Paper: "What Data Benefits My Classifier?" Enhancing Model Performance and Interpretability through Influence-Based Data Selection (ICLR-2024)

Reviewer comment (Novelty):
At the beginning, this paper claims it is related to data valuation, data influence, and data efficiency. Essentially, this paper studies the problem of "coreset selection", which is not a new problem in machine learning. Coreset selection surely is related to the above topics. Therefore, it seems that there is no need to introduce so much redundant content in the main paper.

Author response:
_**Essentially, this paper studies the problem of "coreset selection", which is not a new problem in machine learning. Coreset selection surely is related to the above topics. Therefore, it seems that there is no need to introduce so much redundant content in the main paper.**_

    We would like to emphasize here that coreset selection and our work differ in multiple ways and our main motivation is not aligned with the motivation for coreset selection. We provide more details on these differences below:

    **Motivation and Research Question**: Coreset selection [1,2] is primarily associated with condensing and reducing the size of the training dataset for accelerating training and making training more efficient (for e.g. in deep learning) without sacrificing performance. In contrast, our work is motivated by assessing what type of data is generally beneficial/detrimental for the model by _interpreting the feature space_, and then using _data trimming/selection_ approaches to minimally modify the training set so as to improve performance with respect to _multiple functions of interest_. In our paper currently we had already cited both [1] and [2] (coreset selection approaches) and discussed them at a high-level, but we can describe these differences of motivation in comparison with our work in more detail in the revision.

    **Other Empirical Justification**: To further reinforce the differences in motivation, it can be seen that for coreset selection [1,2] the training data is reduced to be just about 20-30% of the original training dataset size (that is trimming of up to 80%). This is in stark contrast to our work, where we are minimally trimming the training set-- for example, in Figure 2 we only trim up to 5%. Note that we trim minimally in all experiments of our paper, and do not consider removing a large set of samples, as data efficiency is not our main concern.
    
    **Differences in Generality**: Different coreset selection methods generally need to be proposed for each function of interest, as it is non-trivial to apply a coreset selection method for accuracy to fairness [3]. This is also in stark contrast to our work, where we seek to have general and simple approaches that can easily be applied to multiple functions of interest (fairness, accuracy, robustness, etc.) in a trivial way. This makes our approaches more general than coreset selection, which would be more specific. This is also observable by the widespread application of our methods to problem scenarios that extend beyond conventional classification, such as online learning and active learning. For instance, it is not immediately clear how coreset selection could be applied to active learning.

Reviewer comment (Theory):
This paper uses a lot of space to introduce the previous versions of influence functions (Section 2). However, it is not clear that the difference between previous work and this work mathematically.

Author response:
_**This paper uses a lot of space to introduce the previous versions of influence functions (Section 2). However, it is not clear that the difference between previous work and this work mathematically.**_

     - The difference between previous work (such as Koh and Liang's work) is that those provide an approach for computing influence at the sample-level whereas we are trying to derive general "rule sets" at the feature-level for how features affect influence. For example, in Koh and Liang's work, they can obtain influence $I(x_i)$ for a sample $x_i$ (or change in output with sample-level perturbations). In our work, we use a large set of such samples (and their labels) and their influence values, to train an interpretable estimator, such as CART. This can then be used to derive a general interpretation for influence using the feature space-- for example some rules for Accuracy from our 2-D feature space in Figure 1 are: Feature 1 $>= 1.9$ and Feature 2 $< 0.1$ lead to high influence while Feature 1 $< 1.9$ and Feature 2 $< 0.1$ lead to low influence.

    - This sort of feature-level interpretation of performance gain via influence has not been undertaken in the literature previously. This can be useful in many scenarios, some of which we outline below but are not limited to only these: 

        **(1)** When new data needs to be added that we do not have influence scores for yet, we can use the influence estimator to obtain influence scores simply via inference. For Koh and Liang's approach, they would still need to train the model once with all samples under consideration every time a new sample arrives. 

        **(2)** In algorithmic data recourse scenarios [4] such as to describe why an individual was denied a loan, it can be used to provide an interpretable justification for why a decision was made simply by describing the tree rules that were either met or violated by the data sample.

