SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Feature transformation. The initial step involves cropping each image to dimensions of (224, 224, 3). The quickshift method from scikit-image is then employed to segment images into super-pixels, with specific parameters set as follows: kernel_size=4, max_dist=200, ratio=0.2, and random(seed=2023. This approach aligns with the default setting in LIME, except for the modified fixed random seed. Consistency in the random seed ensures that identical images result in the same super-pixels, thereby isolating the source of instability to the calculation of explanations. However, for different images, they are still segmented in different ways.
Computing explanations. The implemented procedure follows the original setup in LIME. The hide_color parameter is configured as None, causing the average value of each super-pixel to act as its reference when the super-pixel is removed. The distance_metric is explicitly set to 12, as recommended for image data in LIME [22]. The default value for alpha in Ridge regression is 1, unless otherwise specified. For each image, the model  $f$  infers the most probable label, and the explanation pertaining to that label is computed. Ten different random seeds are utilized to compute explanations for each image. The random_seed parameter in both the LimeImageExplainer and the explain_instance function is set to these specific random seeds.
A.2 Stability of LIME and GLIME
Figure 7 illustrates the top-1, top-5, top-10, and average Jaccard indices. Importantly, the results presented in Figure 7 closely align with those in Figure 4a. In summary, it is evident that GLIME consistently provides more stable explanations compared to LIME.
A.3 LIME and GLIME-BINOMIAL converge to the same limit
In Figure 8, the difference and correlation between explanations generated by LIME and GLIME-BINOMIAL are presented. With an increasing sample size, the explanations from LIME and GLIME-BINOMIAL become more similar and correlated. The difference between their explanations rapidly converges to zero, particularly when  $\sigma$  is large, such as in the case of  $\sigma = 5$ . While LIME exhibits a slower convergence, especially with small  $\sigma$ , it is impractical to continue sampling until their difference fully converges. Nevertheless, the correlation between LIME and GLIME-BINOMIAL strengthens with an increasing number of samples, indicating their convergence to the same limit as the sample size grows.
A.4 LIME explanations are different for different references.
The earlier work by Jain et al. [14] has underscored the instability of LIME regarding references. As shown in Section 4.2, this instability originates from LIME's sampling distribution, which relies on the chosen reference  $\mathbf{r}$ . Additional empirical evidence is presented in Figure 9. Six distinct references—black, white, red, blue, yellow image, and the average value of the removed super-pixel (the default setting for LIME)—are selected. The average Jaccard indices for explanations computed using these various references are detailed in Figure 9. The results underscore the sensitivity of LIME to different references.
Different references result in LIME identifying distinct features as the most influential, even with a sample size surpassing 2000. Particularly noteworthy is that, with a sample size exceeding 2000, the top-1 Jaccard index consistently remains below 0.7, underscoring LIME's sensitivity to reference variations.
(a) Top-1 Jaccard indices for various methods.
(c) Top-10 Jaccard indices for various methods.
(b) Top-5 Jaccard indices for various methods.
(d) Average Jaccard indices for various methods.
Figure 7: Top-1, 5, 10, and average Jaccard indices are computed for various methods. The average Jaccard index is obtained by averaging the top-1 to top-d Jaccard indices.
Figure 8: Difference and correlation between LIME and GLIME-BINOMIAL explanations. Mean Squared Error (MSE) and Mean Absolute Error (MAE) serve as metrics to evaluate the dissimilarity between explanations provided by LIME and GLIME-BINOMIAL. Pearson and Spearman correlation coefficients are employed to quantify the correlation between these explanations. With an increasing number of samples, the explanations from LIME and GLIME-BINOMIAL tend to show greater similarity. Notably, the dissimilarity and correlation of explanations between LIME and GLIME-BINOMIAL converge more rapidly when  $\sigma$  is higher.
Figure 9: The Top-  $K$  Jaccard index of explanations, computed with different references, consistently stays below 0.7, even when the sample size exceeds 2000.
