Title: Locally Invariant Explanations: Towards Stable and Unidirectional Explanations through Local Invariant Learning
Venue: NeurIPS 2023
Accepted: yes
Keywords: Explainable AI

Game theory

Invariance
Abstract: Locally interpretable model agnostic explanations (LIME) method is one of the most popular methods used to explain black-box models at a per example level. Although many variants have been proposed, few provide a simple way to produce high fidelity explanations that are also stable and intuitive. In this work, we provide a novel perspective by proposing a model agnostic local explanation method inspired by the invariant risk minimization (IRM) principle -- originally proposed for (global) out-of-distribution generalization -- to provide such high fidelity explanations that are also stable and unidirectional across nearby examples. Our method is based on a game theoretic formulation where we theoretically show that our approach has a strong tendency to eliminate features where the gradient of the black-box function abruptly changes sign in the locality of the example we want to explain, while in other cases it is more careful and will choose a more conservative (feature) attribution, a behavior which can be highly desirable for recourse. Empirically, we show on tabular, image and text data that the quality of our explanations with neighborhoods formed using random perturbations are much better than LIME and in some cases even comparable to other methods that use realistic neighbors sampled from the data manifold. This is desirable given that learning a manifold to either create realistic neighbors or to project explanations is typically expensive or may even be impossible. Moreover, our algorithm is simple and efficient to train, and can ascertain stable input features for local decisions of a black-box without access to side information such as a (partial) causal graph as has been seen in some recent works.

Locally Invariant Explanations: Towards Stable and Unidirectional Explanations through Local Invariant Learning

Abstract

Locally interpretable model agnostic explanations (LIME) method is one of the most popular methods used to explain black-box models at a per example level. Although many variants have been proposed, few provide a simple way to produce high fidelity explanations that are also stable and intuitive. In this work, we provide a novel perspective by proposing a model agnostic local explanation method inspired by the invariant risk minimization (IRM) principle - originally proposed for (global) out-of-distribution generalization - to provide such high fidelity explanations that are also stable and unidirectional across nearby examples. Our method is based on a game theoretic formulation where we theoretically show that our approach has a strong tendency to eliminate features where the gradient of the black-box function abruptly changes sign in the locality of the example we want to explain, while in other cases it is more careful and will choose a more conservative (feature) attribution, a behavior which can be highly desirable for recourse. Empirically, we show on tabular, image and text data that the quality of our explanations with neighborhoods formed using random perturbations are much better than LIME and in some cases even comparable to other methods that use realistic neighbors sampled from the data manifold. This is desirable given that learning a manifold to either create realistic neighbors or to project explanations is typically expensive or may even be impossible. Moreover, our algorithm is simple and efficient to train, and can ascertain stable input features for local decisions of a black-box without access to side information such as a (partial) causal graph as has been seen in some recent works.

1 Introduction

Deployment and usage of neural black-box models has significantly grown in industry over the last few years creating the need for new tools to help users understand and trust models [24]. Even wellstudied application domains such as image recognition require some form of prediction understanding in order for the user to incorporate the model into important decisions [46, 33]. An example of this could be a doctor who is advised by a model of a positive cancer diagnosis based on an image

Karthikeyan Natesan Ramamurthy ∗

scan. Since the doctor holds responsibility for the final diagnosis, the model must provide sufficient reason for its prediction. Even new text categorization tasks [21] are becoming important with the growing need for social media companies to provide better monitoring of public content. Twitter was monitoring tweets related to COVID-19 in order to label tweets containing misleading information, disputed claims, or unverified claims [42]. Laws will likely emerge requiring explanations for why red flags were or were not raised in many examples. In fact, the General Data Protection and Regulation (GDPR) [51] act passed in Europe already requires automated systems that make decisions affecting humans to be able to explain them. Given this acute need, a number of methods have been proposed to explain local decisions (i.e. example specific decisions) of classifiers [41, 35, 46, 33, 13]. Locally interpretable model-agnostic explanations (LIME) is arguably the most well-known local explanation method that requires only query (or black-box) access to the model. Although LIME is a popular method, it is known to be sensitive to certain design choices such as i) (random) sampling to create the (perturbation) neighborhood 2 , ii) the size of this neighborhood (number of samples) and iii) (local) fitting procedure to learn the explanation model [37, 54]. The first, most serious issue could lead to nearby examples having drastically different explanations making effective recourse a challenge. One possible mitigation is to increase the neighborhood size but one cannot arbitrarily do so as it not only leads to higher computational cost, but also in today's cloud computing-driven world it could have direct monetary implications where every query to a black-box model has an associated cost [14].

There have been variants suggested to overcome these limitations [8, 45, 39, 15] primarily through mechanisms that create realistic neighborhoods or through adversarial training [32], however, their efficacy is restricted to certain settings and modalities based on their assumptions and training strategies.

In this paper we introduce a new method called Locally INvariant EXplanations (LINEX) inspired by the invariant risk minimization (IRM) principle [7], that produces explanations in the form of feature attributions that are robust to neighborhood sampling and can recover faithful (i.e. mimic black-box behavior), stable (i.e. similar for closeby examples) and unidirectional (i.e. same sign attributions a.k.a. feature importances) for closeby examples, see section 4.1) explanations across tabular, image, and text modalities. In particular, we show that our method performs better than the competitors for random as well as realistic neighborhood generation, where in some cases even with the prior strategy our explanation quality is close to methods that employ the latter. Qualitatively, our method highlights (local) features as important that in the particular locality i) have consistently high gradient with respect to (w.r.t.) the black-box function and ii) where the gradient does not change significantly, especially in sign. Such stable behavior for LINEX is visualized in Figure 1, where we get similar explanations for nearby examples in the IRIS dataset. The (in)fidelity of LINEX is still similar to LIME (see Table 2), but of course our explanations are much more stable.

2 Related Work

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Posthoc explanations can typically be partitioned into two broad categories global and local. Global explainability avers to trying to understand a black-box model at a holistic level where the typical tact is knowledge transfer [28, 17, 16] where (soft/hard) labels of the black-box model are used to train an interpretable model such as a decision tree or rule list [43]. Local explanations on the other hand avers to understanding individual decisions. These explanations are typically in two

forms, either exemplar based or feature based. For exemplar based as the name suggests similar but diverse examples [30, 25] are provided as explanations for the input in question. While for feature based [41, 35, 13, 33, 55], which is the focus of this work, important features are returned as being important for the decision made for the input. There are some methods that do both [39]. Moreover, there are methods which provide explanations that are local, global as well as at a group level [40]. All of these methods though may not still provide stable and robust local feature based explanations which can be desirable in practice [23].

Given this there have been more recent works that try to learn either robust or even causal explanations. In [32] the authors try to learn robust and stable local explanations relative to distribution shifts and adversarial attacks. However, the distribution shifts they consider are linear shifts and adversarial training is performed which can be slow and sometimes unstable [53]. Moreover, the method seems to be applicable primarily to tabular data. There are also works [18, 19] which try to robustify gradient based explanations assuming white box access to the model. Works on causal explanations [22, 27] mainly modify SHAP and assume access to a partial causal graph. Some others [48] assume white-box access. In this work we do not assume availability of such additional information. There are also works which show that creating realistic neighborhoods by learning the data manifold for LIME [8, 45] can lead to better quality explanations, where in a particular work [6] it is suggested that projecting explanations themselves on to the manifold can also make them more robust. The need for stability in a exemplar neighborhood for LIME like methods has been highlighted in [54], with the general desire for stable explanations being also expressed in [52, 49]. Furthermore, it was recently surmised through expert and crowd worker user studies that stability is a key factor when it comes to assessing capability of a model or when learning a new domain [34].

Given that our approach is inspired from IRM we now describe, how it is novel w.r.t. to it. It is important to realize that IRM approaches such as [3, 4] are designed for the out-of-distribution (OOD) generalization, which learn global models directly from the data. The main similarity of these works to ours is only that they also are game theory based approaches, but with the details being quite different. For one, they assume accessibility to environments which (ideally) correspond to different interventional distributions and with assumptions on the structural causal model derive results on how the true causal factors could be divulged. In our case, we propose ways to generate environments as they are not given, and have l 1 and l ∞ constraints on the entire and environment specific parts of the model respectively, which is not the case with these prior works. As such those algorithms do not produce sparse unidirectional models that are also consumable. Moreover, the perspective we provide is novel in the context of local posthoc explanations where a priori it is not obvious that approaches from OOD generalization could be extended and adapted. Additionally, we propose a novel metric Unidirectionality which is not part of any of these works, but as we have argued it is a desirable property for explanations.

3 Preliminaries

Invariant Risk Minimization: Given a collection of training datasets D = { D e } e ∈E tr gathered from a set of environments E tr , where D e = { x i e , y i e } n e i =1 is the dataset gathered from environment e ∈ E tr and n e is the number of points in environment e . The feature value for data point i is x i e ∈ X and the corresponding label is y i e ∈ Y , where X ⊆ R d and Y ⊆ R . Each point ( x i e , y i e ) in environment e is drawn i.i.d from a distribution P e . Define a predictor f : X → R .

The goal of IRM is to use these collection of datasets D to construct a predictor f that performs well across many unseen environments E all , where E all ⊇ E tr . Define the risk achieved by f in environment e as R e ( f ) = E e [ glyph[lscript] ( f ( X e ) , Y e ) ] , where glyph[lscript] is the square loss when f ( X e ) is the predicted value and Y e is the corresponding label, ( X e , Y e ) ∼ P e and the expectation E e is defined w.r.t. the distribution of points in environment e .

An invariant predictor is composed of two parts a representation Φ ∈ R d × n and a predictor (with the constant term) w ∈ R d × 1 . We say that a data representation Φ elicits an invariant predictor w T Φ across the set of environments E tr if there is a predictor w that achieves the minimum risk for all the environments w ∈ argmin ˜ w ∈ R d × 1 R e ( ˜ w T Φ ) , ∀ e ∈ E tr . IRM may be phrased as the following constrained optimization problem [7]:

If w T Φ solves the above, then it is an invariant predictor across the training environments E tr .

glyph[negationslash]

Nash Equilibrium (NE): To understand how certain key aspects of our method function let us revisit the notion of Nash Equilibrium [20]. A standard normal form game is written as a tuple Ω = ( N , { u i } i ∈N , {S i } i ∈N ) , where N is a finite set of players. Player i ∈ N takes actions from a strategy set S i . The utility of player i is u i : S → R , where we write the joint set of actions of all the players as S = Π i ∈N S i . The joint strategy of all the players is given as s ∈ S , the strategy of player i is s i and the strategy of the rest of players is s -i = ( s i ′ ) i ′ = i .