        **(3)** In active learning, traditional influence computation (such as using Koh and Liang's work) is not viable since there is no access to the sample labels. Previous work such as ISAL [5] (one competitive method in our paper) have still utilized influence in these scenarios, but these do not work as well, since they use the base model itself to generate pseudo-labels for prediction. Our approach using the decision tree serves as an effective and much better alternative, as is evident in the active learning experiments of our paper (Section 5.5 - Figure 6E), where we compare with ISAL. 

        **(4)** In some sense, we are bridging feature-interpretation approaches (such as Shapley values and LIME) with sample-interpretation approaches (such as Data Valuation and Influence), and hence, can accommodate any applications that these are used for.

Reviewer comment (Writing):
Could the paper provide more high-level intuitions about the formulas of the overall regression tree prediction and hierarchical shrinkage regularizes?

Author response:
_**Could the paper provide more high-level intuitions about the formulas of the overall regression tree prediction and hierarchical shrinkage regularizes?**_

    We apologize for any lack of clarity and will make these descriptions clearer in the main text. 

    **Overall Regression Tree Prediction**: Since the predictions for a particular node are dependent on where it lies in the tree, the tree model prediction can be denoted as a telescoping sum. We start with the average prediction response at the root node and then keep summing over the individual prediction response at each node (calculated as differences in the average prediction response between successive nodes). This provides us with the prediction at a particular query leaf in the tree. 

    **Hierarchical Shrinkage Regularization**: Without hierarchical shrinkage the tree might have some leaves that noisily influence estimates as they have only a few samples. This can affect both interpretability of the tree as well as predictive capability. To counteract this, hierarchical shrinkage is a powerful yet simple approach. Here, we replace the average prediction response over a leaf with a weighted average of the mean or average responses over the leaf and each of its ancestors. The weights depend on the number of samples in each leaf, and are controlled by the regularization parameter $\lambda$ specified at run-time. Mathematically, this is observable with the slight modification to the summation over individual predictive responses at nodes where we divide the predictive response by the number of samples the node contains as well as $\lambda$, the regularization parameter.

Reviewer comment (Theory):
For the method in Section 3.2, what is its time/space complexity?

Author response:
R2+R3: _**For the method in Section 3.2, what is its time/space complexity?**_

    Our algorithms are computationally efficient and can scale with large datasets. We provide analytical details on the time complexity below.

    **Algorithm 1 Time Complexity**: For calculating computational complexity, we need to analyze the two steps of our influence estimation process. First, the influence computation of training samples and a given base model requires $\mathcal{O}(npm)$ time where $n,p$, and $m$ are number of training samples, model parameters, and test samples respectively. Please refer to Section 3 of Koh and Liang's paper for how influence can be computed in $\mathcal{O}(npm)$ using stochastic estimation or conjugate gradients. Second, the cost of training time of the influence estimation tree in the worst case is $\mathcal{O}(n^2 d)$ where $d$ is the number of features. Thus, the overall complexity is $\mathcal{O}(npm) + \mathcal{O}(n^2 d)$. Note that the quadratic dependence of the second term can be made linear in $n$ simply be using more sophisticated tree-building methods [6] which have a time complexity of $\mathcal{O}(nd)$. Since the overall time complexity will then be linear: $\mathcal{O}(n(pm + d))$, the approaches proposed are amenable to large scale datasets (CelebA in our experiments has 104K samples). Furthermore, our approaches work with very high-dimensional data as well. As shown in experiments on the image (CelebA) and NLP (Jigsaw Toxicity) datasets we consider in the paper, we utilize embeddings obtained by deep learning models as a preprocessing step prior to our approach. This significantly reduces the dataset feature size, and at the same time allows for a rich and powerful feature space that can be used to train our models/methods.

* _**For the method in Section 3.2, what is its time/space complexity?**_ (Continued)

    **Algorithm 2 Time Complexity**: The time complexity of Algorithm 2 is easier to calculate. Note that the influence values provided once at input require $\mathcal{O}(npm)$ to calculate as mentioned before. Then the sorting step requires $\mathcal{O}(n log(n))$ in the worst case. The other operations (lines 8-9) are linear vector operations, leading to a worst case time complexity of $\mathcal{O}(n(pm + log(n))$. This is not a computational bottleneck for large datasets.