Figure 10: The local fidelity of GLIME in the  $\ell_1$  neighborhood
A.5 The local fidelity of GLIME
Figure 5 presents the local fidelity of GLIME, showcasing samples from the  $\ell_2$  neighborhood  $\{\mathbf{z}||\mathbf{z} - \mathbf{x}||_2\leq \epsilon \}$  around  $\mathbf{x}$ . Additionally, Figure 10 and Figure 11 illustrate the local fidelity of GLIME within the  $\ell_1$  neighborhood  $\{\mathbf{z}||\mathbf{z} - \mathbf{x}||_1\leq \epsilon \}$  and the  $\ell_{\infty}$  neighborhood  $\{\mathbf{z}||\mathbf{z} - \mathbf{x}||_{\infty}\leq \epsilon \}$ , respectively.
A comparison between Figure 5 and Figure 10 reveals that, for the same  $\sigma$ , GLIME can explain the local behaviors of  $f$  within a larger radius in the  $\ell_1$  neighborhood compared to the  $\ell_2$  neighborhood. This difference arises from the fact that  $\{\mathbf{z}||\mathbf{z} - \mathbf{x}||_2\leq \epsilon \}$  defines a larger neighborhood compared to  $\{\mathbf{z}||\mathbf{z} - \mathbf{x}||_1\leq \epsilon \}$  with the same radius  $\epsilon$ .
Likewise, the set  $\{\mathbf{z}||\mathbf{z} - \mathbf{x}||_{\infty}\leq \epsilon \}$  denotes a larger neighborhood than  $\{\mathbf{z}||\mathbf{z} - \mathbf{x}||_2\leq \epsilon \}$ , causing the local fidelity to peak at a smaller radius  $\epsilon$  for the  $\ell_{\infty}$  neighborhood compared to the  $\ell_{2}$  neighborhood under the same  $\sigma$ .
Remarkably, GLIME-LAPLACE consistently demonstrates superior local fidelity compared to GLIME-GAUSS and GLIME-UNIFORM. Nevertheless, in cases with larger  $\epsilon$ , GLIME-GAUSS sometimes surpasses the others. This observation implies that the choice of sampling distribution should be contingent on the particular radius of the local neighborhood intended for explanation by the user.
Figure 11: The local fidelity of GLIME in the  $\ell_{\infty}$  neighborhood
A.6 GLIME unifies several previous methods
KernelSHAP [19]. KernelSHAP integrates the principles of LIME and Shapley values. While LIME employs a linear explanation model to locally approximate  $f$ , KernelSHAP seeks a linear explanation model that adheres to the axioms of Shapley values, including local accuracy, missingness, and consistency [19]. Achieving this involves careful selection of the loss function  $\ell(\cdot, \cdot)$ , the weighting function  $\pi(\cdot)$ , and the regularization term  $R$ . The choices for these parameters in LIME often violate local accuracy and/or consistency, unlike the selections made in KernelSHAP, which are proven to adhere to these axioms (refer to Theorem 2 in [19]).
Gradient [2, 27]. This method computes the gradient  $\nabla f$  to assess the impact of each feature under infinitesimal perturbation [2, 27].
SmoothGrad [28]. Acknowledging that standard gradient explanations may contain noise, SmoothGrad introduces a method to alleviate noise by averaging gradients within the local neighborhood of the explained input [28]. Consequently, the feature importance scores are computed as  $\mathbb{E}_{\epsilon \sim \mathcal{N}(0,\sigma^2\mathbf{I})}[\nabla f(\mathbf{x} + \epsilon)]$ .
DLIME [35]. Diverging from random sampling, DLIME seeks a deterministic approach to sample acquisition. In its process, DLIME employs agglomerative Hierarchical Clustering to group the training data, and subsequently utilizes k-Nearest Neighbour to select the cluster corresponding to the explained instance. The DLIME explanation is then derived by constructing a linear model based on the data points within the identified cluster.