Definition 1. A strategy s † ∈ S is said to be a pure strategy Nash equilibrium (NE) if it satisfies, u i ( s † i , s † -i ) ≥ u i ( k, s † -i ) , ∀ k ∈ S i , ∀ i ∈ N , where u i ( s † i , s † -i ) = u i ( s † 1 , s † 2 , ..., s † N ) = u i ( s † ) .

NE thus identifies a state where each player is using the best possible strategy in response to the rest of the players leaving no incentive for any player to alter their strategy. In seminal work by [11] it was shown that for a special class of games called concave games such a pure NE always exists. This is relevant because the game implied by Algorithm 1 falls in this category.

4 Methodology

We first define desirable properties for our explanation methods. The first three have been seen in previous works, while the last Unidirectionality is new. We then describe our method where the goal is to explain a black-box model f : X → R for individual inputs x based on predictors w by looking at their corresponding components, also termed as feature attributions.

We take inspiration from IRM since, our goal here too is to extract robust features that are ideally stable and unidirectional. The main difference is that we do not learn a new (possibly invariant) representation since, we desire interpretability and this new representation may not be interpretable. We hence, are restricted to the provided input or some other interpretable representation. Thus, given that Φ ⊆ X where n = 1 (since local explanations) in our setup, our goal is to find the best predictor w (viz. high fidelity) for an input that will eliminate or at least mitigate the effect of unstable features. In other words, we want to identify features in the input space that will (roughly) have the same importance (i.e. are invariant) in the neighborhood of the example we want to explain. Our approach as we will see is similar in spirit to IRM games [4], where we adopt a game theoretic strategy to obtain such explanations. The differences with IRM games are mentioned in the last paragraph of Section 2.

4.1 Desirable Properties

We now discuss certain properties we would like our explainability method to have in order to provide robust explanations that could potentially be used for recourse. Let D t denote a (test) dataset with examples ( x, y ) where y b ( x ) is the black-box models prediction on x and y x ′ e ( x ) is the prediction on x ( ∈ X ) using the explanation model at x ′ . The feature attributions (or coefficients) for the explanation model at x are denoted by c x e , N x denotes the exemplar neighborhood of x with | . | card denoting cardinality and | . | denoting absolute value.

Fidelity: This is the most standard property which all proxy model based explanation methods are evaluated against [41, 35, 32] as it measures how well the proxy model simulates the behavior of the black-box (i.e. faithfulness to the black box) it is attempting to explain. Here we define inverse of it, that is Infidelity (INFD) , as the MAE between the black-box and explanation model predictions across all the test points:

Wealso define another metric here called Generalized Infidelity (GI) , which also been used in previous works [40] to measure the generalizability of local explanations to neighboring test points. It is defined as:

Stability: This is also a popular notion [26, 40, 52] to evaluate robustness of explanations. Largely, stability can be measured at three levels. One is prediction stability, which measures how much the predictions of an explanation model change for the same example subject to different randomizations within the method or across close by examples. The second is the variance in the feature attributions again for the same or close by examples. It is good for a method to showcase stability w.r.t. both even though in many cases the latter might imply the former. An interesting third notion of stability is the correlation between the feature attributions of an explanation model and average feature values of examples belonging to a particular class. This measures how much does the explanation method pick features that are important for the class, rather than spurious ones that seem important for just the example. We thus define two stability metrics.

Coefficient Inconsistency (CI): This notion has been used before [26] to measure an explanation methods robustness. It can be defined as the MAE between the attributions of the test points and their respective neighbors:

Class-Attribution Consistency (CAC): For local explanations of classification black-boxes, we expect certain important features to be highlighted across most of the explanations of a class. This is codified by this metric which is defined as follows:

where Y denotes the set of class labels in the dataset, µ y the mean (vector) of all inputs in class y ∈ Y , µ y e the mean explanation for class y and r the Pearson's correlation coefficient. This metric quantifies the consistency between the important features for a class and attributions provided by the explanations.

Black-box Invariance: This is the same as implementation invariance defined in [47]. Essentially, if two models have exactly the same behavior on all inputs then their explanations should also be the same. Since, our method is model agnostic with only query access to the model it is easy to see that it satisfies this property if the same environments are created.

Unidirectionality: This is a new property, but as we argue that this is a natural one to have. Loosely speaking, unidirectionality would measure how consistently the sign of the predictor for a feature is maintained for the same or close by examples by an explanation method. This is a natural metric [36], which from an algorithmic recourse [29] perspective is also highly desirable. For instance, recommending a person to increase their salary to get a loan and then recommending to another person with a very similar profile to decrease their salary for the same outcome makes little sense.

We define the unidirectionality Υ as a measure of how consistent the sign of the attribution for a particular feature in a local explanation is when varying neighborhoods for the same example or when considering different close by examples. As such, given m attributions for each of d features denoted by w (1) 1 , ..., w ( d ) m the metric for an example is:

where | . | stands for absolute value. Clearly, the more consistent the signs for the attribution of a particular feature across m attributions the higher the value, where the maximum value can be one. If equal number of attributions have different signs for all features then Υ will be zero, the lowest possible value. This property thus measures how intuitively consistent (ignoring magnitude) the explanations are. Given its sole focus on the sign of the attributions it compliments the above metrics along with attributional robustness metrics [9, 44].

4.2 Method

4.2.1 Description

In Algorithm 1, we show the steps of our method LINEX. The input is the example we want to explain x , the black-box predictor, a few thresholds that we describe next and k (local) environments

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whose creation is described in Section 4.2.2. In the algorithm we iteratively learn a constrained least squares predictor for each environment, where the final (local) linear predictor is the sum of these individual predictors. In each iteration when computing the contribution of environment e i to the final summed predictor, the most recent contributions of the other predictors are summed and the residual is optimized subject to the constraints. The first constraint is a standard lasso type constraint which tries to keep the final predictor sparse as in LIME.

Why l ∞ constraint? The second constraint is more unique and is a l ∞ constraint on the predictor of just the current environment. This constraint as we prove in Section 4.3 is essential for obtaining robust predictors. To intuitively understand why this is the case consider we have two environments. In this case if the optimal predictors for a feature in each environment have opposite signs, then the Nash equilibrium (NE) is when each predictor takes + γ or -γ values as they try to force the sum to have the same sign as them. In other words, features that have a disagreement in even the direction of their impact are eliminated by our method . LIME type methods on the other hand would simply choose some form of average value of the predictors which may be a risky choice especially for actionability/recourse given that the directions change so abruptly. On the other hand, if the optimal predictors for a feature in the two environments have the same sign, the lower absolute valued predictor would be chosen (assuming γ is greater) making it a careful choice. The reasoning for this and a discussion involving more than two environments is given in Section 4.3.

The overall algorithm resembles a (simultaneous) game where each environment is a player trying to find the best predictor for its environment given all other predictors and constraints. Formally, for i ∈ { 1 , ..., k } the players are N = { ξ i } , their strategy space is S i = [ -γ, γ ] d and their utility u i ( ˜ w i , ˜ w + -i ) = -∑ ˜ x ∈ ξ i ( x ) ( f (˜ x ) -˜ w + T -i ˜ x -˜ w T i ˜ x ) 2 . The optimization problem solved by each player is convex as norms are convex.

4.2.2 Creating Local Environments

In standard IRM, environments are assumed to be given. In our case of local explainability we have to decide how to produce them. We offer a few options for the environment creation functions ξ i ∀ i { 1 , ..., k } in Algorithm 1.

Random Perturbation: This simple approach is similar to what LIME employs. We could perturb the input example by adding zero mean gaussian noise to create the base environment (used by LIME) and then perform bootstrap sampling to create the k different environments. This will efficiently create neighbors in each environment, although they may be unrealistic in the sense that they could correspond to low probability points w.r.t. the underlying distribution.

Realistic Generation/Selection: One could also create neighbors using data generators such as done in MeLIME [8] or select neighboring examples from the training set as done in MAPLE [39] to create

the base environment following which bootstrap sampling could be done to form the k different environments. This could provide more realistic neighbors than the previous one, but may be much more computationally expensive. Other than bootstrapping one could also oversample and try to find the optimal hard/soft partition through various clustering type objectives [2, 10].

4.3 Theoretical Results

In this section, we analyze the output of Algorithm 1 with two environments. The extension to multiple environments is discussed following this result, where the general intuition is still maintained but some special cases arise depending on whether there are an even or odd number of environments. To prove our main result we make two assumptions.

Assumption 1 The features of the samples in the local environments are independent.

This assumption is satisfied by the most standard way of creating neighborhoods/environments, where gaussian noise is used to create them as described in Section 4.2.2.

Assumption 2 t ≥ γd , where d is the dimensionality of the feature vector.

Here t is the parameter in the glyph[lscript] 1 penalty and γ in the glyph[lscript] ∞ as noted in Algorithm 1. Making this assumption ensures that we closely analyze the role of the glyph[lscript] ∞ penalty, which is one of our main novelties.

Definition 2 Let the explanation that each environment ξ i arrives at for an example x based on unconstrained least squares minimization be w ∗ i where,

The expectation is taken w.r.t the environment generation distribution.

Theorem 1. The output of Algorithm 1 under Assumptions 1, 2 and equation 7 is given by:

where glyph[circledot] is element wise product and 1 is the indicator function.

Proof Sketch. The above expression describes the NE of the game played between the two local environments each trying to move w towards their least squares optimal solution. Given assumptions 1 and 2, we witness the following behavior of our method. Let the i th feature of the predictors ˜ w 1 and ˜ w 2 from Algorithm 1 be ˜ w 1 i and ˜ w 2 i respectively. Let the corresponding least squares optimal predictors for the i th feature have the following relation: w ∗ 1 i > w ∗ 2 i and | w ∗ 1 i | > | w ∗ 2 i | . Then the two environments will push the ensemble predictor, ˜ w 1 i + ˜ w 2 i , in opposite directions during their turns, with the first environment increasing its weight, ˜ w 1 i , and the second environment decreasing its weight, ˜ w 2 i . Eventually, the environment with a higher absolute value ( ξ 1 = 1 since | w ∗ 1 i | > | w ∗ 2 i | ) reaches the boundary ( ˜ w 1 i = γ ) and cannot move any further due to the l ∞ constraint. The other environment ξ 2 best responds, where it either hits the other end of the boundary ( ˜ w 2 i = -γ ), in which case the weight of the ensemble for component i is zero, a case which occurs if w ∗ 1 i and w ∗ 2 i have opposite signs; or gets close to the other boundary while staying in the interior ( ˜ w 2 i = w ∗ 2 i -γ ), in which case the weight of the ensemble for feature i is w ∗ 2 i , a situation which occurs if w ∗ 1 i and w ∗ 2 i have the same sign.

