods on larger-scale setups with modern networks. ", "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": {}}]

Author rebuttal:

Decision: Reject