ALIME [24]: ALIME leverages an auto-encoder to assign weights to samples. Initially, an auto-encoder, denoted as  $\mathcal{A}\mathcal{E}(\cdot)$ , is trained on the training data. Subsequently, the method involves sampling  $n$  nearest points to  $\mathbf{x}$  from the training dataset. The distances between these samples and the explained instance  $\mathbf{x}$  are assessed using the  $\ell_1$  distance between their embeddings, obtained through the application of the auto-encoder  $\mathcal{A}\mathcal{E}(\cdot)$ . For a sample  $\mathbf{z}$ , its distance from  $\mathbf{x}$  is measured as  $\| \mathcal{A}\mathcal{E}(\mathbf{z}) - \mathcal{A}\mathcal{E}(\mathbf{x})\|_1$ , and its weight is computed as  $\exp (-\| \mathcal{A}\mathcal{E}(\mathbf{z}) - \mathcal{A}\mathcal{E}(\mathbf{x})\|_1)$ . The final explanation is derived by solving a weighted Ridge regression problem.
A.7 Results on tiny Swin-Transformer [18]
The findings on the tiny Swin-Transformer align with those on ResNet18, providing additional confirmation that GLIME enhances stability and local fidelity compared to LIME. Please refer to Figure 12, Figure 13 and Figure 14 for results.
A.8 Comparing GLIME with ALIME
While ALIME [24] improves upon the stability and local fidelity of LIME, GLIME consistently surpasses ALIME. A key difference between ALIME and LIME lies in their methodologies: ALIME employs an encoder to transform samples into an embedding space, calculating their distance from the input to be explained as  $\| \mathcal{A}\mathcal{E}(\mathbf{z}) - \mathcal{A}\mathcal{E}(\mathbf{x})\| _1$ , whereas LIME utilizes a binary vector  $\mathbf{z}\in \{0,1\} ^d$  to represent a sample, measuring the distance from the explained input as  $\| 1 - \mathbf{z}\| _2$ .
Because ALIME relies on distance in the embedding space to assign weights to samples, there is a risk of generating very small sample weights if the produced samples are far from  $\mathbf{x}$ , potentially resulting in instability issues.
In our ImageNet experiments comparing GLIME and ALIME, we utilize the VGG16 model from the repositoryImagenet-autoencoder<sup>3</sup> as the encoder in ALIME. The outcomes of these experiments are detailed in Table 1. The findings demonstrate that, although ALIME demonstrates enhanced stability compared to LIME, this improvement is not as substantial as the improvement achieved by GLIME, particularly under conditions of small  $\sigma$  or sample size.
A.9 Experiment results on IMDb
The DistilBERT model is employed in experimental evaluations on the IMDb dataset, where 100 data points are selected for explanation. The comparison between GLIME-BINOMIAL and LIME
<sup>3</sup>https://github.com/Horizon2333/imagenet-autoencoder
(a) Stability across various methods (tiny SwinTransformer). The reported metric is the Top-20 Jaccard index.  $\mathrm{LIME} + \lambda = 0$  and  $\mathrm{LIME} + \pi = 1$  represent LIME without regularization and without weighting, respectively. LIME exhibits instability, particularly when  $\sigma$  is small, whereas GLIME demonstrates enhanced stability across varying  $\sigma$  values. Notably, in the absence of weighting or regularization, LIME's stability significantly improves when  $\sigma$  is small. Conversely, the impact of regularization and weighting on LIME's stability is marginal when  $\sigma$  is large.
(b)  $R^2$  comparison among LIME and GLIME variants with different sampling distributions (tiny Swin-Transformer). For each image and method, 2048 samples are utilized to compute the explanation and the corresponding  $R^2$ . Notably, LIME yields nearly zero  $R^2$  when  $\sigma = 0.25$  and 0.5, indicating an almost negligible explanation produced by LIME. In contrast, the  $R^2$  values of LIME are consistently lower than those of GLIME, underscoring that GLIME enhances the local fidelity of LIME.
Figure 12: GLIME markedly enhances both stability and local fidelity compared to LIME across various values of  $\sigma$ .
Table 1: Top-20 Jaccard Index of GLIME-BINOMIAL, GLIME-GAUSS, and ALIME. GLIME-BINOMIAL and GLIME-GAUSS exhibit significantly higher stability than ALIME, particularly in scenarios with small  $\sigma$  or limited samples.