Implications of the Theorem 1: The following are the main takeaways from Theorem 1: (1) If the signs of the explanations for unconstrained least squares for the two environments differ for some feature, then the algorithm outputs a zero for that feature attribution. (2) If the signs of the explanations for the two environments are the same, then the algorithm outputs the lesser magnitude of the two. These two properties are highly desirable from an algorithmic recourse or actionability perspective, where the first biases us to not rely on features where the black-box function changes direction rapidly (unidirectionality). The second, provides a reserved estimate so that we do not incorrectly over rely on the particular feature (stability). Based on similar logic presented in the proof sketch the behavior for more than two environments for LINEX is discussed in Suppl. C.

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5 Experiments

We test our method on five real world datasets covering all three modalities: IRIS (Tabular) [12], Medical Expenditure Panel Survey (Tabular) [1], Fashion MNIST (Image) [50], CIFAR10 (Image) [31] and Rotten Tomatoes reviews (Text) [38] with LIME-like random (rand) and MeLIME-like realistic neighborhood generation (real) or MAPLE-like realistic neighborhood selection (mpl). The summary of black-box classifier accuracies, and type of realistic perturbation used for the datasets are provided in Table 3 in the Supplement. In other cases except FMNIST and CIFAR10 which come with their own test partition we randomly split the datasets into 80/20% train/test partition and average results for the local explanations over this test partition. For LINEX we produce two environments where the two environments are formed by performing bootstrap sampling on the base environment which is created either by rand, real or mpl type neighborhood generation. Thus in all cases the union of the environments is the same as a single neighborhood used to produce explanations for the competitors making it a fair comparison. Behavior with more environments is in Suppl. E.

Given the neighborhood generation schemes we compare LINEX with LIME, Smoothed LIME (S-LIME), MeLIME and MAPLE, where for S-LIME we average the explanations of LIME across the LINEX environments. SHAP's results are in Suppl. H, since it is not a natural fit here. Nor are methods such as saliency maps, gradcam, integrated gradients as they are white-box methods requiring access to a differentiable model.

Metrics: We evaluate using five simple metrics: Infidelity (INFD), Generalized Infidelity (GI), Coefficient Inconsistency (CI), Class Attribution Consistency (CAC) and Unidirectionality ( Υ ), which are defined in section 4.1. The first two evaluate faithfulness , the next two stability and the last goodness for recourse .

Wereport the above metrics in Table 2. Each result in Table 2 is mean ± stan-

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dard error of the mean over five kernel sizes τ √ d generally, where τ = { 0 . 05 , 0 . 1 , 0 . 25 , 0 . 5 , 0 . 75 } . Test neighborhoods do not make sense for random perturbations with FMNIST, CIFAR10 and Rotten Tomatoes because the features (viz. superpixels) used by neighboring test examples are different.

Also, we do not use realistic perturbations with MEPS since KDE and VAE generators do not work well with categorical data. In addition, since MEPS data uses regression black-box, CAC cannot be computed. Also for CIFAR10 images in a class are not aligned so CAC is inapplicable. All these justify the missing entries in Table 2. The results were generated on Linux machines with 56 cores and 242 GB RAM. More details regarding the exact perturbation schemes for LIME/MeLIME/MAPLE, the perturbation neighborhood sizes and the time taken by the methods are in Suppl. A and Suppl. D.

Observations: Quantitatively , we see that in terms of CAC, LINEX is better than baselines in all cases which indicates that on average the LINEX explanations highlight the important features characterizing the entire class, making them more stable. This is also verified by looking at Υ and CI metrics where LINEX is similar or better than others. For GI and INFD, the results are more evenly spread which implies that LINEX's key advantage is obtaining stable and unidirectional explanations that are faithful to a similar degree. Ablation studies showing superiority of LINEX over MeLIME on the FMNIST dataset where we have significantly higher INFD than MeLIME are given in Suppl. J.

An interesting observation is that when it comes to the stability metrics (CI and CAC) and unidirectionality LINEX with even random perturbation model is better than MeLIME in some cases. This is very promising as it means LINEX could be potentially be trusted without the need to generate realistic perturbations which may be computationally expensive or not even possible.

Qualitatively , we see in Figures 2 and 3, that LINEX explanations are more coherent and highlight more salient features compared to MeLIME. Even on the text data we see more reasonable attributions in Table 1, where 'masterpiece', 'moving' and 'audacious' are highlighted as the most important words indicative of positive sentiment in the three examples. We also performed qualitative error

analysis on FMNIST where our INFD is much worse than MeLIME and is described in Suppl. I. We see that even where LINEX has high infidelity it invariably still focuses on salient features ignoring superfluous features which may result in lower fidelity but may not be critical for correct identification. The goodness of these features identified by LINEX can be further verified by looking at other metrics such as GI, CAC, CI and Υ in Table 2 where it is either comparable or better than MeLIME.

6 Discussion

In this paper we have provided a method based on a game theoretic formulation and inspired by the invariant risk minimization principle to provide faithful, stable and unidirectional explanations. We have defined the latter property and argued that it is somewhat of a necessity (may not be sufficient) for recourse. We have theoretically shown that our method has a strong tendency to be stable and unidirectional as we will mostly eliminate features where the black-box models gradient changes abruptly and in other cases choose a conservative value. Empirically, we have verified this where we outperform competitors in majority of the cases on these metrics. Interestingly, in some cases our method provides more stable and unidirectional explanations with just a random perturbation model relative to more expensive methods that use realistic neighbors.

We now discuss a real world use case we tested our method on. We worked with a large financial institution to explain the fraud detection model they had built. The Association of Certified Fraud Examiners (ACFE) claims that roughly 5% of a companies revenue is lost to fraud every year. Thus, catching fraud or even non-compliance is extremely important for any organization. Their model (fraud = 1 else 0 ) had ≈ 91% accuracy. The inputs to the model were (transactional) invoices and details corresponding to those invoices such as vendor name, invoice amount, purchase order (PO) or not, vendor address, commodity code, country perception indices (CPI), etc. Since, one of the focuses is to reduce false positives accurate explanations are important. We applied LINEX to this setting to explain why certain invoices were classified as fraudulent. The experts found that in majority of the cases (913 out of 1000) the attributions of LINEX especially in terms of sign made sense. For instance, low CPI implies high risk and so LINEX gave a negative coefficient for this feature for most examples, while LIME gave a positive coefficient for many instances. Going forward their plan is to incorporate such capabilities into their workflow to further improve fraud detection precision.

In the future, it would be worth experimenting with more varied strategies to form environments and if possible find the optimal ones [10], which may lead to picking even more relevant features that are 'causal' to the local decision.

7 Summary of the Supplement

Information about black-box classifier accuracies and realistic perturbation methods used for the datasets are provided in Table 3. Suppl. A has run time comparisons. Suppl. B has proof of Theorem 1. Suppl. C discusses theoretical behavior of LINEX for more than two environments. Suppl. D has dataset details and hyperparameter specifications. Suppl. E has experiments with different hyperparameter combinations (including more than 2 environments). Suppl. F has additional examples of text data attributions. Suppl. G has example feature attributions with image data. Suppl. H has SHAP results. Suppl. I, J and K has error analysis and ablation studies. Suppl. L has additional synthetic experiments. Suppl. M discusses sensitivity to γ . Suppl. N demonstrates convergence of LINEX. Suppl. O discusses limitations of LINEX. Figure 30 depicts SLIME variants using median and median of means which turn out to be worse than using the (typical) mean.

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A Efficiency of LINEX

It is important to note that the query complexity (i.e. number of times we query the black box to obtain an explanation) of LINEX is the same as that of LIME since the union of the environments is the same as a LIME perturbation neighborhood. This is important in todays cloud-driven world where models may exist on different cloud platforms and posthoc explanations are an independent service where each call to the model has an associated cost. In terms of running time for two environments, convergence was fast and running time was approximately 2.5 times that of LIME (LINEX took 2.5 seconds on IRIS for 30 examples as opposed to 1 second by LIME, LINEX took 47 seconds on MEPS for 500 examples as opposed to 18 seconds by LIME), which is very similar to Smoothed LIME (S-LIME) (took 2.3 seconds on IRIS and 40 seconds on MEPS) that we still outperform in majority of the cases.

Realistic neighborhood generation can be time consuming especially for MeLIME since generators have to be trained which may take up to an hour using a single GPU for datasets such as FMNIST. After the generator is trained and neighborhood sampled MeLIME takes the same amount of time as LIME since the model fitting procedure is the same. MAPLE took 1.5 seconds for the IRIS dataset for 30 examples and 27 seconds for 500 MEPS examples.

A way to further speed up LINEX would be to implement it through embarrassing parallelism which can easily be done across explanations. This will prevent scaling of the running time in the number of examples when many explanations are needed. The setting with many explanations is anyway where we would need efficiency because if only few explanations were desired the slightly higher running time of LINEX would not be an issue.

B Proof of Theorem 1

Expanding on the proof sketch provided in the main paper we now provide a case wise analysis to prove Theorem 1.

w ∗ 1 = w ∗ 2 : If the optimal solutions to both environments in the convex set [ -γ, γ ] d are the same, then in the first iteration itself where we fit to the first environment we would have reached the optimal solution to our problem where ˜ w 1 = w ∗ 1 . This is because in the second iteration where we fit the second environment to the residual from the previous fit ˜ w 2 = 0 and the algorithm would terminate. This would imply the output of algorithm 1 would be w = w ∗ 1 .

glyph[negationslash]

w ∗ 1 = w ∗ 2 : When the optimal solutions for the two environments are not equal we consider the following two cases:

Opposite sign attributions: If the i th component of w ∗ 1 and w ∗ 2 have opposite signs, then the i th components of the ensemble predictor, ˜ w 1 i and ˜ w 2 i are both at the boundary γ and -γ respectively if ˜ w 1 i > 0 . This is because both try to push the ensemble (i.e. their sum) towards the sign they have where eventually they reach the boundary ± γ and have no incentive to deviate. Any deviation from these values will lead to a higher least squares error in their environment, thus making this a NE.

Same sign attributions: If the i th component of w ∗ 1 and w ∗ 2 have same signs, then the i th component of ensemble predictor constructed from the NE is set to the least squares attribution with a smaller absolute value, i.e., w i = w ∗ 1 i , where | w ∗ 1 i | ≤ | w ∗ 2 i | . Without loss of generality assume 0 < w ∗ 1 i < w ∗ 2 i , the attribution of the environments' predictors in NE, then ˜ w 1 i and ˜ w 2 i have opposite signs, i.e., ˜ w 2 i = γ and ˜ w 1 i = w ∗ 1 i -γ where the

ensemble predictor for the i th component would be w i = ˜ w 1 i + ˜ w 2 i = w ∗ 1 i -γ + γ = w ∗ 1 i , since any deviation from this would lead to a worse least squares loss for the corresponding environment. This shows that ensemble predictor is conservative and selects the smaller least squares attribution.

