Model Reconstruction Using Counterfactual Explanations: A Perspective From Polytope Theory

Authors: pasan  dissanayake
sanghamitra  dutta
Venue: NeurIPS_cc_2024_Conference
Published: 2024-05-15 21:22:35

Abstract: Counterfactual explanations provide ways of achieving a favorable model outcome with minimum input perturbation. However, counterfactual explanations can also be leveraged to reconstruct the model by strategically training a surrogate model to give similar predictions as the original (target) model. In this work, we analyze how model reconstruction using counterfactuals can be improved by
further leveraging the fact that the counterfactuals also lie quite close to the decision boundary. Our main contribution is to derive novel theoretical relationships between the error in model reconstruction and the number of counterfactual queries required using polytope theory. Our theoretical analysis leads us to propose a strategy for model reconstruction that we call Counterfactual Clamping Attack (CCA) which trains a surrogate model using a unique loss function that treats counterfactuals differently than ordinary instances. Our approach also alleviates the related problem of decision boundary shift that arises in existing model reconstruction approaches when counterfactuals are treated as ordinary instances. Experimental results demonstrate that our strategy improves fidelity between the target and surrogate model predictions on several datasets.

Main paper: 
 introduction: 
 Counterfactual explanations (also called counterfactuals) have emerged as a burgeoning area of research [Wachter et al., 2017 , Guidotti, 2022 , Barocas et al., 2020 , Verma et al., 2022 , Karimi et al., 2022] for providing guidance on how to obtain a more favorable outcome from a machine learning model, e.g., increase your income by 10K to qualify for the loan. Interestingly, counterfactuals can also reveal information about the underlying model, posing a nuanced interplay between model privacy and explainability [Aïvodji et al., 2020 , Wang et al., 2022] . Our work provides the first theoretical analysis between the error in model reconstruction using counterfactuals and the number of counterfactuals queried for, through the lens of polytope theory.Model reconstruction using counterfactuals can have serious implications in Machine Learning as a Service (MLaaS) platforms that allow users to query a model for a specified cost [Gong et al., 2020] . An adversary may be able to "steal" the model by querying for counterfactuals and training a surrogate model to provide similar predictions as the target model, a practice also referred to as model extraction. On the other hand, model reconstruction could also be beneficial for preserving applicant privacy, e.g., if an applicant wishes to evaluate their chances of acceptance from crowdsourced information before formally sharing their own application information with an institution, often due to resource constraints or having a limited number of attempts to apply (e.g., applying for credit cards reduces the credit score [Capital One, 2024] ). Our goal is to formalize how faithfully can one reconstruct an underlying model given a set of counterfactual queries. An existing approach for model reconstruction is to treat counterfactuals as ordinary labeled points and use them for training a surrogate model [Aïvodji et al., 2020] . While this may work for a well-balanced counterfactual queries from the two classes lying roughly equidistant to the decision boundary, it is not the same for unbalanced datasets. The surrogate decision boundary might not always overlap with that of the target model (see Fig. 1 ), a problem also referred to as a decision boundary shift. This is due to the learning process where the boundary is kept far from the training examples (margin) for better generalization [Shokri et al., 2021] .The decision boundary shift is aggravated when the system provides only one-sided counterfactuals, i.e., counterfactuals only for queries with unfavorable predictions. If one were allowed to query for two-sided counterfactuals, the decision boundary shift may be tackled by querying for the counterfactual of the counterfactual [Wang et al., 2022] . However, such strategies cannot be applied when only one-sided counterfactuals are available, which is more common and also a more challenging use case for model reconstruction, e.g., counterfactuals are only available for the rejected applicants to get accepted for a loan but not the other way.In this work, we analyze how model reconstruction using counterfactuals can be improved by leveraging the fact that the counterfactuals are quite close to the decision boundary. We provide novel theoretical analysis for model reconstruction using polytope theory, addressing an important knowledge gap in the existing literature. We demonstrate reconstruction strategies that alleviate the decision-boundary-shift issue for one-sided counterfactuals. In contrast to existing strategies Aïvodji et al. [2020] and Wang et al. [2022] which require the system to provide counterfactuals for queries from both sides of the decision boundary, we are able to reconstruct using only one-sided counterfactuals, a problem that we also demonstrate to be theoretically more challenging than the two-sided case (see Corollary 3.8). In summary, our contributions can be listed as follows:Fundamental guarantees on model reconstruction using counterfactuals: We derive novel theoretical relationships between the error in model reconstruction and the number of counterfactual queries (query complexity) under three settings: (i) Convex decision boundaries and closest counterfactuals (Theorem 3.2): We rely on convex polytope approximations to derive an exact relationship between expected model reconstruction error and query complexity; (ii) ReLU networks and closest counterfactuals (Theorem 3.6): Relaxing the convexity assumption, we provide probabilistic guarantees on the success of model reconstruction as a function of number of counterfactual queries; and (iii) Beyond closest counterfactuals: We provide approximate guarantees for a broader class of models, including ReLU networks and locally-Lipschitz continuous models (Theorem 3.10). Model reconstruction strategy with unique loss function: We devise a reconstruction strategy -that we call Counterfactual Clamping Attack (CCA) -that exploits only the fact that the counterfactuals lie reasonably close to the decision boundary, but need not be exactly the closest. Empirical validation: We conduct experiments on both synthetic datasets, as well as, four realworld datasets, namely, Adult Income [Becker and Kohavi, 1996] , COMPAS [Angwin et al., 2016] , DCCC [Yeh, 2016] , and HELOC [FICO, 2018] . Our strategy outperforms the existing baseline [Aïvodji et al., 2020] over all these datasets (Section 4) using one-sided counterfactuals, i.e., counterfactuals only for queries from the unfavorable side of the decision boundary. We also include additional experiments to observe the effects of model architecture, Lipschitzness, and other types of counterfactual generation methods as well as ablation studies with other loss functions. A python-based implementation is available at: https://github.com/pasandissanayake/ model-reconstruction-using-counterfactuals.Related Works: A plethora of counterfactual-generating mechanisms has been suggested in existing literature [Guidotti, 2022 , Barocas et al., 2020 , Verma et al., 2022 , Karimi et al., 2022 , 2020 , Mothilal et al., 2020 , Dhurandhar et al., 2018 , Deutch and Frost, 2019 , Mishra et al., 2021] . Related works that focus on leaking information about the dataset from counterfactual explanations include membership inference attacks [Pawelczyk et al., 2023] and explanation-linkage attacks [Goethals et al., 2023] . Shokri et al. [2021] examine membership inference from other types of explanations, e.g., featurebased. Model reconstruction (without counterfactuals) has been the topic of a wide array of studies (see surveys Gong et al. [2020] and Oliynyk et al. [2023] ). Various mechanisms such as equation solving [Tramèr et al., 2016] and active learning have been considered [Pal et al., 2020] .Model inversion [Gong et al., 2021 , Struppek et al., 2022 , Zhao et al., 2021] is another form of extracting information about a black box model, under limited access to the model aspects. In contrast to model extraction where the goal is to replicate the model itself, in model inversion an adversary tries to extract the representative attributes of a certain class with respect to the target model. In this regard, Zhao et al. [2021] focuses on exploiting explanations for image classifiers such as saliency maps to improve model inversion attacks. Struppek et al. [2022] proposes various methods based on Generative Adversarial Networks to make model inversion attacks robust (for instance, to distributional shifts) in the domain of image classification. Milli et al. [2019] looks into model reconstruction using other types of explanations, e.g., gradients. Yadav et al. [2023] explore algorithmic auditing using counterfactual explanations, focusing on linear classifiers and decision trees. Using counterfactual explanations for model reconstruction has received limited attention, with the notable exceptions of Aïvodji et al. [2020] and Wang et al. [2022] . Aïvodji et al. [2020] suggest using counterfactuals as ordinary labeled examples while training the surrogate model, leading to decision boundary shifts, particularly for unbalanced query datasets (one-sided counterfactuals). Wang et al. [2022] introduces a strategy of mitigating this issue by further querying for the counterfactual of the counterfactual. However, both these methods require the system to provide counterfactuals for queries from both sides of the decision boundary. Nevertheless, a user with a favorable decision may not usually require a counterfactual explanation, and hence a system providing one-sided counterfactuals might be more common, wherein lies our significance. While model reconstruction (without counterfactuals) has received interest from a theoretical perspective [Tramèr et al., 2016 , Papernot et al., 2017 , Milli et al., 2019] , model reconstruction involving counterfactual explanations lack such a theoretical understanding. Our work theoretically analyzes model reconstruction using polytope theory and proposes novel strategies thereof, also addressing the decision-boundary shift issue. 
preliminaries: 
 Notations: We consider binary classification models m that take an input value x ∈ R d and output a probability between 0 and 1. The final predicted class is denoted by ⌊m(x)⌉ ∈ {0, 1} which is obtained by thresholding the output probability at 0.5 as follows: ⌊m(x)⌉ = 1[m(x) ≥ 0.5] where 1[•] denotes the indicator function. Throughout the paper, we denote the output probability by m(x) and the corresponding thresholded output by ⌊m(x)⌉. Consequently, the decision boundary of the model m is the (d -1)-dimensional hypersurface (generalization of surface in higher dimensions; see Definition 2.5) in the input space, given by ∂M = {x : m(x) = 0.5}. We call the region where ⌊m(x)⌉ = 1 as the favorable region and the region where ⌊m(x)⌉ = 0 as the unfavorable region. We always state the convexity/concavity of the decision boundary with respect to the favorable region (i.e., the decision boundary is convex if the set M = {x ∈ R d : ⌊m(x)⌉ = 1} is convex). We assume that upon knowing the range of values for each feature, the d-dimensional input space can be normalized so that the inputs lie within the set [0, 1] d (the d-dimensional unit hypercube), as is common in literature [Liu et al., 2020 , Tramèr et al., 2016 , Hamman et al., 2023 , Black et al., 2022] . We let g m denote the counterfactual generating mechanism corresponding to the model m. Definition 2.1 (Counterfactual Generating Mechanism). Given a cost function c :[0, 1] d × [0, 1] d → R +0 for measuring the quality of a counterfactual, and a model m, the corresponding counterfactual generating mechanism is the mappingg m : [0, 1] d → [0, 1] d specified as follows: g m (x) = arg min w∈[0,1] d c(x, w), such that ⌊m(x)⌉ ̸ = ⌊m(w)⌉.The cost c(x, w) is selected based on specific desirable criteria, e.g., c(x, w) = ||x -w|| p , with || • || p denoting the L p -norm. Specifically, p = 2 leads to the following definition of the closest counterfactual [Wachter et al., 2017 , Laugel et al., 2017 , Mothilal et al., 2020] .Definition 2.2 (Closest Counterfactual). When c(x, w) ≡ ||x -w|| 2 , the resulting counterfactual generated using g m as per Definition 2.1 is called the closest counterfactual.Given a model m and a counterfactual generating method g m , we define the inverse counterfactual region G for a subset H ⊆ [0, 1] d to be the region whose counterfactuals under g m fall in H. Fid m,Dref ( m) = 1 |D ref | x∈Dref 1 [⌊m(x)⌉ = ⌊ m(x)⌉] .Background on Geometry of Decision Boundaries: Our theoretical analysis employs arguments based on the geometry of the involved models' decision boundaries. We assume the decision boundaries are hypersurfaces. A hypersurface is a generalization of a surface into higher dimensions, e.g., a line or a curve in a 2-dimensional space, a surface in a 3-dimensional space, etc. Definition 2.5 (Hypersurface, Lee [2009] ). A hypersurface is a (d -1)-dimensional sub-manifold embedded in R d , which can also be denoted by a single implicit equation S(x) = 0 where x ∈ R d .We focus on the properties of hypersurfaces which are "touching" each other, as defined next. Definition 2.6 (Touching Hypersurfaces). Let S(x) = 0 and T (x) = 0 denote two differentiable hypersurfaces in R d . S(x) = 0 and T (x) = 0 are said to be touching each other at the point w if and only if S(w) = T (w) = 0, and there exists a non-empty neighborhood B w around w, such that ∀x ∈ B w with S(x) = 0 and x ̸ = w, only one of T (x) > 0 or T (x) < 0 holds. (i.e., within B w , S(x) = 0 and T (x) = 0 lie on the same side of each other).Next, we show that touching hypersurfaces share a common tangent hyperplane at their point of contact. This result is instrumental in exploiting the closest counterfactuals in model reconstruction (proof in Appendix A.1). Lemma 2.7. Let S(x) = 0 and T (x) = 0 denote two differentiable hypersurfaces in R d , touching each other at point w. Then, S(x) = 0 and T (x) = 0 have a common tangent hyperplane at w.Question: Does the paper discuss the limitations of the work performed by the authors?Answer: [Yes] Justification: Assumptions of each theorem are specified on the spot. Paper also includes a discussion on limitations and future work in the appendix.Guidelines:• The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper. • The authors are encouraged to create a separate "Limitations" section in their paper.• The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be. • The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated.• The authors should reflect on the factors that influence the performance of the approach. For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon. • The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size. • If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness. • While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations. 