# samples	128	256	512	1024
σ = 0.25	GLIME-BINOMIAL	0.952	0.981	0.993	0.998
GLIME-GAUSS	0.872	0.885	0.898	0.911
ALIME	0.618	0.691	0.750	0.803
σ = 0.5	GLIME-BINOMIAL	0.596	0.688	0.739	0.772
GLIME-GAUSS	0.875	0.891	0.904	0.912
ALIME	0.525	0.588	0.641	0.688
σ = 1	GLIME-BINOMIAL	0.533	0.602	0.676	0.725
GLIME-GAUSS	0.883	0.894	0.908	0.915
ALIME	0.519	0.567	0.615	0.660
σ = 5	GLIME-BINOMIAL	0.493	0.545	0.605	0.661
GLIME-GAUSS	0.865	0.883	0.898	0.910
ALIME	0.489	0.539	0.589	0.640
is depicted in Figure 15 using the Jaccard Index. Our findings indicate that GLIME-BINOMIAL consistently exhibits higher stability than LIME across a range of  $\sigma$  values and sample sizes. Notably, at smaller  $\sigma$  values, GLIME-BINOMIAL demonstrates a substantial improvement in stability compared to LIME.
(a) Top-1 Jaccard index of different methods (tiny Swin-Transformer).
(b) Top-5 Jaccard index of different methods (tiny Swin-Transformer).
(c) Top-10 Jaccard index of different methods (tiny Swin-Transformer).
(d) Average Jaccard index of different methods (tiny Swin-Transformer).
Figure 13: Top-1, 5, 10, and average Jaccard indices are calculated for various methods, and the average Jaccard index represents the mean of top-1,  $\dots$ ,  $d$  indices for the tiny Swin-Transformer.
Figure 14: Difference and correlation in LIME and GLIME-BINOMIAL explanations (tiny Swin-Transformer). Mean Squared Error (MSE) and Mean Absolute Error (MAE) quantify the divergence between LIME and GLIME-BINOMIAL explanations. Pearson and Spearman correlation coefficients gauge the correlation. With an increasing sample size, the explanations tend to align more closely. Notably, the difference and correlation converge more rapidly with larger values of  $\sigma$ .
(a) Stability comparison between GLIME-BINOMIAL and LIME. The top-20 Jaccard index is reported, illustrating that GLIME displays superior stability compared to LIME across diverse values of  $\sigma$ . Notably, GLIME's stability remains consistently robust, showing limited sensitivity to changes in  $\sigma$ . In contrast, LIME exhibits increased stability as  $\sigma$  values grow larger.
(b) Comparison of  $R^2$  between LIME and GLIME-BINOMIAL under various sampling distributions. Using 2048 samples for explanation computation,  $R^2$  values are computed for each image and method. It is noteworthy that when  $\sigma = 0.25$ , LIME exhibits nearly negligible  $R^2$  values.
Figure 15: GLIME significantly enhances both stability and local fidelity compared to LIME across various  $\sigma$  values.