C Behavior for More than Two Environments

Given Assumptions 1 and 2 we now discuss the behavior of our method for more than two environments. If the number of environments is odd, then using similar logic to that discussed in the proof sketch one can see that the feature attribution would be equal to the median of the feature attributions across all the environments. Essentially, all environments with optimal least squares attributions above the median would be at + γ , while those below it would be at -γ . The one at the median would remain so with no incentive for any environment to alter its attribution making it a NE. This is a stable choice that is also likely to be faithful as we have no more information to decide otherwise. On the other hand if we have an even number of environments the final attribution in this case depends on the middle two environments in the same manner as the two environment case proved in Theorem 1. Thus, if the optimal least squares attributions of the middle two environments have opposite sign, then the final attribution is zero, else its the lower of the two attributions in terms of the numerical value. This happens because the NE for the other environments is ± γ depending on if their optimal least squares attributions are above/below those of the middle two environments. This again is a stable and likely to be faithful choice, where also unidirectionality is preferred.

D Experimental Details

D.1 Dataset Details and Hyperparameter Specifications

We describe the datasets and the hyperparameters used for each. We set perturbation neighborhood sizes 10 (IRIS), 500 (MEPS), 100 (FMNIST-random), 500 (FMNIST-realistic), 100 (CIFAR10random), 500 (CIFAR10-realistic), 100 (Rotten tomatoes) for generating local explanations. We also use 3, 10, 10, 10, 5 as exemplar neighborhood sizes to compute GI, CI and Υ metrics for the five datasets respectively. We also use 5 -sparse explanations for all cases except FMNIST and CIFAR10 with realistic perturbations where we follow MeLIME and generate a dense explanation using ridge penalty with penalty multiplier value of 0 . 001 . The glyph[lscript] ∞ bound γ in Algorithm 1 is set as the maximum absolute value of linear coefficient computed by running LIME/MeLIME in the two individual environments. Please look at IRIS dataset first since it contains some of the common details used across others.

IRIS (Tabular): This dataset has 150 instances with four numerical features representing the sepal and petal width and length in centimeters. The task is to classify instances of Iris flowers into three species: setosa , versicolor , and virginica . A random forest classifier was trained with a train/test split of 0.8/0.2 and yielded a test accuracy of 93%. We provide local explanations for the prediction probabilities for class setosa . For both random and realistic perturbations, we use a perturbation neighborhood size of n . For random perturbations, we used the same approach followed by LIME and sample from a Gaussian around each data point. Realistic perturbations (with the same number n ) were generated using KDEGen [8], a kernel density estimator (KDE) with the Gaussian kernel fitted on the training dataset to sample data around a sample point. For both random and realistic perturbations, we weight the neighborhood using a Gaussian kernel of width τ √ d , where d is the dimension of the feature vector and τ = { 0 . 05 , 0 . 1 , 0 . 25 , 0 . 5 , 0 . 75 } , and this corresponded to kernel widths { 0 . 1 , 0 . 2 , 0 . 5 , 1 . 0 , 1 . 5 } . We also perform a weighted version of realistic selection where we use MAPLE [39] to assign weights to all the test examples and pick the top n weighted examples to use as the perturbation neighborhood. For random/realistic perturbations and realistic selection, the corresponding environments (of size n each) for LINEX are created by drawing k bootstrap samples where k = { 2 , 3 , 4 , 5 } in our experiments. We test for n = { 10 , 20 , 30 , 40 , 50 } with this dataset.

Medical Expenditure Panel Survey (Tabular): The Medical Expenditure Panel Survey (MEPS) dataset is produced by the US Department of Health and Human Services. It is a collection of surveys of families of individuals, medical providers, and employers across the country. We choose Panel 19 of the survey which consists of a cohort that started in 2014 and consisted of data collected over 5

rounds of interviews over 2014 -2015 . The outcome variable was a composite utilization feature that quantified the total number of healthcare visits of a patient. The features used included demographic features, perceived health status, various diagnosis, limitations, and socioeconomic factors. We filter out records that had a utilization (outcome) of 0, and log-transformed the outcome for modeling. These pre-processing steps resulted in a dataset with 11136 examples and 32 categorical features. We train a random forest regressor that has a test R 2 of 0 . 325 in this dataset. We provide local explanations of the predictions. With MEPS, we do not use realistic perturbations since KDE and VAE generators do not work well with categorical data. Otherwise the setting is similar as IRIS data, except that we use n = { 50 , 100 , 200 , 300 , 400 , 500 } . The kernel widths in this case were { 0 . 28 , 0 . 57 , 1 . 41 , 2 . 83 , 4 . 24 } . We use k = { 2 , 3 , 4 , 5 } for this dataset.

Fashion MNIST (Images): This dataset has 28 × 28 grayscale images of fashion articles with 60,000 train and 10,000 test samples. The task is to classify these into 10 classes corresponding to coat, shoe, and so on. A neural network trained with test accuracy of 87%. Explanations are generated for the prediction probabilities corresponding to the predicted class for each example. We choose 1000 test examples to generate explanations. Realistic perturbations were generated using VAEGen [8], a Variational Auto Encoder (VAE) fitted on the training dataset. For random perturbations, we chose n from { 50 , 100 , 200 , 300 , 400 , 500 } and kernel sizes were { 0 . 43 , 0 . 85 , 2 . 14 , 4 . 27 , 6 . 41 } . For realistic perturbations we chose n from { 250 , 500 , 750 , 1000 } and the kernel widths were { 1 . 4 , 2 . 8 , 7 . 0 , 14 . 0 , 21 . 0 } . We use k = { 2 , 3 , 4 , 5 } for this dataset.

CIFAR10 (Images): This dataset has 32 × 32 colored images belonging to 10 different classes. The dataset has 50,000 train and 10,000 test samples. The task is to classify these into 10 classes corresponding to dog, bird, and so on. A residual network with 18 units (ResNet18) was trained with test accuracy of ∼ 95%. Explanations are generated for the prediction probabilities corresponding to the predicted class for each example. We choose 1000 test examples to generate explanations. Realistic perturbations were generated using VAEGen [8], a Variational Auto Encoder (VAE) fitted on the training dataset. For random perturbations, we chose n from { 50 , 100 , 200 , 300 , 400 , 500 } and kernel sizes were { 0 . 43 , 0 . 85 , 2 . 14 , 4 . 27 , 6 . 41 } . For realistic perturbations we chose n from { 250 , 500 , 750 , 1000 } and the kernel widths were { 1 . 4 , 2 . 8 , 7 . 0 , 14 . 0 , 21 . 0 } . We use k = { 2 , 3 , 4 , 5 } for this dataset.

Rotten Tomatoes (Text): This dataset contains 10662 movie reviews from rotten tomatoes website along with their sentiment polarity, i.e., positive or negative reviews and the task is to classify the sentiment of the reviews into positive or negative. The review sentences were vectorized using CountVectorizer and TfidfTransformer and a sklearn Naive Bayes classifier was fitted on training dataset which yielded a test accuracy of 75%. Explanations are generated for the prediction probabilities corresponding to the predicted class for each example. Realistic perturbations were generated using Word2VecGen [8], wherein word2vec embeddings are first trained using the training corpus and new sentences are generated by randomly replacing a sentence word whose distance in the embedding space lies within the radius of the neighbourhood. For both random and realistic perturbations, n was chosen from { 25 , 50 , 75 , 100 } . The kernel sizes were { 0 . 42 , 1 . 06 , 2 . 12 , 3 . 18 } for random perturbations (kernel size 0 . 21 resulted in numerical issues), and { 0 . 21 , 0 . 42 , 1 . 06 , 2 . 12 , 3 . 18 } for realistic perturbations. We use k = { 2 , 3 , 4 , 5 } for this dataset.

E Results with All Datasets and Hyperparameter Combinations for Random and Realistic Perturbations

We present results with all hyperparameter combinations for random and realistic perturbations. Results for LIME with random perturbations (LIME), smoothed LIME (S-LIME), LINEX with random perturbations (LINEX/rand), MeLIME (MeLIME), LINEX with MeLIME-like realistic neighborhoods (LINEX/real), MAPLE (MAPLE), LINEX with MAPLE-like realistic neighborhoods (LINEX/mpl) are presented in figures 5-19. The legend for these figures are given in Figure 4.

For the five datasets, we perform ablations by varying one of perturbation neighborhood size (Figures 5-9), number of environments (Figures 10-14), and kernel width (Figures 15-19). Each point in these figures are averaged over all possible values for the two parameters that are not ablated. For example, each point in Figure 5 is averaged over all possible values for kernel widths and number of

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Neighborhood size environments for a given perturbation neighborhood size. Standard errors of the mean are also plotted in the same color with lesser opacity. Lower values of Infidelity (INFD), Generalized Infidelity (GI), Coefficient Inconsistency (CI) are better whereas for Unidirectionality ( Υ ) and Class Attribution Consistency (CAC) higher values are better.

Figures 5-9 show ablations with respect to perturbation neighborhood sizes. Considering all datasets, the stability/recourse metrics (CI, Υ , CAC) are clearly better for LINEX compared to its counterparts. For LINEX methods (LINEX/rand, LINEX/real, LINEX/mpl), the metrics get better or stays approximately the same generally as perturbation neighborhood size increases keeping with the intuition that larger perturbation neighborhood sizes should produce explanations that are more stable in the exemplar neighborhood. Υ for FMNIST and CIFAR10 are already good for small perturbation neighborhood sizes possibly because of the quality of MeLIME perturbations.

Turning to the fidelity metrics (INFD and GI) in tabular datasets, we see that the results still favor LINEX, but less heavily compared to the stability/recourse metrics. This is in line with what we observe in Table 2. In IRIS and MEPS, LINEX is close to or outperforms the corresponding baselines in the GI measure (except for LINEX/mpl with MEPS). This gap closes a bit with INFD, but we note that GI is a better measure since it estimates how faithful explanations are in a exemplar neighborhood. With the text dataset, LINEX variants are slightly more favored, whereas with the image dataset, the baselines have an edge.

Considering Figures 10-14, we see that variations are less stark with respect to number of environments overall for LINEX variants. Note that except for S-LIME, other baselines do not use multiple environments, and hence stay constant. The slight variations in MAPLE are due to the effect of random seeds. In the stability/recourse metrics, again LINEX variants emerge as the clear winner across datssets. With the faithfulness metrics (GI and INFD), in the text dataset, LINEX variants generally perform better, whereas the baselines have a better performance in the image dataset.