Reviewer comment (Novelty):
Could the paper discuss the difference between this paper and the work [1]?

Author response:
_**Could the paper discuss the difference between this paper and the work (Yang et. al) [7]?**_

    Thank you for referring this work by Yang et al [7]. We had cited it in our paper in Section 2 under *Extension to Non-convex Models* as it utilized influence functions in ResNets. However, it differs significantly from our work with regards to motivation. Based on the reviewer's feedback we will discuss it in the main paper in contrast with our work as follows: 
    
    _The goal of Yang et al differs from ours with regards to motivation. The authors seek to reduce/trim the size of the training dataset as much as possible while ensuring that model performance (utility) on the new training set is relatively similar to the original dataset. In contrast, our goal is to trim a very small percentage of the dataset, while ensuring that model performance is improved with respect to multiple functions of interest (fairness, utility, robustness)._

Reviewer comment (Writing):
When I read feature space, I think of d-dimensions where the variables (features) live. The authors' writing was a bit confusing to me because from the abstract to the introduction, I thought their influence estimation-based method was identifying features from the feature space with positive/negative influence on the model utility (accuracy, fairness, robustness, and so on). However, at the beginning of the background section and throughout the experiment results, the authors focus on only the contribution of the training samples to the utility function. I think the authors should be a bit more clear in the writing or presentation. Although section 3 is fairly written, I would recommend that authors revisit abstract+sections 1-3.

Author response:
Thank you for the concrete suggestion for improving our paper. We will work to improve the writing and readability, and add a section on _Problem Formulation_ that clarifies the presentation of the problem.

Reviewer comment (Experiments):
Since the authors focus on features and samples, it would have been informative to see the difference in selected/excluded features and samples and the consequential contribution to the utility with and without the authors' method.

Author response:
If we are correct in understanding the reviewer's suggestion, you would like to see the difference in performance and utility when our approach is used and when it is not (that is, the performance of the original model). This is currently observable in Figure 2 for each subfigure when the x-axis value = 0 (0% samples trimmed or excluded). Note that this will be a constant value throughout as there is little variance in the output of logistic regression. If the reviewer intended something else, we can provide additional details during the discussion phase.

Reviewer comment (Reproducibility):
Although influences functions are not affected by retraining-related complexity, they have a high incremental complexity due to the computation of the Hessian matrix for each x_{i} valuation, which might worsen (beyond retraining) when n is large. Additionally, using CART as a sub-module further increases model complexity. I would have appreciated looking at the code specific to section E.1 in the appendix (I couldn't find it in the shared code base).

Author response:
- We apologize for not including the Appendix E.1 code with the provided codebase as we primarily used the original code of TMC-Shapley (Ghorbani et al). However, we have added this experiment in the following code repository: <https://anonymous.4open.science/r/ICLR_Rebuttal_Experiments> for the reviewer to go through. This also has implementations of additional baselines which we discuss in a subsequent response. Next, we discuss analytical details on time complexity of our algorithms. 

- The tree-based influence model of Algorithm 1 is computationally efficient and can scale with large datasets. It has a training time of $\mathcal{O}(npm) + \mathcal{O}(n^2 d)$ in the worst case where $n$, $p$, $m$, and $d$ are the number of training samples, the number of model parameters, the number of test samples, and the number of features, respectively. The first term is for the influence computation step (please refer to Section 3 of Koh and Liang's paper for how influence can be computed in $\mathcal{O}(npm)$ using stochastic estimation or conjugate gradients) and the second is for training and constructing the tree model. The biggest factor here is the quadratic dependence on the number of training samples, which can be made linear in $n$ simply by using more sophisticated tree-building methods [1] that have a time complexity of $\mathcal{O}(nd)$. This would result in an improved overall time complexity of $\mathcal{O}(n(pm + d))$. Thus the model can scale easily to larger datasets on modern hardware. This is also evident in our experiments in the paper on the CelebA dataset that has 104K samples and runs computationally efficiently even with a standard tree-building algorithm such as CART (it takes 2.455 seconds to run on average on a Linux server with an Intel Xeon 2.2GHz CPU).