main results: 
 We provide a set of theoretical guarantees on query size of model reconstruction that progressively build on top of each other, starting from stronger guarantees in a somewhat restricted setting towards more approximate guarantees in broadly applicable settings. We discuss important intuitions that can be inferred from the results. Finally, we propose a model reconstruction strategy that is executable in practice, which is empirically evaluated in Section 4.We start out with demonstrating how the closest counterfactuals provide a linear approximation of any decision boundary, around the corresponding counterfactual. Prior work [Yadav et al., 2023] shows that for linear models, the line joining a query instance x and the closest counterfactual w(= g m (x)) is perpendicular to the linear decision boundary. We generalize this observation to any differentiable decision boundary, not necessarily linear. Lemma 3.1. Let S denote the decision boundary of a classifier and x ∈ [0, 1] d be any point that is not on S. Then, the line joining x and its closest counterfactual w is perpendicular to S at w. The proof follows by showing that the d-dimensional ball with radius ||x-w|| 2 touches (as in Definition2.6) S at w, and invoking Lemma 2.7. For details see Appendix A.1. As a direct consequence of Lemma 3.1, a user may query the system and calculate tangent hyperplanes of the decision boundary drawn at the closest counterfactuals. This leads to a linear approximation of the decision boundary at the closest counterfactuals (see Fig. 3 ).If the decision boundary is convex, such an approximation will provide a set of supporting hyperplanes. The intersection of these supporting hyperplanes will provide a circumscribing convex polytope approximation of the decision boundary. We show that the average fidelity of such an approximation, evaluated over uniformly distributed input instances, tends to 1 when the number of queries is large. Theorem 3.2. Let m be the target binary classifier whose decision boundary is convex (i.e., the set {x ∈ [0, 1] d : ⌊m(x)⌉ = 1} is convex) and has a continuous second derivative. Denote by Mn , the convex polytope approximation of m constructed with n supporting hyperplanes obtained through i.i.d. counterfactual queries. Assume that the fidelity is evaluated with respect to D ref which is uniformly distributed over [0, 1] d . Then, when n → ∞ the expected fidelity of Mn with respect to m is given byE Fid m,Dref ( Mn ) = 1 -ϵ (1)where ϵ ∼ O n -2 d-1and the expectation is over both Mn and D ref .Theorem 3.2 provides an exact relationship between the expected fidelity and number of queries.The proof utilizes a result from random polytope theory [Böröczky Jr and Reitzner, 2004] which provides a complexity bound on volume-approximating smooth convex sets by convex polytopes. The proof involves observing that the volume of the overlapping decision regions of m and Mn (for example, regions A and C in Fig. 3 ) translates to the expected fidelity when evaluated under a uniformly distributed D ref . Appendix A.2 provides the detailed steps. Remark 3.3 (Relaxing the Convexity Assumption). This strategy of linear approximations can also be extended to a concave decision boundary since the closest counterfactual will always lead to a tangent hyperplane irrespective of convexity. Now the rejected region can be seen as the intersection of these half-spaces (Lemma 3.1 does not assume convexity). However, it is worth noting that approximating a concave decision boundary is, in general, more difficult than approximating a convex region. To obtain equally-spaced out tangent hyperplanes on the decision boundary, a concave region will require a much denser set of query points (see Fig. 4 ) due to the inverse effect of length contraction discussed in Aleksandrov [1967, Chapter III Lemma 2] . Deriving similar theoretical guarantees for a decision boundary which is neither convex nor concave is much more challenging as the decision regions can no longer be approximated as intersections of half-spaces. The assumption of convex decision boundaries may only be satisfied under limited scenarios such as input-convex neural networks [Amos et al., 2017] . However, we observe that this limitation can be circumvented in case of ReLU networks to arrive at a probabilistic guarantee as discussed next.Rectified Linear Units (ReLU) are one of the most used activation functions in neural networks [Zeiler et al., 2013 , Maas et al., 2013 , Ronneberger et al., 2015 , He et al., 2016] . A deep neural network that uses ReLU activations can be represented as a Continuous Piece-Wise Linear (CPWL) function [Chen et al., 2022, Hanin and Rolnick, 2019] . A CPWL function comprises of a union of linear functions over a partition of the domain. Definition 3.4 below provides a precise characterization. Definition 3.4 (Continuous Piece-Wise Linear (CPWL) Function [Chen et al., 2022] ). A function ℓ : R d → R is said to be continuous piece-wise linear if and only if 1. There exists a finite set of closed subsets of R d , denoted as {U i } i=1,2,...,q such that ∪ q i=1 U i = R n 2. ℓ(x) is affine over each U i i.e., over eachU i , ℓ(x) = ℓ i (x) = a T i x + b i with a i ∈ R d , b i ∈ R.This definition can be readily applied to the models of our interest, of which the domain is the unit hypercube [0, 1] d . A neural network with ReLU activations can be used as a classifier by appending a Sigmoid activation σ(z) = 1 1+e -z to the final output. We denote such a classifier by m(x) = σ(ℓ(x)) where ℓ(x) is CPWL. It has been observed that the number of linear pieces q of a trained ReLU network is generally way below the theoretically allowed maximum [Hanin and Rolnick, 2019] .We first show that the decision boundaries of such CPWL functions are collections of polytopes (not necessarily convex). The proof of Lemma 3.5 is deferred to Appendix A.3. Next we will analyze the probability of successful model reconstruction using counterfactuals. Consider a uniform grid N ϵ over the unit hypercube [0, 1] d , where each cell is a smaller hypercube with side length ϵ (see Fig. 5 ). For this analysis, we make an assumption: If a cell contains a part of the decision boundary, then that part is completely linear (affine) within that small cell2 . White color depicts the accepted region. Pale-colored are the inverse counterfactual regions of the H i 's with the matching color. In this case k(ϵ) = 7 and v * (ϵ) is the area of lower amber region. Now, as the decision boundary becomes linear for each small cell that it passes through, having just one closest counterfactual in each such cell is sufficient to get the decision boundary in that cell (recall Lemma 3.1). We formalize this intuition in Theorem 3.6 to obtain a probabilistic guarantee on the success of model reconstruction. A proof is presented in Appendix A.4. P [Reconstruction] ≥ 1 -k(ϵ)(1 -v * (ϵ)) n (2)where v * (ϵ) = min i=1,...,k(ϵ) v i (ϵ) and n is the number of queries.Remark 3.7. Here k(ϵ) and v * (ϵ) depend only on the nature of the model being reconstructed and are independent of the number of queries n. The value of k(ϵ) roughly grows with the surface area of the decision boundary (e.g., length when input is 2D), showing that models with more convoluted decision boundaries might need more queries for reconstruction. Generally, k(ϵ) lies within the interval A(∂M)√ 2ϵ d-1 ≤ k(ϵ) ≤ 1 ϵ d where A(.) denotes the surface area in d-dimensional space. The lower bound is due to the fact that the area of any slice of the unit hypercube being at-most √ 2 [Ball, 1986] . Upper bound is reached when the decision boundary traverses through all the cells in the grid which is less likely in practice. When the model complexity increases, we get a larger k(ϵ) as well as a smaller v * (ϵ), requiring a higher number of queries to achieve similar probabilities of success.Corollary 3.8 (Linear Models). For linear models with one-sided counterfactuals,P [Reconstruction] = 1 -(1 -v) nwhere v is the volume of the unfavorable region. However, with two-sided counterfactuals, P [Reconstruction] = 1 with just one single query.This result mathematically demonstrates that allowing counterfactuals from both accepted and rejected regions (as in Aïvodji et al. [2020] , Wang et al. [2022] ) is easier for model reconstruction, when compared to the one-sided case. It effectively increases each v i (ϵ) (volume of the inverse counterfactual region). As everything else remains unaffected, for a given n, P[Reconstruction] is higher when counterfactuals from both regions are available. For a linear model, this translates to a guaranteed reconstruction with a single query since v = 1.However, we note that all of the aforementioned analysis relies on the closest counterfactual which can be challenging to generate. Practical counterfactual generating mechanisms usually provide counterfactuals that are reasonably close but may not be exactly the closest. This motivates us to now propose a more general strategy assuming local Lipschitz continuity of the models involved.Lipschitz continuity, a property that is often encountered in related works [Bartlett et al., 2017 , Gouk et al., 2021 , Pauli et al., 2021 , Hamman et al., 2023 , 2024 , Liu et al., 2020 , Marques-Silva et al., 2021] , demands the model output does not change too fast. Usually, a smaller Lipschitz constant is indicative of a higher generalizability of a model [Gouk et al., 2021 , Pauli et al., 2021] . Definition 3.9 (Local Lipschitz Continuity). A model m is said to be locally Lipschitz continuous if for everyx 1 ∈ [0, 1] d there exists a neighborhood B x1 ⊆ [0, 1] d around x 1 such that for all x 2 ∈ B x1 , |m(x 1 ) -m(x 2 )| ≤ γ||x 1 -x 2 || 2 for some γ ∈ R + 0 .For analyzing model reconstruction under local-Lipschitz assumptions, we consider the difference of the model output probabilities (before thresholding) as a measure of similarity between the target and surrogate models because it would force the decision boundaries to overlap. The proposed strategy is motivated from the following observation: the difference of two models' output probabilities corresponding to a given input instance x can be bounded by having another point with matching outputs in the affinity of the instance x. This observation is formally stated in Theorem 3.10. See Appendix A.5 for a proof. Theorem 3.10. Let the target m and surrogate m be ReLU classifiers such that m(w) = m(w) for every counterfactual w. For any point x that lies in a decision boundary cell,| m(x) -m(x)| ≤ √ d(γ m + γ m)ϵ holds with probability p ≥ 1 -k(ϵ)(1 -v * (ϵ)) n .Note that within each decision boundary cell, models are affine and hence locally Lipschitz for some γ m , γ m ∈ R + 0 . Local Lipschitz property assures that the approximation is quite close (γ m , γ m are small) except over a few small ill-behaved regions of the decision boundary. This result can be extended to any locally Lipschitz pair of models as stated in following corollary. √ d(γ m + γ m)ϵgiven that all the counterfactuals satisfy m(w) = m(w). Knowing that m(w) = 0.5, we may design a loss function which clamps m(w) to be 0.5. Consequently, with a sufficient number of well-spaced counterfactuals to cover ∂M, we may achieve arbitrarily small | m(x) -m(x)| at the decision boundary of m (Fig. 6 ).However, we note that simply forcing m(w) to be exactly equal to 0.5 is quite unstable since in practice the target model's output m(w) is known to be close to 0.5, while being greater but may not be exactly equal. Thus, we propose a unique loss function for training the surrogate neural networks that does not penalize the counterfactuals that are already inside the favorable region of the surrogate model. For 0 < k ≤ 1, we define the Counterfactual Clamping loss function asL k ( m(x), y x ) = 1 [y x = 0.5, m(x) ≤ k] {L( m(x), k) -h(k)} + 1 [y x ̸ = 0.5] L( m(x), y x ). (3)Here, y x denotes the label assigned to the input instance x by the target model, received from the API. L(ŷ, y) is the binary cross-entropy loss and h(•) denotes the binary entropy function. We assume that the counterfactuals are distinguishable from the ordinary instances, and assign them a label of y x = 0.5. The first term accounts for the counterfactuals, where they are assigned a non-zero loss if the surrogate model's prediction is below k. This term ensures m(w) ≳ k for the counterfactuals w while not penalizing the ones that lie farther inside the favorable region. The second term is the ordinary binary cross-entropy loss, which becomes non-zero only for ordinary query instances. Note that substituting k = 1 in L k ( m(x), y x ) yields the ordinary binary cross entropy loss. Algorithm 1 summarizes the proposed strategy. Query API with x to get y x {y x ∈ {0, 1}} 4:A ← A ∪ {(x, y x } 5: if y x = 0 then 6:Query API for counterfactual w of x 7:A ← A ∪ {(w, 0.5)} {Assign w a label of 0.5} 8:end if 9: end for 10: Train m on A with L k ( m(x), y x ) as the loss 11: return mIt is noteworthy that this approach is different from the broad area of soft-label learning Nguyen et al. [2011a,b] in two major aspects: (i) the labels in our problem do not smoothly span the interval [0,1] -instead they are either 0, 1 or 0.5; (ii) labels of counterfactuals do not indicate a class probability; even though a label of 0.5 is assigned, there can be counterfactuals that are well within the surrogate decision boundary which should not be penalized. Nonetheless, we also perform ablation studies where we compare the performance of our proposed loss function with another potential loss which simply forces m(w) to be exactly 0.5, inspired from soft-label learning (see Appendix B.2.10 for results).Counterfactual Clamping overcomes two challenges beset in existing works; (i) the problem of decision boundary shift (particularly with one-sided counterfactuals) present in the method suggested by Aïvodji et al. [2020] and (ii) the need for counterfactuals from both sides of the decision boundary in the methods of Aïvodji et al. [2020] and Wang et al. [2022] .Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?Answer: [Yes] Justification: Assumptions corresponding to each theoretical result are stated immediately before the theorems. Proofs for all the results are presented in the appendix.Guidelines:• The answer NA means that the paper does not include theoretical results.• All the theorems, formulas, and proofs in the paper should be numbered and crossreferenced. • All assumptions should be clearly stated or referenced in the statement of any theorems.• The proofs can either appear in the main paper or the supplemental material, but if they appear in the supplemental material, the authors are encouraged to provide a short proof sketch to provide intuition. • Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material. • Theorems and Lemmas that the proof relies upon should be properly referenced. 
experiments: 
 We carry out a number of experiments to study the performance of our proposed strategy Counterfactual Clamping. We include some results here and provide further details in Appendix B.All the classifiers are neural networks unless specified otherwise and their decision boundaries are not necessarily convex. The performance of our strategy is compared with the existing attack presented in Aïvodji et al. [2020] that we refer to as "Baseline", for the case of one-sided counterfactuals.As the initial counterfactual generating method, we use an implementation of the Minimum Cost Counterfactuals (MCCF) by Wachter et al. [2017] .Performance metrics: Fidelity is used for evaluating the agreement between the target and surrogate models. It is evaluated over both uniformly generated instances (denoted by D uni ) and test data instances from the data manifold (denoted by D test ) as the reference dataset D ref .A summary of the experiments is provided below with additional details in Appendix B.(i) Visualizing the attack using synthetic data: First, the effect of the proposed loss function in mitigating the decision boundary shift is observed over a 2-D synthetic dataset. Fig. 7 1 provides some of the results over four real-world datasets. We refer to Appendix B.2.2 (specifically Fig. 11 ) for additional results. In all cases, we observe that CCA has either better or similar fidelity as compared to Baseline. (v) Studying other counterfactual generating methods: Effects of counterfactuals being sparse, actionable, realistic, and robust are observed. Sparse counterfactuals are generated by using L 1 -norm as the cost function. Actionable counterfactuals are generated using DiCE [Mothilal et al., 2020] by defining a set of immutable features. Realistic counterfactuals (that lie on the data manifold) are generated by retrieving the 1-Nearest-Neighbor from the accepted side for a given query, as well as using the autoencoder-based method C-CHVAE [Pawelczyk et al., 2020] . Additionally, we generate robust counterfactuals using ROAR [Upadhyay et al., 2021] . We evaluate the attack performance on the HELOC dataset (Table 2 ). Moreover we observe the distribution of the counterfactuals generated using each method w.r.t. the target model's decision boundary using histograms (Fig. 8 ). Additional details are given in Appendix B.2.6.(vi) Comparison with DualCFX: DualCFX proposed by Wang et al. [2022] is a strategy that utilizes the counterfactual of the counterfactuals to mitigate the decision boundary shift. We compare CCA with DualCFX in Table 6 , Appendix B.2.7.(vii) Studying alternate loss functions: We explore using binary cross-entropy loss function directly with labels 0, 1 and 0.5, in place of the proposed loss. However, experiments indicate that this scheme performs poorly when compared with the CCA loss (see Fig. 18 and Appendix B.2.10).(viii) Validating Theorem 3.2: Empirical verification of the theorem is done through synthetic experiments, where the model has a spherical decision boundary since they are known to be more difficult for polytope approximation [Arya et al., 2012] . Fig. 20 presents a log-log plot comparing the theoretical and empirical query complexities for several dimensionality values d. The empirical approximation error decays faster than n -2/(d-1) as predicted by the theorem (see Appendix B.3). Figure 8 : Histograms of the target model's predictions on different types of input instances. Counterfactual generating methods except MCCF with L 2 norm often generate counterfactuals that are farther inside the favorable region, hence having a target model prediction much greater than 0.5. We count all the query results across all the target models in the ensembles used to compute the average fidelities corresponding to each counterfactual generating method.Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?Answer: [Yes] Justification: Details pertaining to all the experiments, including model architectures, datasets, querying strategy, no. of training epochs are presented in the appendix. 
conclusion: 
 Our work provides novel insights that bridge explainability and privacy through a set of theoretical guarantees on model reconstruction using counterfactuals. We also propose a practical model reconstruction strategy based on the analysis. Experiments demonstrate a significant improvement in fidelity compared to the baseline method proposed in Aïvodji et al. [2020] for the case of onesided counterfactuals. Results show that the attack performs well across different model types and counterfactual generating methods. Our findings also highlight an interesting connection between Lipschitz constant and vulnerability to model reconstruction.Limitations and Future Work: Even though Theorem 3.6 provides important insights about the role of query size in model reconstruction, it lacks an exact characterization of k(ϵ) and v i (ϵ). Moreover, local Lipschitz continuity might not be satisfied in some machine learning models such as decision trees. Any improvements along these lines can be avenues for future work. Utilizing techniques in active learning in conjunction with counterfactuals is another problem of interest. Extending the results of this work for multi-class classification scenarios can also be explored. The relationship between Lipschitz constant and vulnerability to model reconstruction may have implications for future work on generalization, robustness, etc. Another direction of future work is to design defenses for model reconstruction attacks, leveraging and building on strategies for robust counterfactuals [Dutta et al., 2022 , Hamman et al., 2024 , Upadhyay et al., 2021 , Black et al., 2022 , Jiang et al., 2023 , Pawelczyk et al., 2020] .Broader Impact: We demonstrate that one-sided counterfactuals can be used for perfecting model reconstruction. While this can be beneficial in some cases, it also exposes a potential vulnerability in MLaaS platforms. Given the importance of counterfactuals in explaining model predictions, we hope our work will inspire countermeasures and defense strategies, paving the way toward secure and trustworthy machine learning systems.Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material?Answer: [Yes] Justification: While all the datasets used in the experiments are publicly available, we have made the code public as a Github repository, of which the link is given in the Introduction.Guidelines:• The answer NA means that paper does not include experiments requiring code. • Providing as much information as possible in supplemental material (appended to the paper) is recommended, but including URLs to data and code is permitted. 
experimental setting/details: 
 Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?Answer: [Yes] Justification: Details pertaining to each experiment are given in the corresponding sections of the appendix.Guidelines:• The answer NA means that the paper does not include experiments.• The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them. • The full details can be provided either with the code, in appendix, or as supplemental material. 7. Experiment Statistical Significance Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments? Answer: [Yes] Justification: In the main experiments supporting the claim, we report standard deviations or variance along with the results. Guidelines:• The answer NA means that the paper does not include experiments.• The authors should answer "Yes" if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper. • The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions). • The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.) • The assumptions made should be given (e.g., Normally distributed errors).• It should be clear whether the error bar is the standard deviation or the standard error of the mean. • It is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have a 96% CI, if the hypothesis of Normality of errors is not verified. • For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out of range (e.g. negative error rates). • If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text. 
experiments compute resources: 
 Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments? Answer: [No] Justification: While the GPU models on which the experiments were done are listed in the appendix, compute times have not been reported. Guidelines:• The answer NA means that the paper does not include experiments.• The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage. • The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute. • The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't make it into the paper). 9. Code Of Ethics Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics https://neurips.cc/public/EthicsGuidelines? Answer: [Yes] Justification: The research conforms with the NeurIPS Code of Ethics. Guidelines:• The answer NA means that the authors have not reviewed the NeurIPS Code of Ethics.• If the authors answer No, they should explain the special circumstances that require a deviation from the Code of Ethics. • The authors should make sure to preserve anonymity (e.g., if there is a special consideration due to laws or regulations in their jurisdiction). 10. Broader Impacts Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed? Answer: [Yes] Justification: Implications of the theoretical results on privacy of the models are discussed next to the theorems. Appendix contains a separate section on the broader impact.Guidelines:• The answer NA means that there is no societal impact of the work performed.• If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact. • Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations. • The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is a direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster. • The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology. • If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML). 
safeguards: 
 Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)? Answer: [NA] Justification: This research does not release any data or models. Guidelines:• The answer NA means that the paper poses no such risks.• Released models that have a high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters. • Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images. • We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort. • We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution. • For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review. 
 