Finally, we study the variation of the performance measures with respect to kernel width in Figures 15-19. We see that the stability/recourse metrics flatten out in all cases with large kernel widths. This behaviour holds true for faithfulness metrics (GI and INFD) as well except in some cases. GI and INFD measures also increase before they flatten out since the fit becomes poorer at larger kernel widths. The stability/recourse metrics become better or remain approximately the same since explanations generally improve or preserve their stability properties as kernel widths increase. Note that very small kernel widths can lead to unexpected behavior that does not fit the trend as seen with the tabular datasets since explanations can become hyper-local. MAPLE and LINEX/mpl stay the same at different kernel widths since they use a different weighting scheme. As with other ablations, we see that LINEX variants are similar or better in stability/recourse metrics overall, while with the faithfulness metrics the results are more mixed.

Note that we do not compute MeLIME perturbations with MEPS since KDE and VAE generators do not work well with categorical data, and do not use compute CAC since the task is regression. Further, the features used in explanations for different test examples are not comparable for random perturbations with FMNIST, CIFAR10 and Rotten Tomatoes, hence we cannot compute CAC for those cases as well. This explains the missing curves/plots.

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300

400

500

Neighborhood size

0.075

0.100

0.125

0.150

0.175

0.200

0.225

Generalized infidelity

(b) MEPS

300

400

500

600

700

800

900

1000

Neighborhood size

0.26

0.28

0.30

0.32

0.34

0.36

Generalized infidelity

(c) FMNIST

30

40

50

60

70

80

90

100

Neighborhood size

0.35

0.36

0.37

0.38

0.39

Generalized infidelity

300

400

500

600

700

800

900

1000

Neighborhood size

0.250

0.275

0.300

0.325

0.350

0.375

0.400

0.425

Generalized infidelity

(e) CIFAR10

10

15

20

25

30

35

40

45

50

Neighborhood size

0.005

0.010

0.015

0.020

0.025

0.030

0.035

0.040

Infidelity

100

200

300

400

500

Neighborhood size

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

Infidelity

(b) MEPS

100

200

300

400

500

Neighborhood size

0.15

0.20

0.25

0.30

0.35

Infidelity

(c) FMNIST (random)

300

400

500

600

700

800

900

1000

Neighborhood size

0.000

0.025

0.050

0.075

0.100

0.125

0.150

0.175

0.200

Infidelity

(d) FMNIST (realistic)

30

40

50

60

70

80

90

100

Neighborhood size

0.05

0.06

0.07

0.08

0.09

Infidelity

(random)

30

40

50

60

70

80

90

100

Neighborhood size

0.0200

0.0225

0.0250

0.0275

0.0300

0.0325

0.0350

0.0375

0.0400

Infidelity

(realistic)

300

400

500

600

700

800

900

1000

Neighborhood size

0.08

0.09

0.10

0.11

0.12

0.13

0.14

Infidelity

(g) CIFAR10

2.0

2.5

3.0

3.5

4.0

4.5

5.0

Number of environments

0.05

0.10

0.15

0.20

0.25

0.30

Coefficient inconsistency

2.0

2.5

3.0

3.5

4.0

4.5

5.0

Number of environments

0.004

0.005

0.006

0.007

0.008

0.009

Coefficient inconsistency

(b) MEPS

2.0

2.5

3.0

3.5

4.0

4.5

5.0

Number of environments

0.002

0.003

0.004

0.005

0.006

0.007

Coefficient inconsistency

(c) FMNIST

2.0

2.5

3.0

3.5

4.0

4.5

5.0

Number of environments

0.00017

0.00018

0.00019

0.00020

0.00021

0.00022

0.00023

0.00024

Coefficient inconsistency

2.0

2.5

3.0

3.5

4.0

4.5

5.0

Number of environments

0.006

0.007

0.008

0.009

0.010

0.011

0.012

0.013

0.014

Coefficient inconsistency

(e) CIFAR10

2.0

2.5

3.0

3.5

4.0

4.5

5.0

Number of environments

0.5

0.6

0.7

0.8

0.9

Class attribution consistency

2.0

2.5

3.0

3.5

4.0

4.5

5.0

Number of environments

0.35

0.40

0.45

0.50

0.55

0.60

0.65

Class attribution consistency

(b) FMNIST

2.0

2.5

3.0

3.5

4.0

4.5

5.0

Number of environments

0.920

0.925

0.930

0.935

0.940

0.945

0.950

0.955

0.960

Class attribution consistency

F Example Feature Attributions in Text Data: MeLIME vs LINEX

Below we see sample attributions by the two methods along with the magnitude of the attributions. Attribution magnitudes are printed with a precision of 10 -3 and shown along with the corresponding words in descending order.

F.1 Positive Sentiment

enticing and often funny documentary . MeLIME: documentary funny and enticing often LINEX : documentary funny often enticing and MeLIME: 0.517 0.446 0.333 0.317 0.311

2.0

2.5

3.0

3.5

4.0

4.5

5.0

Number of environments

0.65

0.70

0.75

0.80

0.85

0.90

2.0

2.5

3.0

3.5

4.0

4.5

5.0

Number of environments

0.945

0.950

0.955

0.960

0.965

0.970

0.975

(b) MEPS

2.0

2.5

3.0

3.5

4.0

4.5

5.0

Number of environments

0.768

0.770

0.772

0.774

0.776

0.778

0.780

0.782

(c) FMNIST

2.0

2.5

3.0

3.5

4.0

4.5

5.0

Number of environments

0.99949

0.99950

0.99951

0.99952

0.99953

0.99954

0.99955

0.99956

0.99957

2.0

2.5

3.0

3.5

4.0

4.5

5.0

Number of environments

0.54

0.56

0.58

0.60

0.62

0.64

0.66

(e) CIFAR10

2.0

2.5

3.0

3.5

4.0

4.5

5.0

Number of environments

0.02

0.04

0.06

0.08

0.10

0.12

Generalized infidelity

2.0

2.5

3.0

3.5

4.0

4.5

5.0

Number of environments

0.08

0.10

0.12

0.14

0.16

0.18

0.20

0.22

Generalized infidelity

(b) MEPS

2.0

2.5

3.0

3.5

4.0

4.5

5.0

Number of environments

0.28

0.29

0.30

0.31

0.32

0.33

0.34

0.35

Generalized infidelity

(c) FMNIST

2.0

2.5

3.0

3.5

4.0

4.5

5.0

Number of environments

0.35

0.36

0.37

0.38

Generalized infidelity

2.0

2.5

3.0

3.5

4.0

4.5

5.0

Number of environments

0.26

0.28

0.30

0.32

0.34

0.36

0.38

0.40

0.42

Generalized infidelity

(e) CIFAR10

2.0

2.5

3.0

3.5

4.0

4.5

5.0

Number of environments

0.010

0.015

0.020

0.025

0.030

Infidelity

2.0

2.5

3.0

3.5

4.0

4.5

5.0

Number of environments

0.06

0.08

0.10

0.12

0.14

0.16

Infidelity

(b) MEPS

2.0

2.5

3.0

3.5

4.0

4.5

5.0

Number of environments

0.16

0.18

0.20

0.22

0.24

0.26

0.28

0.30

0.32

Infidelity

(c) FMNIST (random)

2.0

2.5

3.0

3.5

4.0

4.5

5.0

Number of environments

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

Infidelity

(d) FMNIST (realistic)

2.0

2.5

3.0

3.5

4.0

4.5

5.0

Number of environments

0.050

0.055

0.060

0.065

0.070

0.075

0.080

0.085

Infidelity

(random)

2.0

2.5

3.0

3.5

4.0

4.5

5.0

Number of environments

0.0200

0.0225

0.0250

0.0275

0.0300

0.0325

0.0350

0.0375

Infidelity

(realistic)

2.0

2.5

3.0

3.5

4.0

4.5

5.0

Number of environments

0.085

0.090

0.095

0.100

0.105

0.110

0.115

0.120

Infidelity

(g) CIFAR10

LINEX : 0.416 0.377 0.342 0.331 0.330

one-of-a-kind near-masterpiece .

MeLIME:

kind near masterpiece

LINEX :

masterpiece kind one

MeLIME:

0.832 0.695 0.182

LINEX :

0.712 0.384 0.381

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Kernel width

0.5

0.6

0.7

0.8

0.9

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Kernel width

0.93

0.94

0.95

0.96

0.97

0.98

0.99

1.00

(b) MEPS

2.5

5.0

7.5

10.0 12.5 15.0 17.5 20.0

Kernel width

0.768

0.770

0.772

0.774

0.776

0.778

0.780

(c) FMNIST

0.5

1.0

1.5

2.0

2.5

3.0

Kernel width

0.99949

0.99950

0.99951

0.99952

0.99953

0.99954

0.99955

2.5

5.0

7.5

10.0

12.5

15.0

17.5

20.0

Kernel width

0.54

0.56

0.58

0.60

0.62

0.64

0.66

0.68

(e) CIFAR10

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Kernel width

0.0

0.1

0.2

0.3

0.4

0.5

0.6

Coefficient inconsistency

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Kernel width

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.010

Coefficient inconsistency

(b) MEPS

2.5

5.0

7.5

10.0 12.5 15.0 17.5 20.0

Kernel width

0.002

0.003

0.004

0.005

0.006

0.007

Coefficient inconsistency

(c) FMNIST

0.5

1.0

1.5

2.0

2.5

3.0

Kernel width

0.00017

0.00018

0.00019

0.00020

0.00021

0.00022

0.00023

0.00024

Coefficient inconsistency

2.5

5.0

7.5

10.0

12.5

15.0

17.5

20.0

Kernel width

0.006

0.007

0.008

0.009

0.010

0.011

0.012

0.013

0.014

Coefficient inconsistency

(e) CIFAR10

Figure 16: Coefficient inconsistency (CI) vs. Kernel width.

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Kernel width

0.2

0.4

0.6

0.8

1.0

Class attribution consistency

2.5

5.0

7.5

10.0 12.5 15.0 17.5 20.0

Kernel width

0.35

0.40

0.45

0.50

0.55

0.60

Class attribution consistency

(b) FMNIST

0.5

1.0

1.5

2.0

2.5

3.0

Kernel width

0.920

0.925

0.930

0.935

0.940

0.945

0.950

0.955

Class attribution consistency

Figure 17: Class attribution consistency (CAC) vs. Kernel width.