Reviewer comment (Presentation):
I am not entirely sure, probably it's the figure, but I find the almost constant utility values with random deletion somewhat unrealistic. Could the authors also explain Figure 2C? The scale for accuracy on some figures in 2 is not intuitive. Is it possible for authors to adopt similar scales for similar utilities across datasets?

Author response:
We understand the reviewer's concern regarding the figure scales; unfortunately, employing same scales across the different subfigures for a function of interest would make it challenging to observe the differences in the trends across different datasets. For example consider accuracy-- for Adult the values vary from 0.826 to 0.832, for Bank they vary from 0.80 to 0.90, and for Jigsaw Toxicity they vary from 0.735 to 0.744. Thus, using the same scale (say 0.735 to 0.90) would make all the trends for datasets except Bank look like a constant line. This is because the metric values are highly dataset-dependent and comparative analysis necessitates individual scales for each dataset.

Reviewer comment (Evaluation):
Figure 2 specific questions: I find the almost constant utility values with random deletion somewhat unrealistic. Could the authors also explain Figure 2C?

Author response:
- We understand the reviewer's point of view regarding constant utility values, but this is not as surprising as the very small number of randomly sampled points that are then deleted (less than 5%) might not necessarily be contributing to utility in a significant way. This trend is also observable in other work on data valuation. For example in the Data Shapley paper by Ghorbani et al, observe Figure 1 (a) and (b) and only view the scale till 5% (as the authors there consider up to 40% removals). It is evident that random deletion does not change the results that much up until a very large portion of the data is removed.

- Regarding Figure 2C, since CelebA is a high-dimensional complex image dataset and we are using a logistic regression model the accuracy value obtained is most likely an upper bound and the model cannot improve upon it further. That is why even after removing low influence samples, we do not see performance improvements (but it also does not drop). With random deletion, since the scale is fairly zoomed in between 0.853 and 0.854 the variance is possibly leading to more of an observable trend (it drops below the original value as well at times). Note that when we use the MLP model on CelebA which has a non-linear decision boundary we can actually improve upon accuracy much more significantly (as is observable in Figure 8C in the Appendix), backing the above hypothesis.

Reviewer comment (Evaluation):
Figure 10 in the appendix: If you're removing low-value samples, I wouldn't expect TMC-Shapley to behave like that — accuracy would increase with the removal of low-value samples. If you're trimming high-value examples, then this graph would make sense but would mean influence-based trimming is performing poorly.

Author response:
We understand the reviewer's concern. We are removing low-influence samples. We had used the original code provided by Ghorbani et al for TMC-Shapley and obtained the trends shown. However, to verify our results, we have run their approach again with and reduced the error tolerance parameter to 0.05 (the original default was 0.1) but it still does not perform satisfactorily. **We have provided new results for the subsampled Bank dataset in Figure 27 which we have added on Page 33 (Section R3) of our revised paper PDF**. It can be seen that in this instance with the smaller subsampled Bank dataset (training set has 100 samples, and test set is of the original Bank), influence-based data selection approaches perform better than TMC-Shapley. The code for these experiments is located here: <https://anonymous.4open.science/r/ICLR_Rebuttal_Experiments>.

Reviewer comment (Experiments):
Instead of TMC-Shapley and random, it would have been more informative to see how the proposed approach compares with other influence estimation-based approaches, including vanilla (without CART) influence estimation.

Author response:
Thank you for the concrete suggestion for improvement. To undertake this experiment, we have considered the original/vanilla influence computation approach by Koh and Liang, alongside our influence tree estimation implementation. We have also considered a more recent yet simplistic approach based on model self-confidence [2] that has been used to detect issues with training samples (such as label errors) as a baseline for comparison along with TMC-Shapley. **These new results are provided for the subsampled Bank dataset in Figure 27 which we have added on Page 33 (Section R3) of our revised paper PDF**. It is evident that our performance (red line) is superlative to the other baselines. The code for these experiments is located here: <https://anonymous.4open.science/r/ICLR_Rebuttal_Experiments>.