 Appendix: A.1 Proof of Lemma 2.7 and Lemma 3.1 Lemma 2.7. Let S(x) = 0 and T (x) = 0 denote two differentiable hypersurfaces in R d , touching each other at point w. Then, S(x) = 0 and T (x) = 0 have a common tangent hyperplane at w.Proof. From Definition 2.6, there exists a non-empty neighborhood B w around w, such that ∀x ∈ B w with S(x) = 0 and x ̸ = w, only one of T (x) > 0 or T (x) < 0 holds. Let x = (x 1 , x 2 , . . . , x d ) and x [p] denote x without x p for 1 ≤ p ≤ d. Then, within the neighborhood B w , we may re-parameterize S(x) = 0 as x p = S(x [p] ). Note that a similar re-parameterization denoted by x p = T (x [p] ) can be applied to T (x) = 0 as well. Let A w = x [p] : x ∈ B w \ {w} . From Definition 2.6, allx ∈ B w \ {w} satisfy only one of T (x) < 0 or T (x) > 0, and hence without loss of generality the re-parameterization of T (x) = 0 can be such that S(x[p] ) < T (x [p] ) holds for all x [p] ∈ A w . Now, define F (x [p] ) ≡ T (x [p] ) -S(x [p] ). Observe that F (x [p]) has a minimum at w and hence,∇ x [p] F (w [p] ) = 0. Consequently, ∇ x [p] T (w [p] ) = ∇ x [p] S(w [p]), which implies that the tangent hyperplanes to both hypersurfaces have the same gradient at w. Proof concludes by observing that since both tangent hyperplanes go through w, the two hypersurfaces should share a common tangent hyperplane at w.Lemma 3.1. Let S denote the decision boundary of a classifier and x ∈ [0, 1] d be any point that is not on S. Then, the line joining x and its closest counterfactual w is perpendicular to S at w.Proof. The proof utilizes the following lemma.Lemma A.1. Consider the d-dimensional ball C x centered at x, with w lying on its boundary (hence C x intersects S at w). Then, S lies completely outside C x .The proof of Lemma A.1 follows from the following contradiction. Assume a part of S lies within C x . Then, points on the intersection of S and the interior of C x are closer to x than w. Hence, w can no longer be the closest point to x, on S.From Lemma A.1, C x is touching the curve S at w, and hence, they share the same tangent hyperplane at w by Lemma 2.7. Now, observing that the line joining w and x, being a radius of C x , is the normal to the ball at w concludes the proof (see Fig. 9 ).Figure 9 : Line joining the query and its closest counterfactual is perpendicular to the decision boundary at the counterfactual. See Lemma 3.1 for details.We present the following corollary as an additional observation resulting from Lemma A.1.Corollary A.2. Following Lemma A.1, it can be seen that all the points in the d-dimensional ball with x as the center and w on boundary lies on the same side of S as x.A.2 Proof of Theorem 3.2 Theorem 3.2. Let m be the target binary classifier whose decision boundary is convex (i.e., the set {x ∈ [0, 1] d : ⌊m(x)⌉ = 1} is convex) and has a continuous second derivative. Denote by Mn , the convex polytope approximation of m constructed with n supporting hyperplanes obtained through i.i.d. counterfactual queries. Assume that the fidelity is evaluated with respect to D ref which is uniformly distributed over [0, 1] d . Then, when n → ∞ the expected fidelity of Mn with respect to m is given byE Fid m,Dref ( Mn ) = 1 -ϵ (1)whereϵ ∼ O n -2 d-1and the expectation is over both Mn and D ref .Proof. We first have a look at Böröczky Jr and Reitzner [2004, Theorem 1 (restated as Theorem A.3 below)] from the polytope theory. Let M be a compact convex set with a second-order differentiable boundary denoted by ∂M. Let a 1 , . . . , a n be n randomly chosen points on ∂M, distributed independently and identically according to a given density d ∂M . Denote by H + (a i ) the supporting hyperplane of ∂M at a i . Assume C to be a large enough hypercube which contains M in its interior.Mn = n i=1 H + (a i ) ∩ C (4)which is the polytope created by the intersection of all the supporting hyperplanes. The theorem characterizes the expected difference of the volumes of M and Mn .Theorem A.3 (Random Polytope Approximation, [Böröczky Jr and Reitzner, 2004] ). For a convex compact set M with second-order differentiable ∂M and non-zero continuous density d ∂M ,EQUATIONas n → ∞, where V (•) denotes the volume (i.e., the Lebesgue measure), and τ (∂M, d) is a constant that depends only on the boundary ∂M and the dimensionality d of the space.Let x i , i = 1, . . . , n be n i.i.d queries from the ⌊m(x)⌉ = 0 region of the target model. Then, their corresponding counterfactuals g m (x i ) are also i.i.d. Furthermore, they lie on the decision boundary of m. Hence, we may arrive at the following result.Corollary A.4. Let M = {x ∈ [0, 1] d : ⌊m(x)⌉ = 1} and Mn = {x ∈ [0, 1] d : Mn (x) = 1}.Then, by Theorem A.3,EQUATIONwhen n → ∞. Note that M ⊆ Mn and hence, the left-hand side is always non-negative.From Definition 2.4, we may writeEQUATION= Mn P ⌊m(x)⌉ = Mn (x) Mn (x) = mn (x) P Mn (x) = mn (x) d mn (10)where M n is the set of all possible mn 's. Now, by noting that P ⌊m(x)⌉ = Mn (x) Mn (x) = mn (x) = 1 -P ⌊m(x)⌉ ̸ = Mn (x) Mn (x) = mn (x) , (11) we may obtain E Fid m,Dref ( Mn ) = 1 -Mn P ⌊m(x)⌉ ̸ = Mn (x) Mn (x) = mn (x)EQUATION)The above result, in conjunction with Corollary A.4, concludes the proof.A.3 Proof of Lemma 3.5 Lemma 3.5. Let m(x) = σ(ℓ(x)) be a ReLU classifier, where ℓ(x) is CPWL and σ(.) is the Sigmoid function. Then, the decision boundary ∂M = {x ∈ [0, 1] d : m(x) = 0.5} is a collection of (possibly non-convex) polytopes in [0, 1] d , when considered along with the boundaries of the unit hypercube.Proof. Consider the i th piece m i (x) of the classifier defined over U i . A part of the decision boundary exists within U i only if ∃x ∈ U i such that m i (x) = 0.5. When it is the case, at the decision boundary,EQUATION⇐⇒ 1 1 + e -ℓi(x) = 0.5 ( 16)⇐⇒ e -ℓi(x) = 1 (17) ⇐⇒ ℓ i (x) = 0 (18) ⇐⇒ a T i x + b i = 0 (19)which represents a hyperplane restricted to U i . Moreover, the continuity of the ℓ(x) demands the decision boundary to be continuous across the boundaries of U i 's. This fact can be proved as follows:Note that within each region U i , exactly one of the following three conditions holds:(a) ∀x ∈ U i , ℓ i (x) > 0 → U i does not contain a part of the decision boundary (b) ∀x ∈ U i , ℓ i (x) < 0 → U i does not contain a part of the decision boundary (c) ∃x ∈ U i , ℓ i (x) = 0 → U i contains aIn case when (c) holds for some region U i , the decision boundary within U i is affine and it extends from one point to another on the region boundary. Now let U s and U t , s, t ∈ {1, . . . , q}, s ̸ = t be two adjacent regions sharing a boundary. Assume that U s contains a portion of the decision boundary, which intersects with a part of the shared boundary between U s and U t (note that U i 's are closed and hence they include their boundaries). Let x 0 be a point in the intersection of the decision boundary within U s and the shared region boundary. Now, continuity of ℓ(x) at x 0 requires ℓ t (x 0 ) = ℓ s (x 0 ) = 0. Hence, condition (c) holds for U t . Moreover, this holds for all the points in the said intersection. Therefore, if such a shared boundary exists between U s and U t , then the decision boundary continues to U t . Applying the argument to all U s -U t pairs show that each segment of the decision boundary either closes upon itself or ends at a boundary of the unit hypercube. Hence, when taken along with the boundaries of the unit hypercube, the decision boundary is a collection of polytopes.A.4 Proof of Theorem 3.6Theorem 3.6. Let m be a target binary classifier with ReLU activations. Let k(ϵ) be the number of cells through which the decision boundary passes. Define {H i } i=1,...,k(ϵ) to be the collection of affine pieces of the decision boundary within each decision boundary cell where eachH i is an open set. Let v i (ϵ) = V (G m,gm (H i ))where V (.) is the d-dimensional volume (i.e., the Lebesgue measure) and G m,gm (.) is the inverse counterfactual region w.r.t. m and the closest counterfactual generator g m .Then the probability of successful reconstruction with counterfactual queries distributed uniformly over [0, 1] d is lower-bounded asP [Reconstruction] ≥ 1 -k(ϵ)(1 -v * (ϵ)) n (2)where v * (ϵ) = min i=1,...,k(ϵ) v i (ϵ) and n is the number of queries.Proof. Note that Let M i denote the event "i th decision boundary cell does not have a counterfactual". At the end of n queries,P[Reconstruction] = P[EQUATIONTherefore, Proof. Note that from Theorem 3.6, with probability p ≥ 1 -k(ϵ)(1 -v * (ϵ)) n at least one counterfactual exists within each decision boundary cell. When this is the case, we haveEQUATION| m(x) -m(x)| = | m(x) -m(w) -(m(x) -m(w)) | (27) = | m(x) -m(w) -(m(x) -m(w)) | (28) ≤ | m(x) -m(w)| ≤γ m||x-w||2 + |m(x) -m(w)| ≤γm||x-w||2 (29) ≤ (γ m + γ m)||x -w|| 2 (30) ≤ √ d(γ m + γ m)ϵ (31)where the first inequality is a result of applying the triangle inequality and the second follows from the definition of local Lipschitz continuity (Definition 3.9). The final inequality is due to the availability of a counterfactual within each decision boundary cell, which ensures ||x -w|| 2 ≤ √ dϵ. Corollary 3.11 follows directly from the second inequality, since the δ-cover of w's ensure ||x -w|| 2 ≤ δAll the experiments were carried-out on two computers, one with a NVIDIA RTX A4500 GPU and the other with a NVIDIA RTX 3050 Mobile.We use four publicly available real-world tabular datasets (namely, Adult Income, COMPAS, DCCC, and HELOC) to evaluate the performance of the attack proposed in Section 3.3. The details of these datasets are as follows:• Adult Income: The dataset is a 1994 census database with information such as educational level, marital status, age and annual income of individuals [Becker and Kohavi, 1996] . The target is to predict "income", which indicates whether the annual income of a given person exceeds $50000 or not (i.e., y = 1[income ≥ 0.5]). It contains 32561 instances in total (the training set), comprising of 24720 from y = 0 and 7841 from y = 1. To make the dataset class-wise balanced we randomly sample 7841 instances from class y = 0, giving a total effective size of 15682 instances. Each instance has 6 numerical features and 8 categorical features. During pre-processing, categorical features are encoded as integers.All the features are then normalized to the range [0, 1]. • Home Equity Line of Credit (HELOC): This dataset contains information about customers who have requested a credit line as a percentage of home equity FICO [2018] . It contains 10459 instances with 23 numerical features each. Prediction target is "is_at_risk" which indicates whether a given customer would pay the loan in the future. Dataset is slightly unbalanced with class sizes of 5000 and 5459 for y = 0 and y = 1, respectively. Instead of using all 23 features, we use the following subset of 10 for our experiments; "estimate_of_risk", "net_fraction_of_revolving_burden", "percent-age_of_legal_trades", "months_since_last_inquiry_not_recent", "months_since_last_trade", "percentage_trades_with_balance", "number_of_satisfactory_trades", "aver-age_duration_of_resolution", "nr_total_trades", "nr_banks_with_high_ratio".All the features are normalized to lie in the range [0, 1].