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Kernel width

0.05

0.10

0.15

0.20

0.25

Generalized infidelity

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Kernel width

0.10

0.15

0.20

0.25

0.30

Generalized infidelity

(b) MEPS

2.5

5.0

7.5

10.0 12.5 15.0 17.5 20.0

Kernel width

0.28

0.29

0.30

0.31

0.32

Generalized infidelity

(c) FMNIST

0.5

1.0

1.5

2.0

2.5

3.0

Kernel width

0.35

0.36

0.37

0.38

0.39

Generalized infidelity

2.5

5.0

7.5

10.0

12.5

15.0

17.5

20.0

Kernel width

0.26

0.28

0.30

0.32

0.34

0.36

0.38

0.40

0.42

Generalized infidelity

(e) CIFAR10

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Kernel width

0.00

0.02

0.04

0.06

0.08

Infidelity

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Kernel width

0.00

0.05

0.10

0.15

0.20

0.25

0.30

Infidelity

(b) MEPS

1

2

3

4

5

6

Kernel width

0.16

0.18

0.20

0.22

0.24

0.26

Infidelity

(c) FMNIST (random)

2.5

5.0

7.5

10.0 12.5 15.0 17.5 20.0

Kernel width

0.00

0.02

0.04

0.06

0.08

0.10

0.12

Infidelity

(d) FMNIST (realistic)

0.5

1.0

1.5

2.0

2.5

3.0

Kernel width

0.000

0.025

0.050

0.075

0.100

0.125

0.150

0.175

Infidelity

(random)

0.5

1.0

1.5

2.0

2.5

3.0

Kernel width

0.022

0.023

0.024

0.025

0.026

0.027

0.028

0.029

Infidelity

(realistic)

2.5

5.0

7.5

10.0

12.5

15.0

17.5

20.0

Kernel width

0.085

0.090

0.095

0.100

0.105

0.110

0.115

0.120

0.125

Infidelity

(g) CIFAR10

LINEX : 0.403 0.335 0.245 0.239 this charming but slight tale has warmth , wit and interesting characters compassionately portrayed . MeLIME: charming compassionately and interesting portrayed LINEX : charming compassionately has tale portrayed MeLIME: 0.690 0.507 0.456 0.444 0.424 LINEX : 0.464 0.435 0.431 0.430 0.429 thoughtful , provocative and entertaining . MeLIME: thoughtful entertaining and provocative LINEX : thoughtful entertaining and provocative MeLIME: 0.612 0.517 0.402 0.395 LINEX : 0.505 0.461 0.415 0.404 the film is quiet , threatening and unforgettable . MeLIME: quiet unforgettable and film the LINEX : unforgettable quiet film and is MeLIME: 0.597 0.483 0.412 0.325 0.303 LINEX : 0.421 0.416 0.388 0.378 0.338 a moving tale of love and destruction in unexpected places , unexamined lives . MeLIME: unexpected moving love tale lives LINEX : moving unexpected places lives in MeLIME: 0.692 0.662 0.577 0.538 0.499 LINEX : 0.538 0.530 0.521 0.513 0.501 though frodo's quest remains unfulfilled , a hardy group of determined new zealanders has proved its creative mettle . MeLIME: creative group proved has new LINEX : creative quest its proved determined MeLIME: 0.602 0.441 0.424 0.402 0.393 LINEX : 0.410 0.392 0.390 0.385 0.381 F.2 Negative Sentiment originality is sorely lacking . MeLIME: lacking sorely is originality LINEX : lacking sorely originality is MeLIME: 0.543 0.381 0.296 0.278 LINEX : 0.430 0.356 0.314 0.271 an ugly , pointless , stupid movie . MeLIME: stupid pointless ugly movie an LINEX : stupid pointless ugly movie an MeLIME: 0.543 0.499 0.385 0.365 0.276 LINEX : 0.446 0.411 0.373 0.360 0.350 so devoid of pleasure or sensuality that it cannot even be dubbed hedonistic . MeLIME: devoid even be dubbed of LINEX : devoid so dubbed be cannot MeLIME: 0.666 0.416 0.413 0.372 0.344 LINEX : 0.400 0.392 0.387 0.380 0.368 neither revelatory nor truly edgy--merely crassly flamboyant and comedically labored . MeLIME: edgy neither nor labored revelatory LINEX : edgy neither nor labored truly MeLIME: 1.256 0.338 0.277 0.204 0.021

LINEX : 0.439 0.398 0.398 0.369 0.349

occasionally funny , sometimes inspiring , often boring MeLIME: boring occasionally inspiring sometimes often LINEX : boring occasionally sometimes often inspiring MeLIME: 0.669 0.242 0.218 0.210 0.182

LINEX :

0.377 0.266 0.266 0.250 0.236

.

a cumbersome and cliche-ridden movie greased with every emotional device known to man . MeLIME: cliche every device movie with LINEX : cliche every man cumbersome emotional MeLIME: 0.695 0.449 0.327 0.280 0.268 LINEX : 0.385 0.361 0.354 0.349 0.309

ponderous , plodding soap opera disguised as a feature film MeLIME: plodding soap ponderous opera disguised LINEX : plodding soap film ponderous feature MeLIME: 0.579 0.522 0.421 0.408 0.382 LINEX : 0.442 0.440 0.418 0.406 0.377

kitschy , flashy , overlong soap opera . MeLIME: soap flashy opera overlong kitschy LINEX : soap flashy opera overlong kitschy MeLIME: 0.499 0.397 0.391 0.358 0.230

LINEX :

0.389 0.362 0.360 0.346 0.300

[a] poorly executed comedy . MeLIME: poorly comedy executed LINEX : poorly comedy executed MeLIME: 0.653 0.348 0.257 LINEX : 0.502 0.335 0.309

a bad movie that happened to good actors . MeLIME: bad happened movie to that LINEX : bad happened to movie actors MeLIME: 0.692 0.396 0.371 0.367 0.242 LINEX : 0.442 0.384 0.367 0.361 0.344

a complete waste of time . MeLIME: waste complete time of LINEX : waste complete time of MeLIME: 0.614 0.425 0.313 0.247

LINEX : 0.480 0.381 0.348 0.278

don't waste your money .

MeLIME: waste money don your LINEX : waste money don your MeLIME: 0.592 0.497 0.408 0.309 LINEX : 0.483 0.450 0.411 0.337

witless and utterly pointless . MeLIME: pointless witless and utterly LINEX : pointless witless utterly and MeLIME: 0.652 0.491 0.263 0.245

LINEX : 0.506 0.444 0.311 0.269

.

0

10

20

0

5

10

15

20

25

(1a) MeLIME:

r

=0.279

0

10

20

0

5

10

15

20

25

(1b) LINEX:

r

=0.603

0

10

20

0

5

10

15

20

25

(1c) Original

0

10

20

0

5

10

15

20

25

(2a) MeLIME:

r

=0.434

0

10

20

0

5

10

15

20

25

(2b) LINEX:

r

=0.689

0

10

20

0

5

10

15

20

25

(2c) Original

0

10

20

0

5

10

15

20

25

(3a) MeLIME:

r

=0.259

0

10

20

0

5

10

15

20

25

(3b) LINEX:

r

=0.711

0

10

20

0

5

10

15

20

25

(3c) Original

0

10

20

0

5

10

15

20

25

(4a) MeLIME:

r

=0.292

0

10

20

0

5

10

15

20

25

(4b) LINEX:

r

=0.633

0

10

20

0

5

10

15

20

25

(4c) Original

0

10

20

0

5

10

15

20

25

(5a) MeLIME:

r

=0.271

0

10

20

0

5

10

15

20

25

(5b) LINEX:

r

=0.701

0

10

20

0

5

10

15

20

25

(5c) Original

0

10

20

0

5

10

15

20

25

(6a) MeLIME:

r

=0.297

0

10

20

0

5

10

15

20

25

(6b) LINEX:

r

=0.962

0

10

20

0

5

10

15

20

25

(6c) Original

0

10

20

0

5

10

15

20

25

(7a) MeLIME:

r

=0.482

0

10

20

0

5

10

15

20

25

(7b) LINEX:

r

=0.820

0

10

20

0

5

10

15

20

25

(7c) Original

0

10

20

0

5

10

15

20

25

(8a) MeLIME:

r

=0.177

0

10

20

0

5

10

15

20

25

(8b) LINEX:

r

=0.788

0

10

20

0

5

10

15

20

25

(8c) Original

0

10

20

0

5

10

15

20

25

(9a) MeLIME:

r

=0.247

0

10

20

0

5

10

15

20

25

(9b) LINEX:

r

=0.729

0

10

20

0

5

10

15

20

25

(9c) Original

0

10

20

0

5

10

15

20

25

(10a) MeLIME:

r

=0.381

0

10

20

0

5

10

15

20

25

(10b) LINEX:

r

=0.792

0

10

20

0

5

10

15

20

25

(10c) Original

G Example Feature Attributions in Image Data: MeLIME vs LINEX

We show feature attributions for individual example images with MeLIME and LINEX with MeLIME perturbations in Figure 20. In Figure 21 we show class-wise mean feature attributions along with mean images. In Figure 22, we see examples from CIFAR10. LINEX explanations seem to provide more meaningful feature attributions.

H Results for All Methods Including SHAP

In Table 4, we provide the results for SHAP along with all methods for easy comparison. Note that SHAP does not have standard errors since it is computed only once per test point. The INFD values for SHAP are miniscule since SHAP values add up to the predictions by definition. In order to compute GI, CI, Υ , CAC, we convert the SHAP values to SHAP attributions [5] first and follow the same approach used by other explanation methods.

0

10

20

0

5

10

15

20

25

(1a) MeLIME:

r

=0.269

0

10

20

0

5

10

15

20

25

(1b) LINEX:

r

=0.549

0

10

20

0

5

10

15

20

25

(1c) Original

0

10

20

0

5

10

15

20

25

(2a) MeLIME:

r

=0.486

0

10

20

0

5

10

15

20

25

(2b) LINEX:

r

=0.809

0

10

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(3a) MeLIME:

r

=0.311

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r

=0.619

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(3c) Original

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(4a) MeLIME:

r

=0.418

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(4b) LINEX:

r

=0.860

0

10

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5

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(5a) MeLIME:

r

=0.323

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(5b) LINEX:

r

=0.696

0

10

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5

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(5c) Original

0

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(6a) MeLIME:

r

=0.419

0

10

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0

5

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20

25

(6b) LINEX:

r

=0.800

0

10

20

0

5

10

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20

25

(6c) Original

0

10

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(7a) MeLIME:

r

=0.379

0

10

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5

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25

(7b) LINEX:

r

=0.783

0

10

20

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5

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(7c) Original

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(8a) MeLIME:

r

=0.432

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(8b) LINEX:

r

=0.840

0

10

20

0

5

10

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(8c) Original

0

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(9a) MeLIME:

r

=0.373

0

10

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5

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20

25

(9b) LINEX:

r

=0.757

0

10

20

0

5

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(9c) Original

0

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(10a) MeLIME:

r

=0.441

0

10

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0

5

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r

=0.852

0

10

20

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5

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(10c) Original

I Error Analysis of LINEX

We perform error analysis for LINEX to gain better understanding about the method. We choose FMNIST dataset for doing this since, LINEX/real under performs MeLIME in terms of the INFD measure here (see Table 2) more heavily compared to other datasets and so we wanted to investigate the reasons for this. This also happens to be one of the higher dimensional datasets that is intuitive to visualize and understand.