Reviewer comment (Presentation):
The scale for accuracy on some figures in 2 is not intuitive. Is it possible for authors to adopt similar scales for similar utilities across datasets?

Author response:
We understand the reviewer's concern regarding the figure scales; unfortunately, employing same scales across the different subfigures for a function of interest would make it challenging to observe the differences in the trends across different datasets. For example consider accuracy-- for Adult the values vary from 0.826 to 0.832, for Bank they vary from 0.80 to 0.90, and for Jigsaw Toxicity they vary from 0.735 to 0.744. Thus, using the same scale (say 0.735 to 0.90) would make all the trends for datasets except Bank look like a constant line. This is because the metric values are highly dataset-dependent and comparative analysis necessitates individual scales for each dataset.

Reviewer comment (Theory):
While the focus on convex loss is understandable, it might lead to sub-optimal influence value estimation due to the model parameters not being at a stationary point or the model not converging. This might then be a net negative and misleading data value estimation.

Author response:
We agree with the reviewer, the fundamental limitation of the convexity assumption is that for non-convex models it is not possible to derive theoretical guarantees for the efficacy of influence functions. However, despite this lack of provable theoretical guarantee, by adding a damping term to the model's Hessian or using convex surrogates, our experiments on non-convex models such as MLPs and (preliminary results on) BERT in the appendix, we obtain significant performance improvements using our proposed approaches for multiple functions of interest. Despite this, as the reviewer pointed out there might be cases where they do not work as expected.

Reviewer comment (Experiments):
It looks like the authors do one utility at a time. Due to often competing utilities, for example, key features and samples for fairness might not necessarily be the same for accuracy, and in most cases might have a negative influence. It would be interesting to see an interplay of various utilities.

Author response:
Thank you for the great suggestion, we can do this using our current obtained results. By observing the joint overlapping influence regions (or influence tree rules) for multiple functions of interest we can assess what samples contribute positively or negatively to both the functions of interest being considered (fairness, utility, etc.). In the current version of our paper, we did not do this since visualization for this is somewhat non-trivial and messy; however, in practice this is easy to do using our approach. **To exemplify this, we provide new analysis for the synthetic toy dataset of Section 4, for both accuracy + fairness in Figure 25 (Section R1, Page 32) of our revised paper PDF, and for accuracy + robustness in Figure 26 (Section R2, Page 32) of our revised paper PDF**. It can be seen that there are certain samples that are beneficial to remove for both accuracy and robustness, but for fairness and accuracy there is no overlap. Through this, we would like to emphasize that making joint improvements along multiple functions of interest is a dataset-dependent property and at times it might not be possible to make improvements along both functions (for e.g. the fairness-accuracy tradeoff [3]).

Reviewer comment (Experiments):
Although authors use several datasets, all of them are binary settings. Value computation increases with classes, so I am curious to know if this is the reason authors only focused on binary settings or if there is another reason behind this design choice.

Author response:
The choice for utilizing binary classification datasets are based on three factors. **First**, a number of approaches and work in fairness explicitly consider binary classification [3-5] as it is easier to interpret outcomes, and since we wanted to compare with these fairness baselines, we sought to utilize binary datasets throughout the paper. For example for works in both Section 5.1 (fairness distribution shift) and Section 5.2 (fairness poisoning attacks) datasets have binary classes. **Second**, we believe and hope that since our approaches are general, the binary case can be empirically extended to the multi-class case in future work without too much of a challenge. **Third**, a minor consideration was to have _neater_ looking trees when visualized. In the binary case, the class label can only take on two values, but in the multi-class case it would take on many different combinations and interplay with features in numerous ways leading to more complex trees.

Reviewer comment (Theory):
Limitation on model class: The authors provide a discussion on why the influence function evaluations are limited to convex classifiers, possible remedies, and recent applications to deep neural networks. However,

Author response:
It seems that perhaps this question was accidentally cut-off. If the reviewer can provide more details, we will aim to provide a targeted response and justification during the discussion phase.

Reviewer comment (Theory):
Theoretical analysis: the estimation of the influence function is based on the trees with hierarchical shrinkage regularization. However, there is no analysis on the credibility, time complexity of the proposed Algorithm 1 and Algorithm 2. It seems that these algorithms are not scalable to large-scale datasets.