• Correctional Offender Management Profiling for Alternative Sanctions (COMPAS): This dataset has been used for investigating racial biases in a commercial algorithm used for evaluating reoffending risks of criminal defendants [Angwin et al., 2016] . It includes 6172 instances and 20 numerical features. The target variable is "is_recid". Class-wise counts are 3182 and 2990 for y = 0 and y = 1, respectively. All the features are normalized to the interval [0, 1] during pre-processing. • Default of Credit Card Clients (DCCC): The dataset includes information about credit card clients in Taiwan Yeh [2016] . The target is to predict whether a client will default on the credit or not, indicated by "default.payment.next.month". The dataset contains 30000 instances with 24 attributes each. Class-wise counts are 23364 from y = 0 and 6636 from y = 1. To alleviate the imbalance, we randomly select 6636 instances from y = 0 class, instead of using all the instances. Dataset has 3 categorical attributes, which we encode into integer values. All the attributes are normalized to [0, 1] during pre-processing.In this section, we provide details about our experimental setup with additional results. For convenience, we present the neural network model architectures by specifying the number of neurons in each hidden layer as a tuple, where the leftmost element corresponds to the layer next to the input; e.g.: a model specified as (20, 30, 10) has the following architecture:Input → Dense(20, ReLU) → Dense(30, ReLU) → Dense(10, ReLU) → Output(Sigmoid)Other specifications of the models, as detailed below, are similar across most of the experiments. Changes are specified specifically. The hidden layer activations are ReLU and the layer weights are L 2 -regularized. The regularization coefficient is 0.001. Each model is trained for 200 epochs, with a batch size of 32.Fidelity is evaluated over a uniformly sampled set of input instances (uniform data) as well as a held-out portion of the original data (test data). The experiments were carried out as follows:1. Initialize the target model. Train using D train . 2. For t = 1, 2, . . . , T : (a) Sample N × t data points from the dataset to create D attack .(b) Carry-out the attack given in Algorithm 1 with D attack . Use k = 1 for "Baseline" models and k = 0.5 for "Proposed" models. (c) Record the fidelity over D ref along with t. 3. Repeat steps 1 and 2 for S number of times and calculate average fidelities for each t, across repetitions.Based on the experiments of Aïvodji et al. [2020] and Wang et al. [2022] , we select T = 20, 50, 100; N = 20, 8, 4 and S = 100, 50, in different experiments. We note that the exact numerical results are often variable due to the multiple random factors affecting the outcome such as the test-train-attack split, target and surrogate model initialization, and the randomness incorporated in the counterfactual generating methods. Nevertheless, the advantage of CCA over the baseline attack is observed across different realizations.This experiment is conducted on a synthetic dataset which consists of 1000 samples generated using the make_moons function from the sklearn package. Features are normalized to the range [0, 1] before feeding to the classifier. The target model has 4 hidden layers with the architecture (10, 20, 20, 10) . The surrogate model is 3-layered with the architecture (10, 20, 20) . Each model is trained for 100 epochs. Since the intention of this experiment is to demonstrate the functionality of the modified loss function given in (3), a large query of size 200 is used, instead of performing multiple small queries. Fig. 7 shows how the original model reconstruction proposed by Aïvodji et al. [2020] suffers from the boundary shift issue, while the model with the proposed loss function overcomes this problem. Fig. 10 illustrates the instances misclassified by the two surrogate models. Figure 14 : A comparison of the query complexity derived in Theorem 3.2 with the empirical query complexities obtained on the Adult Income and HELOC datasets. The graphs are on a log-log scale. We observe that the analytical query complexity is an upper bound for the empirical query complexities. All the graphs are recentered with an additive constant for presentational convenience. However, this does not affect the slope of the graph, which corresponds to the complexity.For this experiment, we use a target model having 3 hidden layers with the architecture (20, 10, 5) and a surrogate model having 2 hidden layers with the architecture (20, 10). The surrogate model layers are L 2 -regularized with a fixed regularization coefficient of 0.001. We achieve different Lipschitz constants for the target models by controlling their L 2 -regularization coefficients during the target model training step. Following Gouk et al. [2021] , we approximate the Lipschitz constant of target models by the product of the spectral norms of the weight matrices. We observe the effect of the model architectures on the attack performance over Adult Income, COMPAS and HELOC datasets. Tables 3, 4 , and 5, respectively, present the results. Counterfactuals can be generated such that they satisfy additional desirable properties such as actionability, sparsity and closeness to the data manifold, other than the proximity to the original instance. In this experiment, we observe how counterfactuals with above properties affect the attack performance. HELOC is used as the dataset. Target model has the architecture (20, 30, 10) and the architecture of the surrogate model is (10, 20).To generate actionable counterfactuals, we use Diverse Counterfactual Explanations (DiCE) by Mothilal et al. [2020] with the first four features, i.e., "estimate_of_risk", "months_since_last_trade", "average_duration_of_resolution", and "number_of_satisfactory_trades" kept unchanged. The diversity factor of DiCE generator is set to 1 in order to obtain only a single counterfactual for each query. Sparse counterfactuals are obtained by the same MCCF generator used in other experiments, but now with L 1 norm as the cost function c(x, w). Counterfactuals from the data manifold (i.e., realistic counterfactuals, denoted by 1-NN) are generated using a 1-Nearest-Neighbor algorithm. We use ROAR [Upadhyay et al., 2021] and C-CHVAE [Pawelczyk et al., 2020] to generate robust counterfactuals. Table 2 summarizes the performance of the attack. Fig. 8 shows the distribution of the counterfactuals generated using each method w.r.t. the decision boundary of the target model. We observe that the sparse, realistic, and robust counterfactuals have a tendency to lie farther away from the decision boundary, within the favorable region, when compared to the closest counterfactuals under L 2 norm.Wang et al. [2022] is one of the few pioneering works studying the effects of counterfactuals on model extraction, which proposes the interesting idea of using counterfactuals of counterfactuals to mitigate the decision boundary shift. This requires the API to provide counterfactuals for queries originating from both sides of the decision boundary. However, the primary focus of our work is on the one-sided scenario where an institution might be giving counterfactuals only to the rejected applicants to help them get accepted, but not to the accepted ones. Hence, a fair comparison cannot be achieved between CCA and the strategy proposed in Wang et al. [2022] in the scenario where only one-sided counterfactuals are available.Therefore, in the two-sided scenario, we compare the performance of CCA with the DualCFX strategy proposed in Wang et al. [2022] under two settings:1. only one sided counterfactuals are available for CCA (named CCA1) 2. CCA has all the data that DualCFX has (named CCA2)We also include another baseline (following Aïvodji et al. [2020] ) for the two-sided scenario where the models are trained only on query instances and counterfactuals, but not the counterfactuals of the counterfactuals. Results are presented in Table 6 . Note that even for the same number of initial query instances, the total number of actual training instances change with the strategy being used (CCA1 < Baseline < DualCFX = CCA2 -e.g.: queries+CFs for the baseline but queries+CFs+CCFs for DualCFX). We explore using binary cross-entropy loss function directly with labels 0, 1 and 0.5 in place of the proposed loss. Precisely, the surrogate loss is now defined asEQUATIONwhich is symmetric around 0.5 for y(x) = 0.5. Two surrogate models are observed, with architectures (20, 10) for model 0 and (20, 10, 5) for model 1. The target model's architecture is similar to that of model 0. The ensemble size is S = 20.The results (in Fig. 18 ) indicate that the binary cross-entropy loss performs worse than the proposed loss. The reason might be the following: As the binary cross-entropy loss is symmetric around 0.5 for counterfactuals, it penalizes the counterfactuals that are farther inside the favorable region. This in turn pulls the surrogate decision boundary towards the favorable region more than necessary, causing a decision boundary shift. Justification: Abstract and introduction correctly summarize and highlight the main claims made.Guidelines:• The answer NA means that the abstract and introduction do not include the claims made in the paper. • The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers. • The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings. • It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper.• The answer NA means that the paper does not include experiments.• If the paper includes experiments, a No answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not. • If the contribution is a dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable. • Depending on the contribution, reproducibility can be accomplished in various ways.For example, if the contribution is a novel architecture, describing the architecture fully might suffice, or if the contribution is a specific model and empirical evaluation, it may be necessary to either make it possible for others to replicate the model with the same dataset, or provide access to the model. In general. releasing code and data is often one good way to accomplish this, but reproducibility can also be provided via detailed instructions for how to replicate the results, access to a hosted model (e.g., in the case of a large language model), releasing of a model checkpoint, or other means that are appropriate to the research performed. • While NeurIPS does not require releasing code, the conference does require all submissions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution. (d) We recognize that reproducibility may be tricky in some cases, in which case authors are welcome to describe the particular way they provide for reproducibility.In the case of closed-source models, it may be that access to the model is limited in some way (e.g., to registered users), but it should be possible for other researchers to have some path to reproducing or verifying the results.38th Conference on Neural Information Processing Systems (NeurIPS 2024).Performance can be evaluated using accuracy or fidelity[Jagielski et al.,2020]. Accuracy is a measure of how well the surrogate model can predict the true labels over the data manifold of interest. While attacks based on both measures have been proposed in literature, fidelity-based attacks have been deemed more useful as a first step in designing future attacks[Jagielski et al., 2020, Papernot et al., 2017].This is violated only for the cells containing parts of the edges of the decision boundary. However, we may assume that ϵ is small enough so that the total number of such cells is negligible compared to the total cells containing the decision boundary. Generally, ϵ can be made sufficiently small such that for most of the cells, if a cell contains a part of the decision boundary then that part is affine within the cell. 