B.1 Equivalent GLIME formulation without  $\pi (\cdot)$
By integrating the weighting function into the sampling distribution, the problem to be solved is
$$
\begin{array}{l} \mathbf {w} ^ {\text {G L I M E}} = \underset {\mathbf {v}} {\arg \min } \mathbb {E} _ {\mathbf {z} ^ {\prime} \sim \mathcal {P}} [ \pi (\mathbf {z} ^ {\prime}) \ell (f (\mathbf {z}), g (\mathbf {z} ^ {\prime})) ] + \lambda R (\mathbf {v}) \\ = \arg \min  _ {\mathbf {v}} \int_ {\mathbb {R} ^ {d}} \pi (\mathbf {z} ^ {\prime}) \ell (f (\mathbf {z}), g (\mathbf {z} ^ {\prime})) \mathcal {P} (\mathbf {z}) d \mathbf {z} + \lambda R (\mathbf {v}) \\ = \frac {\arg \min  _ {\mathbf {v}} \int_ {\mathbb {R} ^ {d}} \ell (f (\mathbf {z}) , g (\mathbf {z} ^ {\prime})) \pi (\mathbf {z} ^ {\prime}) \mathcal {P} (\mathbf {z}) \mathrm {d} \mathbf {z} + \lambda R (\mathbf {v})}{\int_ {\mathbb {R} ^ {d}} \pi (\mathbf {u} ^ {\prime}) \mathcal {P} (\mathbf {u}) \mathrm {d} \mathbf {u}} \\ = \arg \min  _ {\mathbf {v}} \frac {\int_ {\mathbb {R} ^ {d}} \ell (f (\mathbf {z}) , g \left(\mathbf {z} ^ {\prime}\right)) \pi \left(\mathbf {z} ^ {\prime}\right) \mathcal {P} (\mathbf {z}) \mathrm {d} \mathbf {z}}{\int_ {\mathbb {R} ^ {d}} \pi \left(\mathbf {u} ^ {\prime}\right) \mathcal {P} (\mathbf {u}) \mathrm {d} \mathbf {u}} + \frac {\lambda R (\mathbf {v})}{\int_ {\mathbb {R} ^ {d}} \pi \left(\mathbf {u} ^ {\prime}\right) \mathcal {P} (\mathbf {u}) \mathrm {d} \mathbf {u}} \\ = \arg \min  _ {\mathbf {v}} \int_ {\mathbb {R} ^ {d}} \ell (f (\mathbf {z}), g (\mathbf {z} ^ {\prime})) \widetilde {\mathcal {P}} (\mathbf {z}) d \mathbf {z} + \frac {\lambda}{Z} R (\mathbf {v}) \quad \widetilde {\mathcal {P}} (\mathbf {z}) = \frac {\pi (\mathbf {z} ^ {\prime}) \mathcal {P} (\mathbf {z})}{Z}, Z = \int_ {\mathbb {R} ^ {d}} \pi (\mathbf {u} ^ {\prime}) \mathcal {P} (\mathbf {u}) d \mathbf {u} \\ = \underset {\mathbf {v}} {\arg \min} \mathbb {E} _ {\mathbf {z} ^ {\prime} \sim \widetilde {\mathcal {P}}} [ \ell (f (\mathbf {z}), g (\mathbf {z} ^ {\prime})) ] + \frac {\lambda}{Z} R (\mathbf {v}) \\ \end{array}
$$
B.2 Equivalence between LIME and GLIME-BINOMIAL
For LIME,  $\mathcal{P} = \mathrm{Uni}(\{0,1\}^d)$  and thus  $\mathcal{P}(\mathbf{z}',\| \mathbf{z}'\| _0 = k) = \frac{1}{2^d},k = 0,1,\dots ,d,$  so that
$$
Z = \int_ {\mathbb {R} ^ {d}} \pi (\mathbf {u} ^ {\prime}) \mathcal {P} (\mathbf {u}) \mathrm {d} \mathbf {u} = \sum_ {k = 0} ^ {d} e ^ {(k - d) / \sigma^ {2}} \frac {\binom {d} {k}}{2 ^ {d}} = \frac {e ^ {- d / \sigma^ {2}}}{2 ^ {d}} (1 + e ^ {1 / \sigma^ {2}}) ^ {d}
$$
$$
\widetilde {\mathcal {P}} (\mathbf {z}) = \frac {\pi (\mathbf {z} ^ {\prime}) \mathcal {P} (\mathbf {z})}{Z} = \frac {e ^ {(k - d) / \sigma^ {2}} 2 ^ {- d}}{Z} = \frac {e ^ {k / \sigma^ {2}}}{(1 + e ^ {1 / \sigma^ {2}}) ^ {d}}
$$
Therefore, GLIME-BINOMIAL is equivalent to LIME.
B.3 LIME requires many samples to accurately estimate the expectation term in Equation 1.