We start by observing that even though LINEX/real underperforms in the INFD metric, the gap is not so great in the GI metric, which suggests that MeLIME may be overfitting explanations here. We also note that in terms of CI, Υ , and CAC metrics, LINEX/real clearly outperforms MeLIME.

We now choose a sample of images from the dataset where LINEX/real has highest instancelevel infidelity numbers and display them in Figure 23. Just looking at the explanations and the corresponding original images visually, it is evident that LINEX/real highlights the prominent features like sleeves and collar in a shirt, handles of the bags, outlines of the boots/shoes, even though the infidelity values are high. However, MeLIME misses out on some of these prominent features and

Original

LINEX

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1.0

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0.6

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focuses only on optimizing the local fit. The fact that LINEX zeroes in on important features also provides additional evidence for the closeness of GI metrics between the two methods, and the better performance of LINEX/real with CI, Υ , and CAC metrics.

This conclusion is also verified when we look at the performance of LINEX at a class level. In Figure 24, we see two classes one where the infidelity of LINEX is low (i.e. Trousers class) and the other where its infidelity is high (i.e Shirt class). As can be seen since the Trousers class has examples with less superfluous features (viz. varied designs) focusing on which might reduce infidelity but are not critical for determination of the class, LINEX does better in terms of infidelity on the prior. However, although infidelity is higher for the latter Shirt class it does much better on other metrics such as GI, CAC, CI and Υ indicating that LINEX truly focuses on robust features.

J Ablation Analysis of Important Features for Various Explanation Methods

We wanted to analyze the most challenging case for us in the reported experiments which is on the FMNIST dataset where we are more worse than MeLIME in terms of INFD than any of the other setups. We thus assess if the features deemed important - those with the largest coefficients - by the explanation methods are indeed important for the black box model to make their predictions. To assess this, we set the we set a fraction of features (pixel values) corresponding to the top coefficients of MeLIME and LINEX/realistic to a baseline value and run the modified images again through the black box model - this is what we mean by ablation here. The baseline value here was chosen to be -1 since that is the value of the background pixels. We then used two measures to assess the quality of explanations - higher values being better for both. The first measure is mean absolute error between the predicted scores before and after ablation, corresponding to the original predicted class. The second measure is the fraction of images that changed their predicted class after ablation. We

see from Figure 25 that LINEX/realistic substantially outperforms MeLIME in both these measures, clearly demonstrating the relevance of features chosen by our method to the black box.

K Error Analysis of LINEX based on Ablation

Highlighting stable features for examples near non-linearities is a key strength of LINEX. However, in some cases for examples near class boundaries it may ignore sensitive features as we show in this demonstration.

In Figure 26, we show 6 examples that are appear to be close to class boundaries. We ablate pixels corresponding to top 15% of important features chosen by MeLIME and LINEX/realistic using the approach discussed in Section J. Ablation based on MeLIME importances meaningfully changes classes, whereas ablation by LINEX importances does not. The changes in prediction for MeLIME ablation for the six images are respectively from Dress to Trouser , Sneaker to Sandal , Pullover to Dress , Sneaker to Sandal , Bag to Pullover , and Sneaker to Sandal . The new class assignment looks

reasonable looking at the ablated images. We also see that the changes in class probabilities for the original class ( p ) are much higher after MeLIME ablation compared to LINEX/realistic ablation.

MeLIME ablated images for the first example has structures that look like trouser legs, for the second, fourth and sixth examples the area around the heel is more open making the original sneaker look like a sandal, for the third example, there is a hole in the hooded part of the pullover making it resemble a dress. The fifth example is classified as a pullover possibly because of the elongated structures on the sides that look like hands.

Note that such cases of LINEX under performing are rare though as is confirmed by its superior performance in Figure 25.

L Understanding Behavior of LIME and LINEX with Synthetic Data

We consider explaining the behavior of a function of two variables x and y with Class 1 sandwiched between Class 0 (see Figure 27). The third (or vertical) axis denotes the probability of being in Class 1. Clearly, x is the only important feature here that determines the class label.

From Figure 27 (left), we see that the LIME (here MeLIME would be the same as LIME since the space is flat and all points are realistic) feature attributions at points a , b , and c will provide importance to x feature for small as well as large kernel width (1 and 2 respectively) neighborhoods. For point c , in the interior of the Class 0, the attributions are stable across kernel widths. However for points a and b close to the boundary of classes, the attributions for small kernel width and large kernel width neighborhoods differ significantly along the x direction. This shows the instability of LIME explanations near boundaries of classes for different kernel widths.

In contrast in Figure 27 (right), we see that the LINEX explanation constructed for the two kernel widths provides stable feature attributions for all points a , b , c . For a and b , LINEX will conservatively pick a smaller feature attribution along the x direction since the function changes rapidly in its neighborhood. As such though LINEX will still pick the feature in the x direction in this scenario.

M Variation of feature attributions with γ

Based on the proof of Theorem 1, if for a feature the optimal attributions have opposite sign for each of the two environments, then γ can be made arbitrarily small (except 0) or large and the output of Algorithm 1 should still be the same which is 0 as the Nash Equilibrium is ± γ . If the optimal attributions are the same sign then we should still get the same output from Algorithm 1 as long as γ ≥ min( | w 1 i | , | w 2 i | ) since the attribution from our algorithm is the minimum of those values. When γ < min( | w 1 i | , | w 2 i | ) then the feature attributions will smoothly reduce as γ reduces.

We demonstrate this behavior in Figure 28 using an example from the IRIS dataset with random perturbations using the same setting as in Section 5. In the experiments in Section 5, we set γ = 0 . 329 which is the maximum absolute value based on a linear fit to each environment. As γ increases beyond 0 . 329 , the attributions are unchanged demonstrating robustness. Same holds true while reducing γ up to 0 . 165 beyond which we see smooth reduction in the attribution values. Qualitatively, similar behavior is seen for other examples too. Because we set γ pessimistically (ignoring constraints) to a high value, we can expect our reported performances in the paper to be robust across many values of γ .

N Convergence of LINEX procedure and comparisons

We demonstrate based on a synthetic example how Algorithm 1 and provides a unidirectional explanation. We generate synthetic data using a function in R 2 (Figure 29(left)). The function gently rises with increasing y values, and along x it is flat first, then rises abruptly and then falls gradually. We want to obtain robust attributions of this function at the point x = 1 . 0 , y = 0 . 0 , which is close to the end of the rising edge along x direction.

As we can imagine, since the slope changes abruptly along x direction near the point, it should be ideally excluded from an explanation intended towards recourse based on a linear proxy. Otherwise, the explanation will not generalize in the neighborhood of this point. On the other hand, the y direction should be included since the function changes smoothly along y throughout.

To generate explanations We first create two environments centered at the example to explain with variances 0 . 5 and 2 . 0 . Now independently fitting to these environments leads to feature attributions that are {-0 . 033 , 0 . 098 } and { 0 . 084 , 0 . 102 } . Appending the two environments the attributions are { 0 . 029 , 0 . 095 } , whereas with LINEX, the attributions would be { 0 . 0 , 0 . 093 } . Thus, LINEX effectively eliminates the feature with high variability or abrupt changes. The behavior of the coefficients for each environment as LINEX converges is shown in Figure 29(right). As such, one can also see the convergence is fast.

O Limitations

Like any other posthoc explainable AI method there is no way to surely say that LINEX exactly reflects the true reasoning behind a black box classifier in arbitrary applications. It also is somewhat slower than LIME as shown in section A given the game theoretic nature of the algorithm, where its stability and unidirectionality hopefully offsets the additional time required. On the flip side, given its favorable properties in terms of recovering explanations it could be used to violate privacy which may be concerning from a social standpoint.

MeLime: r=0.121, INFD=0.001 LINEX/real: r=0.334, INFD=0.169

Original

MeLime: r=0.093, INFD=0.001 LINEX/real: r=0.338, INFD=0.216

Original

MeLime: r=0.112, INFD=0.001 LINEX/real: r=0.539, INFD=0.201

Original

MeLime: r=0.114, INFD=0.001 LINEX/real: r=0.412, INFD=0.189

Original

MeLime: r=0.393, INFD=0.001 LINEX/real: r=0.713, INFD=0.180

Original

MeLime: r=0.162, INFD=0.001 LINEX/real: r=0.528, INFD=0.161

Original

Trouser class - less infidelity (INFD)

Shirt class - more infidelity (INFD)

error

probs)

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Average predicted

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changes

(originaland

0.4

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ablatedpred.

0.3

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class

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LINEX/real Coeffs.

Original - p=0.9987, cls. 3

MeLIME ablate - p=0.1472, cls. 1

LINEX/real ablate - p=0.7400, cls. 3

MeLime Coeffs.

LINEX/real Coeffs.

Original - p=0.9509, cls. 7

MeLIME ablate - p=0.3041, cls. 5

LINEX/real ablate - p=0.5137, cls. 7

MeLime Coeffs.

LINEX/real Coeffs.

Original - p=0.9105, cls. 2

MeLIME ablate - p=0.3437, cls. 3

LINEX/real ablate - p=0.8116, cls. 2

MeLime Coeffs.

LINEX/real Coeffs.

Original - p=0.9905, cls. 7

MeLIME ablate - p=0.4782, cls. 5

LINEX/real ablate - p=0.9862, cls. 7

MeLime Coeffs.

LINEX/real Coeffs.

Original - p=0.9228, cls. 8

MeLIME ablate - p=0.1243, cls. 2

LINEX/real ablate - p=0.2858, cls. 8

MeLime Coeffs.

LINEX/real Coeffs.