Decision: This paper Counterfactual Clamping that approximates the boundary of a convex classification region by counterfactual queries. The paper contains the performance guarantees for the fidelity measured between the true boundary and the approximated boundary. All four reviewers appreciate the new perspective of polytope theory for the (counterfactual) explanation problem. I thus recommend to accept this paper.

Reviews: review: paper_summary: paper_summary: This paper studies model reconstruction attacks by using the proximity of counterfactuals to the decision boundary. The authors aim to establish theoretical guarantees for such attacks. To this end, they characterize the number of queries required for the attacker to achieve a given error in model approximation using results from polytope theory (Theorem 2). The authors’ main result from relies on the decision boundary being convex. To relax the convexity assumption, the paper additionally studies the case of underlying relu networks to be attacked and provides probabilistic bound on the reconstruction rate. Finally, the authors propose a strategy for model extraction.

The paper offers strengths in terms of proposing new tools to analyze model extraction attacks. The main theoretical results are covered for two models classes that are commonly used in recourse literature (linear models, and NNs with relu activations) Overall, the paper provides a good starting point for future research in the area of model extraction attacks through counterfactual explanations but further improvements would be necessary to generealize the analysis to general commonly used models such a tree based classifiers.
strength_weakness: strength_weakness: *Missing variety of recourse methods*: The paper would greatly benefit from a more comprehensive set of experiments that examine the viability of model reconstruction under various recourse settings. Specifically, incorporating experiments that generate counterfactuals with data manifold constraints [3,4] or consider robustness [1,2] would be highly valuable. Such experiments would clarify the conditions under which the proposed attacks are likely to succeed or fail, and could indicate potential defenses. For instance, if the attacks fail under certain counterfactual attack methods, this would highlight key characteristics necessary to ensure the safe use of recourse or counterfactual explanations. I strongly encourage the authors to include additional experiments addressing these aspects. Demonstrating the impact of these conditions would significantly strengthen the paper, and I would be happy to increase my score if such experiments are provided.