In Figure 2, it is evident that a lot of samples generated by LIME possess considerably small weights. Consequently, the sample estimation of the expectation in Equation 1 tends to be much smaller than the true expectation with high probability. In such instances, the regularization term would have a dominant influence on the overall objective.
Consider specific parameters, such as  $\sigma = 0.25$ ,  $n = 1000$ ,  $d = 20$  (where  $\sigma$  and  $n$  are the default values in the original implementation of LIME). The probability of obtaining a sample  $\mathbf{z}'$  with  $\| \mathbf{z}' \|_0 = d - 1$  or  $d$  is approximately  $\frac{d}{2^d} + \frac{1}{2^d} = \frac{d + 1}{2^d} \approx 2 \times 10^{-5}$ . Let's consider a typical scenario where  $|f(\mathbf{z})| \in [0,1]$ , and  $(f(\mathbf{z}) - \mathbf{v}^\top \mathbf{z}')^2$  is approximately 0.1 for most  $\mathbf{z}, \mathbf{z}'$ . In this case,  $\mathbb{E}_{\mathbf{z}' \sim \mathrm{Uni}(\{0,1\}^d)} [\pi(\mathbf{z}') (f(\mathbf{z}) - \mathbf{v}^\top \mathbf{z}')^2] \approx 0.1 \cdot \sum_{k=0}^{d} \frac{e^{(k-d)/\sigma^2}}{2^d} \approx 10^{-7}$ . However, if we lack samples  $\mathbf{z}'$  with  $\| \mathbf{z}' \|_0 = d - 1$  or  $d$ , then all samples  $\mathbf{z}_i'$  with  $\| \mathbf{z}_i' \|_0 \leq d - 2$  have weights  $\pi(\mathbf{z}_i') \leq \exp(-\frac{2}{\sigma^2}) \approx 1.26 \times 10^{-14}$ . This leads to the sample average  $\frac{1}{n} \sum_{i=1}^{n} \pi(\mathbf{z}_i') (f(\mathbf{z}_i) - \mathbf{v}^\top \mathbf{z}_i')^2 \leq 1.26 \times 10^{-15} \ll 10^{-7}$ . The huge difference between the magnitude of the expectation term in Equation 1 and the sample average of this expectation indicates that the sample average is not an accurate estimation of  $\mathbb{E}_{\mathbf{z}' \sim \mathrm{Uni}(\{0,1\}^d)} [\pi(\mathbf{z}') (f(\mathbf{z}) - \mathbf{v}^\top \mathbf{z}')^2]$  (if we do not get enough samples). Additionally, under these circumstances, the regularization term is likely to dominate the sample average term, leading to an underestimation of the intended value of  $\mathbf{v}$ . In conclusion, the original sampling method for LIME, even with extensively used default parameters, is not anticipated to yield meaningful explanations.
Theorem B.1. Suppose samples  $\{\mathbf{z}_i^{\prime}\}_{i = 1}^{n}\sim \mathrm{Uni}(\{0,1\}^{d})$  are used to compute LIME explanation. For any  $\epsilon >0,\delta \in (0,1)$  , if  $n = \Omega (\epsilon^{-2}d^{5}2^{4d}e^{4 / \sigma^{2}}\log (4d / \delta)),\lambda \leq n$  , we have  $\mathbb{P}(\| \hat{\mathbf{w}}^{LIME} - \mathbf{w}^{LIME}\|_2 < \epsilon)\geq 1 - \delta$  .  $\mathbf{w}^{LIME} = \lim_{n\to \infty}\hat{\mathbf{w}}^{LIME}$
Proof. To compute the LIME explanation with  $n$  samples, the following optimization problem is solved:
$$
\hat {\mathbf {w}} ^ {\mathrm {L I M E}} = \arg \min _ {\mathbf {v}} \frac {1}{n} \sum_ {i = 1} ^ {n} \pi (\mathbf {z} _ {i} ^ {\prime}) (f (\mathbf {z} _ {i}) - \mathbf {v} ^ {\top} \mathbf {z} _ {i} ^ {\prime}) ^ {2} + \frac {\lambda}{n} \| \mathbf {v} \| _ {2} ^ {2}.