Author response:
Both our proposed algorithms are very computationally efficient and can scale with large datasets. We provide more analytical details on time complexity below.

**Algorithm 1 Time Complexity**: For calculating computational complexity, we need to analyze the two steps of our influence estimation process. First, the influence computation of training samples and a given base model requires $\mathcal{O}(npm)$ time where $n,p$, and $m$ are number of training samples, model parameters, and test samples respectively. Please refer to Section 3 of Koh and Liang's paper for how influence can be computed in $\mathcal{O}(npm)$ using stochastic estimation or conjugate gradients. Second, the cost of training time of the influence estimation tree in the worst case is $\mathcal{O}(n^2 d)$ where $d$ is the number of features. Thus, the overall complexity is $\mathcal{O}(npm) + \mathcal{O}(n^2 d)$. Note that the quadratic dependence of the second term can be made linear in $n$ simply be using more sophisticated tree-building methods [1] which have a time complexity of $\mathcal{O}(nd)$. Since the overall time complexity will be linear, the approaches proposed are amenable to large scale datasets (CelebA in our experiments has 104K samples). Furthermore, our approaches work with very high-dimensional data as well. As shown in experiments on the image (CelebA) and NLP (Jigsaw Toxicity) datasets we consider in the paper, we utilize embeddings obtained by deep learning models as a preprocessing step prior to our approach. This significantly reduces the dataset feature size, and at the same time allows for a rich and powerful feature space that can be used to train our models/methods.

**Algorithm 2 Time Complexity**: The time complexity of Algorithm 2 is easier to calculate. Note that the influence values provided once at input require $\mathcal{O}(npm)$ to calculate as mentioned before. Then the sorting step requires $\mathcal{O}(n log(n))$ in the worst case. The other operations (lines 8-9) are linear vector operations, leading to a worst case time complexity of $\mathcal{O}(n(pm + log(n))$. This is not a computational bottleneck for large datasets as the time complexity is linear or logarithmic for each factor.

Reviewer comment (Presentation):
The utility, fairness and adversarial robustness are important performance metrics for a classifier; however, there is a lack of a unifying story to connect all three and therefore the discussion and experiments may seem distracted.

Author response:
Thank you for the great suggestion, we can do this using our current obtained results. By observing the joint overlapping influence regions (or influence tree rules) for multiple functions of interest we can assess what samples contribute positively or negatively to both the functions of interest being considered (fairness, utility, etc.). In the current version of our paper, we did not do this since visualization for this is somewhat non-trivial and messy; however, in practice this is easy to do using our approach. **To exemplify this, we provide new analysis for the synthetic toy dataset of Section 4, for both accuracy + fairness in Figure 25 (Section R1, Page 32) of our revised paper PDF, and for accuracy + robustness in Figure 26 (Section R2, Page 32) of our revised paper PDF**. It can be seen that there are certain samples that are beneficial to remove for both accuracy and robustness, but for fairness and accuracy there is no overlap. Through this, we would like to emphasize that making joint improvements along multiple functions of interest is a dataset-dependent property and at times it might not be possible to make improvements along both functions (for e.g. the fairness-accuracy tradeoff [2]).

Reviewer comment (Novelty):
Feature explanation is a key aspect in this paper; however, the connection of feature explanation using the influence function with existing explainable AI literature is lacking.

Author response:
Thank you for the great suggestion for strengthening the paper. We will add a paragraph for discussion on this in the main paper as follows:

_Most explainable and interpretable ML/AI approaches leverage feature importances to understand why models decided on a particular class label for a sample. For instance in [3], the goal is to obtain what regions of the image contribute most to the model picking that label for the given image. This interpretability by means of explaining sample-level model predictions contrasts with our work on interpretability, where we aim to interpret what samples/features benefit the model or lead to its detriment with respect to a function of interest (fairness, utility, etc.)._

Reviewer comment (Experiments):
Should not the influence estimator have the same architecture of the classifier?