---------
**References**

[1] "Probabilistically Robust Recourse: Navigating the Trade-offs between Costs and Robustness in Algorithmic Recourse", ICLR, https://arxiv.org/abs/2203.06768

[2] "Towards robust and reliable algorithmic recourse.", NeurIPS, https://arxiv.org/abs/2102.13620

[3] "Learning model-agnostic counterfactual explanations for tabular data", WWW, https://arxiv.org/abs/1910.09398

[4] "Towards realistic individual recourse and actionable explanations in black-box decision making systems", arxiv, https://arxiv.org/abs/1907.09615
questions: questions: Could the authors provide insights into their attempts to estimate the Lipschitz constant in practical applications? Considering the inherent difficulty in obtaining low Lipschitz constants for neural networks of reasonable size, the practicality of the Lipschitz result may be questionable. Clarifying this aspect would help in understanding the real-world applicability of their theoretical findings.
limitations: limitations: The authors did not mention any limitations of their work.
paper_summary: paper_summary: The paper presents an approach to mitigate the decision boundary shift problem of counterfactual-based model reconstruction algorithms. They leverage the fact that counterfactuals differ from ordinary instances since they exist relatively close to the decision boundary, to derive a novel loss function for model reconstruction that provides special treatment to counterfactuals. The theoretical foundations of this work are based on polytope theory, using which the authors derive a relationships between the error in model reconstruction and the number of required counterfactual samples. Experimental results show that the proposed approach surpasses the baseline on a number of model reconstruction benchmarks.
strength_weakness: strength_weakness: 1. Is there any dependency of the proposed method on the counterfactual generating method MCCF, or in other words, the quality/nature of the counterfactuals used?

2. A comparison of the proposed method with regular approaches (that treat counterfactuals and ordinary samples the same) like [Wang et al., 2022] using two sided counterfactuals could help solidify the contributions. I can understand that it might sound like an unfair comparison, but one should not necessarily expect the proposed CCA to outperform models using two-sided counterfactuals. This is just to get an understanding of how close can one get (using a model like CCA) to such two-sided counter-factual based approaches, by just using one-sided counterfactuals.

3. Comparing just with a single baseline seems a bit insufficient. Perhaps a discussion on how this work relates to model inversion attacks (such as [a,b]) would add to the completeness of the paper.

References:

[a] Zhao et al., "Exploiting Explanations for Model Inversion Attacks", ICCV 2021.

[b] Struppek et al. "Plug & Play Attacks: Towards Robust and Flexible Model Inversion Attacks", ICML 2022.
questions: questions: Please refer to the Weaknesses section.
limitations: limitations: The limitations of this work have been adequately discussed.
paper_summary: paper_summary: The paper proposed a model reconstruction methodology by using one-sided counterfactual explanations. After generating counterfactuals (assumed to be the closest counterfactual to the observation), the authors reconstruct the original model using a piece-wise linear approximation (with a bunch of hyperplanes). The key idea is that such hyperplanes can be identified by the pair of observation and its closest counterfactual, of which the the joining-line is straightforwardly perpendicular to the model's decision boundary. The paper discussed convex and non-convex decision regions, and demonstrated that the general non-convex case is challenging for being approximated with finite number of counterfactual explanations. Nevertheless, the authors showed that given sufficiently many counterfactual explanations (queries), a ReLU neural network can still be reconstructed. The experiment results demonstrated improvement over one literature (served as baseline), for both the known architecture and unknown architecture of the ReLU network.

The paper is well-written and organized. The idea, to my knowledge, is novel. The paper goes into good depth for analysis, showing interesting insights.
strength_weakness: strength_weakness: The key limitation of this paper is that all the theories hold if the counterfactuals are the closest ones to the observations, which is difficult to be guaranteed utilizing counterfactual explanations algorithms. One way to deal with this is to consider in the derived theory some tolerance on the counterfactual's quality. Namely, if an algorithm is able to find counterfactuals within a given radius epsilon of the closest counterfactual, then how does this affect model construction.

The connection between the proposed theory and the effectiveness of the authors model construction methodology is not stated. Namely, algorithm 1 could stand alone without any of the derived theories.

Theorem 3.6 relies on the assumption that each cell only contains completely linear part of the model's decision boundary, with one closest counterfactual in each cell. In order to make the first part (that only linear part is contained in a cell) holds for most of the cells, one has to make the edge of the cell (varepsilon) really small, which requires a massive amount of queries to be done. Hence, the assumption that is required to make this Theorem work, is too strong.
questions: questions: How does the "polytope theory" help the model reconstruction algorithm design? Namely, the foundation (Lemma 3.1), the fidelity expectation for the convex case (Theorem 3.2), and the reconstruction probability lower bound (Theorem 3.6) are proposed, but how could we use them to better reconstruct the model than existing counterfactual explanations based reconstruction methodologies?