$$
Let  $L = \frac{1}{n}\sum_{i=1}^{n}\pi(\mathbf{z}_i')\left(f(\mathbf{z}_i) - \mathbf{v}^\top \mathbf{z}_i'\right)^2 + \frac{\lambda}{n}\|\mathbf{v}\|_2^2$ . Setting the gradient of  $L$  with respect to  $\mathbf{v}$  to zero, we obtain:
$$
- 2 \frac {1}{n} \pi (\mathbf {z} _ {i} ^ {\prime}) (f (\mathbf {z} _ {i}) - \mathbf {v} ^ {\top} \mathbf {z} _ {i} ^ {\prime}) \mathbf {z} _ {i} ^ {\prime} + \frac {2}{n} \lambda \mathbf {v} = 0,
$$
which leads to:
$$
\hat {\mathbf {w}} ^ {\text {L I M E}} = \left(\frac {1}{n} \sum_ {i = 1} ^ {n} \pi (\mathbf {z} _ {i} ^ {\prime}) \mathbf {z} _ {i} ^ {\prime} (\mathbf {z} _ {i} ^ {\prime}) ^ {\top} + \frac {\lambda}{n}\right) ^ {- 1} \left(\frac {1}{n} \sum_ {i = 1} ^ {n} \pi (\mathbf {z} _ {i} ^ {\prime}) \mathbf {z} _ {i} ^ {\prime} f (\mathbf {z} _ {i})\right).
$$
Denote  $\pmb{\Sigma}_{n} = \frac{1}{n}\sum_{i = 1}^{n}\pi (\mathbf{z}_{i}^{\prime})\mathbf{z}_{i}^{\prime}(\mathbf{z}_{i}^{\prime})^{\top} + \frac{\lambda}{n}$ ,  $\Gamma_{n} = \frac{1}{n}\sum_{i = 1}^{n}\pi (\mathbf{z}_{i}^{\prime})\mathbf{z}_{i}^{\prime}f(\mathbf{z}_{i})$ ,  $\pmb{\Sigma} = \lim_{n\to \infty}\pmb{\Sigma}_n$ , and  $\pmb{\Gamma} = \lim_{n\to \infty}\pmb{\Gamma}_n$ . Then, we have:
$$
\hat {\mathbf {w}} ^ {\text {L I M E}} = \boldsymbol {\Sigma} _ {n} ^ {- 1} \boldsymbol {\Gamma} _ {n}, \quad \mathbf {w} ^ {\text {L I M E}} = \boldsymbol {\Sigma} ^ {- 1} \boldsymbol {\Gamma}.
$$
To prove the concentration of  $\hat{\mathbf{w}}^{\mathrm{LIME}}$ , we follow the proofs in [8]: (1) First, we prove the concentration of  $\boldsymbol{\Sigma}_n$ ; (2) Then, we bound  $\| \boldsymbol{\Sigma}^{-1}\|_F^2$ ; (3) Next, we prove the concentration of  $\Gamma_n$ ; (4) Finally, we use the following inequality:
$$
\| \boldsymbol {\Sigma} _ {n} ^ {- 1} \boldsymbol {\Gamma} _ {n} - \boldsymbol {\Sigma} ^ {- 1} \boldsymbol {\Gamma} \| \leq 2 \| \boldsymbol {\Sigma} ^ {- 1} \| _ {F} \| \boldsymbol {\Gamma} _ {n} - \boldsymbol {\Gamma} \| _ {2} + 2 \| \boldsymbol {\Sigma} ^ {- 1} \| _ {F} ^ {2} \| \boldsymbol {\Gamma} \| \| \boldsymbol {\Sigma} _ {n} - \boldsymbol {\Sigma} \|,
$$
when  $\| \pmb{\Sigma}^{-1}(\pmb{\Sigma}_n - \pmb{\Sigma})\| \leq 0.32$  [8].
Before establishing concentration results, we first derive the expression for  $\Sigma$ .
Expression of  $\Sigma$