Original - p=0.5980, cls. 7

MeLIME ablate - p=0.2787, cls. 5

LINEX/real ablate - p=0.5855, cls. 7

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smoothing

(d) GI - S-LIME - mean smoothing

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smoothing

(f) GI - S-LIME - MoM smoothing

(g) CI - S-LIME - mean smoothing

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0.035

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0.55

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Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis work extends [LIME](https://arxiv.org/abs/1602.04938) to generate stable and unidirectional feature attributions through the [invariant risk minimization framework](https://arxiv.org/abs/1907.02893). Their method, LINEX, ensures that it follows the desiderata of faithfulness, robustness to neighborhood sampling, stability (similar to neighbors), and the proposed unidirectionality (closeby examples have the same sign for its attributions). They design LINEX as a concave game (pure Nash equilibrium always exists), thereby ensuring stability and unidirectionality. In the experiments, they show that they can find feature attributions that stay consistent to its neighbors across various modalities (tabular, image, text) and mostly outperform baselines.\n\nSTRENGTHS:\n- The (concave) game theoretic design of the feature attribution method is novel and sound.\n- The unidirectionality desideratum is original and very sensible.\n- The theoretical results provide sound support for their design of the proposed feature attribution method.\n- The proposed method empirically improves upon previous query-based methods.\n- Stable & robust feature attributions, such as generated by their proposed method, are important for researchers as well as practitioners to gain better insights into black-box models.\n- The paper is clearly written and easy to follow.\n- Code is provided.\n\nWEAKNESSES:\n- The largest weakness of this work is missing any (novel) qualitative insights into how model work (e.g., similar to Sec. 6.3 and/or 6.4 of [LIME](https://arxiv.org/abs/1602.04938)). The present submission only contains comparisons to previous works and does not show any interesting application of their approach to understand models.\n- While unidirectionality may be a valid property for most cases, there will be edge cases for which minor differences for close by examples may be discriminative (e.g., a person that has paid back previous loans and another has not, while all other features remain the same).\n- The trustworthiness of the proposed feature attribution method is not shown. It would be helpful to conduct a user study similar to Sec. 5.4 of [LIME](https://arxiv.org/abs/1602.04938).\n\nQUESTIONS:\n- How is t is set during the experiments? Does it violate Assumption 2?\n- Are the results for MeLIME correct for CIFAR results since their attributions often just seem to be flipped from LINEX?\n\nLIMITATIONS:\nThe limitations are adequately addressed.", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-06-21"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper proposes Locally Invariant Explanations (LINEX) as a variation of the well-known LIME explanation method. While LIME is widely used in interpretability, it has been demonstrated to be dependent on the sampling procedure to create neighbouring points as well as the choice of the explanation model. The manuscript aims to mitigate the dependence on the sampling procedure by combining LIME with invariant risk minimisation. \n\nThe method is evaluated on various datasets, such as IRIS, Fashion MNIST, CIFAR10, Rotten Tomatoes, as well as Medical Expenditure Survey and compared to number of previously proposed neighbourhood selection procedures. Both qualitative and quantitative comparisons using a number of metrics are discussed and it is demonstrated that the proposed method leads to a statistically albeit mild improvement.\n\nSTRENGTHS:\n- The proposed neighbourhood selection mechanism seems reasonable and theoretically grounded. \n- The quantitative experimental analysis is through. I particularly like that t-tests are used to evaluate statistical significance. \n- Experiments cover various datasets and modalities.\n\nWEAKNESSES:\n- The proposed method modifies the neighbourhood sampling procedure of LIME and is thus rather limited in scope. \n\n- The paper is a very hard read. This is not because the covered material is particularly unaccessible but rather a result of a suboptimally structured and at times rather confusing presentation. I will list a few concrete suggestions of how to improve the structure. I want to encourage the authors to thoroughly revise the presentation. This would make the ms a much more accessible and valuable contribution.\n\nConcrete suggestions for improving structure:\n- Explain IRM in more accessible manner (either in the main text or appendix). For example, an environment is currently not properly defined. Furthermore avoid imprecise mathematical definitions such as \"the expectation \\mathbb{E}_e is defined w.r.t. the distribution over points in the environment\" where it is not obvious what is meant by \"distribution over points in the environment\".\n- I find the brief introduction to the Nash equilibrium very hard to parse (L 134-145). For example, the utility is defined over joint set actions but in L141 it takes a player k as an input. \n- Metrics for evaluation, as outlined in Section 4.1, should be moved to the experimental section. Separate paragraphs with corresponding equations and a brief explanation for each of the metrics used in Table 2 should be added.\n- Assumptions 1 and 2 are hard to parse. Assumption 1 is rather convoluted and should be split up into at least two sentences. Assumption 2 should clearly state that t and \\gamma are thresholds and refer to their definitions. \n- Similarly, refer to the definitions of the environment generation distribution in Definition 2 and explain what is meant by |.| in the indicator functions of equation 2.\n- The proof sketch L264-277 is very hard to follow and adds little to the overall presentation. Consider to not include it. \n\nMinor suggestions for improving writing:\n- L28 reads as if the doctor is given a cancer diagnosis. I think what is meant is that a patient receives a cancer diagnosis and the doctor has to validate it?\n- L57 - L63 is rather convoluted. Splitting this up in separate sentences would help.\n- There are some ill-placed whitespaces such as in L302.\n- Be consistent with abbreviations, e.g. Supplement vs Suppl. \n- Avoid a.k.a. in favour of i.e.\n\nQUESTIONS:\n- In L 302, it is stated that SHAP is not a \"natural fit\". Why is this so?\n- What was your logic for selecting the particular kernel sizes in L317?\n\nLIMITATIONS:\nLimitations are addressed briefly but adequately in the conclusions.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-04"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nExplanation methods are known to be unstable with flickering attributions on inputs with mild perturbations.\nThis work addresses the explanation instability problem by re-posing it as computing attributions that are invariant across different environments (that are obtained through input perturbations).\nTheir proposed scheme exploits Nash Equilibrium guarantees for convex objectives (of LIME) to obtain an explanation scheme with some desirable properties, which they also demonstrate empirically through multi-modal evaluation.\n\nAlthough I found the idea interesting, I did not like the story and have some critical concerns regarding their evaluation. I elaborate them below.\n\nSTRENGTHS:\n- Datasets used for experimental evaluation span various modalities: Image/Text/Tabular.\n- Connection with NE (Nash Equilibrium) is refreshing and the algorithm is simple.\n- Theoretical insight: their theorem is intuitive and I like that their estimator regresses all disagreeing attributions to 0, which is clean.\n\nWEAKNESSES:\n1. Connection to IRM. I do not see any connection to IRM (Inv. Risk Minimization) setting or method except for the fact that they both use \"environments\" of some kind. Neither does their estimator borrow techniques from IRMv1 algorithm nor does it impose \"invariance\" across environments. In that regard, I found their story muddled.\n2. Game theoretic perspective. The paper also did not justify well why we should view the estimation problem from game-theoretic standpoint. More specifically, why does the utility function as defined in L230 lead to an explanation method with desired properties. Why should the environments be viewed as agents in a game with competing objectives (and limited resources) at all? \n3. I have many evaluation concerns that are described in \"questions\".\n4. The paper is not easy to read because the story is cluttered (first two points), contributions and motivation not straight-forward.\n\nQUESTIONS:\n1. Baselines and ablation. The problem resembles robust mean estimation. I would like to comparisons with the following baselines:\n   1. S-LIME but with attributions clipped at $[-\\gamma, \\gamma]$.\n   2. S-LIME but with attributions aggregated using median instead of mean.\n   3. (1) + (2)\n   4. Median-of-Mean estimation instead of mean estimation with S-LIME without increasing the number of function evaluations.\n   5. (1) + (4) \n2. Metrics.\n   1. Has CAC been used before? I cannot understand well why/how the mean vector matching indicates recourse utility.\n   2. What happens if we slightly change the defn. of GI to aggregate $|y_b(x') - y_e^{x'}(x')|$? Although similar to INFD, we would then be able to check if the estimator is simply smoothing the explanations while the underlying model is not that smooth.\n3. Theoretical analysis and explanation. LINEX performed well on CAC and CI metrics, but I do not understand what is contributing to the improvement. We need ablation study to understand the rrelative contribution of different aspects of LINEX: (a) $\\gamma$ clipping, (b) setting non-zero attribution ($w_i$) only when $\\sum_e \\frac{|w_{ji}|}{w_{ji}}>0$ etc., (c) setting attribution value to the smallest among environments. In other words, a crisp intuitive explanation for why NE leads to empirical gains helps.\n4. Figure 3 is hard to follow. LINEX too is highlighting features in the background.\n\nLIMITATIONS:\nI do not see them mentioned anywhere.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 2, "DATE": "2023-07-05"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nIn this paper, the authors put forward a local, attribution-based explanation method that fits a local, explainable surrogate (a linear model in this case) to perturbation output of the original model.  In essence, this is exact scenario as the immensely popular LIME explanation method; however, this work puts forward LINEX which seeks to improve on the robustness of LIME (where robustness is measured as Fidelity, Stability, Invariance, and a new property put forward as Unidirectionality). LINEX proceeds by learning a collection of linear classifiers, each fitting to a different perturbation “environment” and the authors provide an interpretation of their least-squares fitting approach as a multiplayer game. In addition the authors provide some minor theoretical analysis (Theorem 1) which justifies the modeling choices allowing them to give an intuition for why their method works well.\n\nSTRENGTHS:\nThe paper provides clear motivation for the need for its method, provides a straight forward development of the method, and provides salient theoretical and experimental arguments for the method. \n\nThe experimental evidence in favor of the method is strong, and the method is simple enough that I expect it could be widely adopted into post-hoc explanation tools without too much experimental effort.\n\nThe paragraph “Implications of Theorem 1” is nice as it allows the reader to very easily digest an interpretation of why Algorithm 1 achieves the goal of the method.\n\nWEAKNESSES:\nThe method seems to rely on the diversity of the “environments” in a way that is quite unclear to me. Simply picking gaussians with different covariances seems like it will not give rise to the desired benefits of the method (i.e., it is unclear that simply selecting an ensemble of random environments should be enough to improve over using a single superset random environment). I understand that in general we the method picks the lowest magnitude attribution, which can in turn provide greater stability. We observe, indeed, in table 1 that the method provides little benefit in the cases of FNIST, and CIFAR 10, where perhaps these random environments are not enough. However, in IRIS and MEPS, the random environments have significantly better performance than S-LIME, and I am having trouble fully understanding why this is the case. (See questions section). \n\nWhile I understand the authors general point about the desirability of unidirectionality, it seems like it is already generally satisfied in LIME, S-LIME, and MeLIME. The only place LINEX has a considerable advantage in unidirectionality is on IRIS (and to some extent CIFAR). Perhaps to underscore this point, the authors could run experiments on something like Adult or any other financial dataset where their motivating example for using recourse is relevant and see if there is a considerable difference. To be clear, I am not suggesting the authors carry out this experiment during the rebuttal phase, but I am simply stating that having a more clear experimental example demonstrating  the importance of unidirectionality would be good in future versions of the paper.\n\nThere are several works in robustness of explanations (albeit gradient-based and not LIME-based) that are very related to this work but are not included as prior works:\n[1] Dombrowski et. al. - https://arxiv.org/abs/2012.10425\n[2] Wicker et. al. - https://arxiv.org/abs/2212.08507\nThese are just two of the more recent works that cover robustness of grad. Explanations. I think the more seminal work might be a single, sufficient citation to provide readers with more context: \n[3] Dombrowski et. al. - https://proceedings.neurips.cc/paper/2019/file/bb836c01cdc9120a9c984c525e4b1a4a-Paper.pdf\n\nQUESTIONS:\nCan the authors further elaborate precisely what they mean by “perform bootstrap sampling to create k different environments.” I would like to understand exactly how is this done step-by-step such that you get such an incredibly meaningful difference between the output of your method and that of S-LIME.\n\nWhy is there a trailing 1 in Equation (2)?\n\nLIMITATIONS:\nSee weaknesses.", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-07"}