The results in Table 1 have large confidence intervals. For example, in the column Architecture unknown (model 1), D_{uni}, and row DCCC, the baseline is 95+-2.2 and the proposed CCA is 95+-11.8 etc. Does this imply that the proposed method is not very stable? 

What if we assume that sufficient many queries can be made and then increase the number of queries from 400 to 4000? Will the performance of CCA go close to the baseline?
limitations: limitations: Yes, the authors have stated clearly some limitations, and foresee potential non-positive impact.
paper_summary: paper_summary: This paper presents a novel and valuable theoretical analysis leveraging polytope theory to provide novel insights into using counterfactual explanations for model reconstruction and significant contribution to the field of model reconstruction/interpretability of black box models.  The key contributions are:
Providing a geometric interpretation of how counterfactuals relate to the decision boundary using polytope geometry.
Deriving bounds on the expected approximation error of the reconstructed model as a function of the number of counterfactual queries .
Extending the analysis to handle cases where only approximate closest counterfactuals are available using Lipschitz continuity.
Proposing the Counterfactual Clamping Attack (CCA) which treats counterfactuals differently in the loss function to prevent decision boundary shift during training of the surrogate model.
Demonstrating CCA's improved performance over baselines on multiple real-world datasets, including with one-sided counterfactuals.
strength_weakness: strength_weakness: Limited discussion on the applicability of CCA to model families beyond neural networks like tree based models XGBoost, Random Forest which are most commonly used for problems discussed like loan decisions. The loss function and counterfactual generation process may need adaptations.
Lack of analysis on the computational complexity and scalability of CCA, especially for high-dimensional data or large datasets. Generating closest counterfactuals can be computationally expensive.
No exploration of the effect of counterfactual quality aspects like sparsity, actionability or realness on CCA's performance. The paper does not analyze how enforcing quality constraints on counterfactuals impacts CCA's ability to accurately reconstruct the target model's decision boundary.
questions: questions: 1. Have you explored the application of CCA to other models like tree ensembles, XGBoost or others which might have non continuous / non differentiable decision boundaries? 
2. Have you explored into the computational complexity and memory requirements of CCA, especially when scaling to high-dimensional data or large datasets for example in comparison to other standard model reconstruction methods for practical applications of the approach?
3. Any exploration of sensitivity of CCA's performance to the quality (e.g., sparsity, realistic) of the counterfactual instances used?
limitations: limitations: The theoretical analysis is limited to neural network architectures and assumes convex/concave decision boundaries. The authors should discuss the applicability to other model families like tree ensembles.
While bounds on approximation error are derived, there is limited analysis of the computational complexity and scalability of the proposed Counterfactual Clamping Attack (CCA) to high-dimensional data or large datasets.
The paper does not explore the effect of counterfactual quality criteria like sparsity and similarity on CCA's model extraction performance, which is important for real-world applications.
The empirical evaluation is focused on classification tasks. More analysis on regression problems or tasks with complex decision boundaries is needed.

Comments: title: Response to Rebuttal
Author Response acknowledged
Reply by Authors
comment: Thank you very much for raising these important questions and the detailed review. We are ready to address any other questions the reviewer might have.
Thank you very much for the insightful comments. We remain open to address any other comments and to provide any clarifications required.
Thank you very much for updating the score. We will definitely include the additional results and a discussion in the updated version of the manuscript. We greatly appreciate the insightful review.
I thank the authors for their detailed response to my comments, which addresses all of my present concerns. I will carefully consider them to arrive at my final decision.
I very much appreciate the additional results provided by the authors. As the response has addressed all my concerns, I am increasing my score. I would expect the additional results to be discussed in the final version of the paper as well.
I have read the authors response.
We appreciate that the reviewer finds that “there are shining and inspiring points in this paper.” We will definitely include a detailed discussion on the assumptions that we have to make for analytical tractability and the limitations pointed out by the Reviewer. 

**_Clarifications on the grid-size ($\epsilon$) assumption:_** In general, approximating a non-convex decision boundary becomes challenging since it is impossible to express it as an intersection of half-planes. However, for a ReLU network, the problem becomes analytically tractable because the input domain can be partitioned into **q** regions such that the model is linear in each region (leading to polytope decision boundaries with at most **q** linear pieces) [Chen et al., Hanin and Rolnick]. In fact, if the number of such partitioned regions is q and **one assumes that the adversary knows each of these partitions**, we can derive another result similar to our Theorem 3.6 by constructing only q inverse counterfactual regions and the probability will depend on only q (and not the grid-size). 

$$\mathbb{P}[\text{Reconstruction}] \geq 1-q(1-v^*)^n$$ 
where $v^* = \min_i v_i$ with $v_i$ being the volume of the $i^\text{th}$ inverse counterfactual region ($i=1,2,..,q$).


However, instead of assuming that the adversary exactly knows the partitions of the ReLU network they are trying to reconstruct (which we feel would be a much stronger assumption for our problem), we assume there is a uniform grid such that the ReLU network’s decision boundary is linear across each small grid-cell that it passes through. 

We agree that the grid-size would need to be reduced in order for this to hold: otherwise, there would be more “edge” cells where two or more facets intersect (e.g., cells containing the corners in 2D). Nonetheless, it is still a weaker assumption than assuming the adversary exactly knows the partitions of the ReLU network. 

Furthermore, the grid-size would still be determined by the complexity of the ReLU network’s decision boundary (essentially goes back to the number of original partitions q). What we wanted to highlight is that this number of partitions for a ReLU network observed in practice is far less than the theoretically possible maximum [Hanin and Rolnick], holding promise that the grid size might also not need to be too small. 

Moreover, [Jordan et al.] suggests that the decision regions can be represented as polyhedral complexes, which are a specific type of unions of convex polytopes. This may further reduce the number of high-dimensional edges that would occur in practice, as opposed to the worst-case scenario, and consequently holds promise that the number of such “edge” cells (cells containing corners in 2D) would be fewer. Therefore, the required size of the grid might actually depend largely on the complexity of the classification problem.

Relaxing the said assumptions is the main goal of our future work. In particular, we will study the possibility of allowing the adversary to know alternate partitions such that the ReLU network is linear over each region.


**_On Algorithm 1:_** Algorithm 1 can stand on its own, without the basis of theorems 3.2 and 3.6 (Theorem 3.10 and Corollary 3.11 provide the main intuition for Algorithm 1). However we see theorems 3.2 and 3.6 as important parts of the journey where we start from a more constrained but mathematically-tractable setting and move to a less constrained but not-so-mathematically-tractable setting.

Thank you very much for the detailed review and the valuable insights.


[Chen et al.] Chen, K. L., Garudadri, H., & Rao, B. D. (2022). Improved bounds on neural complexity for representing piecewise linear functions. Advances in Neural Information Processing Systems, 35, 7167-7180.

[Hanin and Rolnick] Hanin, B., & Rolnick, D. (2019, May). Complexity of linear regions in deep networks. In International Conference on Machine Learning (pp. 2596-2604). PMLR.

[Jordan et al.] Jordan, M., Lewis, J., & Dimakis, A. G. (2019). Provable certificates for adversarial examples: Fitting a ball in the union of polytopes. Advances in neural information processing systems, 32.
Thank you for your response. 
> The number of linear regions of a ReLU network in practice has been observed to be far less than the theoretically achievable maximum [Hanin and Rolnick]

Do you want to say that the "corners" in Figure 5 would be quite many such that in reality we need many grids to properly use Theorem 3.6, or the opposite? It reads a bit confusion to me.

> Moreover, [Jordan et al.] suggests that the decision regions can be represented as polyhedral complexes, which are a specific type of unions of convex polytopes. This may further reduce the number of high-dimensional edges that would occur in practice, as opposed to the worst-case scenario. Therefore, the required size of the grid might actually depend largely on the complexity of the classification problem.

I believe it worths a discussion somewhere in the paper, especially for how the size of the grid could be influenced in practice. One of the major concerns of the proposed method, from my point of view, lies on the two strong assumptions (though they are not assumed simultaneously):
1. "Closeness" for convex case
2. "Grids num" required for non-convex case
 
Still, I'm not fully convinced by your response (I meant, I believe in sentences your argument, but they seem not persuasive as rebuttal for why these two assumptions are not too strong.)

> Algorithm 1 is motivated from the need to clamp the counterfactuals to the decision boundary as per our analysis in Theorem 3.10 and Corollary 3.11 (which will have further relaxations now using the tolerance ball).
And yet, I don't think this answers my question fully. My original question is how does the "polytope theory" help the model reconstruction algorithm design, since "A perspective from polytope theory" is part of your title? Namely, how do the bounds you have derived, contribute to Algorithm 1? To me, Algorithm 1 seems quite independent to other parts of the paper and can stand alone very well.

But I believe there are shining and inspiring points in this paper, even though with above weaknesses. So, I would like to keep my score.