Author response:
Since we would like to obtain interpretable rule sets/decisions for what features contribute positively or negatively to the function of interest, we opt for a decision tree regressor. However, since the learning problem is in general a regression problem where the inputs are sample feature, label tuples and targets are influence values, any interpretable classifier can be used. It is not necessary to utilize the same model or architecture as the original base model.

Reviewer comment (Experiments):
For the fairness experiments in Section 5.1, would the authors justify the choice of the fairness intervention baselines?

Author response:
The baseline approaches for comparison in Section 5.1 were selected based on multiple factors. **First**, due to the vast literature on fairness, we aimed to select methods that were recently proposed (2020-2023). **Second**, our aim was to compare with approaches that belonged to the preprocessing category of fairness interventions, since our approach is also preprocessing based. Except for the Robust Fair Logloss Classifier (RLL) which is an in-processing approach, all baselines belong to this category. **Third**, our aim was to select approaches that have been specifically used for mitigating fairness distribution shift or have robustness properties so as to counter fairness shift. All the approaches considered (Correlation Shift, Influence-based Reweighing, FairBatch, and RLL) meet this criteria. Additionally, Influence-based Reweighing is a recent approach from 2022 that also utilizes influence functions, so our aim was to show how our proposed approach can improve performance even more when used in conjunction with it. We are happy to add more as well if the reviewer would like to point out any key missing approaches for comparison.

Reviewer comment (Experiments):
**Scalability Concerns**: The use of tree-based influence estimation models might indeed pose scalability issues. Tree-based models can become computationally expensive as the size of the dataset increases, especially if the influence estimation requires building trees for many subsets of data or for complex feature interactions. This could limit the method's applicability to big data scenarios or require significant computational resources, which may not always be feasible.

Author response:
The tree-based influence models are computationally efficient and can scale with large datasets. They have a training time of $\mathcal{O}(npm) + \mathcal{O}(n^2 d)$ in the worst case where $n$, $p$, $m$, and $d$ are the number of training samples, model parameters, test samples, and features, respectively. The first term is for the influence computation step (please refer to Section 3 of Koh and Liang's paper for how influence can be computed in $\mathcal{O}(npm)$ using stochastic estimation or conjugate gradients) and the second is for training and constructing the tree model. The biggest factor here is the quadratic dependence on the number of training samples for constructing the tree, which can be made linear in $n$ simply by using more sophisticated tree-building methods [1], which have a time complexity of $\mathcal{O}(nd)$. This will result in an overall time complexity of $\mathcal{O}(n(pm + d))$. Thus the model can scale easily to larger datasets on modern hardware. This is also evident in our experiments in the paper on the CelebA dataset that has 104K samples and runs computationally efficiently even with a standard tree-building algorithm such as CART (it takes 2.455 seconds to run on average on a Linux server with an Intel Xeon 2.2GHz CPU).

Reviewer comment (Experiments):
**Hard to Adopt Data with High-Dimensional Features**: For example, image data presents unique challenges due to its high dimensionality and the spatial relationships between pixels. Influence functions and feature space interpretations that work well for tabular data may not translate directly to image data.

Author response:
Our approaches can be used efficiently with high-dimensional image and text datasets simply by employing a feature extraction preprocessing step on the raw data such as by obtaining embeddings from a deep learning model or some other approach. This is how we conduct experiments on CelebA and Jigsaw Toxicity datasets. For e.g., we use the Mini-LM transformer model to get embeddings for the Jigsaw Toxicity dataset and then conduct experiments on it. This significantly reduces the dataset feature size which further reduces the overall worst case time complexity. Additionally, we would also like to mention that past work has found that pixel-level interpretation of images is not akin to human interpretation/reasoning, and hence, prototype-based explanations are generally preferred [2]. Hence, for raw image data, using a feature extraction step for preprocessing is a viable option for improved interpretation.

Reviewer comment (Experiments):
How do the tree-based influence estimation models proposed by the authors scale with very large datasets, and what are the computational costs associated with these models?

Author response:
The tree-based influence models are computationally efficient and can scale with large datasets. They have a training time of $\mathcal{O}(npm) + \mathcal{O}(n^2 d)$ in the worst case where $n$, $p$, $m$, and $d$ are the number of training samples, model parameters, test samples, and features, respectively. The first term is for the influence computation step (please refer to Section 3 of Koh and Liang's paper for how influence can be computed in $\mathcal{O}(npm)$ using stochastic estimation or conjugate gradients) and the second is for training and constructing the tree model. The biggest factor here is the quadratic dependence on the number of training samples for constructing the tree, which can be made linear in $n$ simply by using more sophisticated tree-building methods [1], which have a time complexity of $\mathcal{O}(nd)$. This will result in an overall time complexity of $\mathcal{O}(n(pm + d))$. Thus the model can scale easily to larger datasets on modern hardware. This is also evident in our experiments in the paper on the CelebA dataset that has 104K samples and runs computationally efficiently even with a standard tree-building algorithm such as CART (it takes 2.455 seconds to run on average on a Linux server with an Intel Xeon 2.2GHz CPU).

Reviewer comment (Theory):
Could the authors provide insights into the computational complexity of their influence estimation approach, and are there any strategies they recommend for scaling it to big data applications?

Author response:
Our approaches can be used with big data. For calculating computational complexity, please refer to the previous response. As mentioned before, even though the overall time complexity is $\mathcal{O}(npm) + \mathcal{O}(n^2 d)$, using more sophisticated tree-building methods [1] we can improve time complexity to $\mathcal{O}(n(pm + d))$ which is linear in $n$. Furthermore, for very high-dimensional data ($d$ is large), a feature extraction step can reduce time complexity considerably. For the image (CelebA) and NLP (Jigsaw Toxicity) datasets we consider in the paper, we utilize embeddings obtained by deep learning models as a preprocessing step prior to our approach. This significantly reduces the dataset feature size, and at the same time allows for a rich and powerful feature space that can be used to train our models/methods.

Reviewer comment (Experiments):
How does the tree model handle high-dimensional data, such as images, where feature interactions are more complex?

Author response:
Our approaches can be used efficiently with high-dimensional image and text datasets simply by employing a feature extraction preprocessing step on the raw data such as by obtaining embeddings from a deep learning model or some other approach. This is how we conduct experiments on CelebA and Jigsaw Toxicity datasets. For e.g., we use the Mini-LM transformer model to get embeddings for the Jigsaw Toxicity dataset and then conduct experiments on it. This significantly reduces the dataset feature size which further reduces the overall worst case time complexity. Additionally, we would also like to mention that past work has found that pixel-level interpretation of images is not akin to human interpretation/reasoning, and hence, prototype-based explanations are generally preferred [2]. Hence, for raw image data, using a feature extraction step for preprocessing is a viable option for improved interpretation.

Reviewer comment (Experiments):
Could the authors elaborate on any modifications or extensions to their approach that might be necessary to apply it effectively to image data or other high-dimensional datasets?

Author response:
Our approaches can be used efficiently with high-dimensional image and text datasets simply by employing a feature extraction preprocessing step on the raw data such as by obtaining embeddings from a deep learning model or some other approach. This is how we conduct experiments on CelebA and Jigsaw Toxicity datasets. For e.g., we use the Mini-LM transformer model to get embeddings for the Jigsaw Toxicity dataset and then conduct experiments on it. This significantly reduces the dataset feature size which further reduces the overall worst case time complexity. Additionally, we would also like to mention that past work has found that pixel-level interpretation of images is not akin to human interpretation/reasoning, and hence, prototype-based explanations are generally preferred [2]. Hence, for raw image data, using a feature extraction step for preprocessing is a viable option for improved interpretation.

Reviewer comment (Theory):
The work presented focuses on convex models or convex surrogates for non-convex models. Could the authors discuss the potential limitations of this?

Author response:
The most fundamental limitation of the convexity assumption is that for non-convex models it is not possible to derive theoretical guarantees for the efficacy of influence functions. However, despite this lack of provable theoretical guarantee, in practice this does not invalidate the use of influence functions in non-convex models. By adding a damping term to the model's Hessian or using convex surrogates, influence functions can obtain satisfactory results. For instance, in our experiments on non-convex models such as MLPs and (preliminary results on) BERT in the appendix, we obtain performance improvements using our proposed approaches for multiple functions of interest.

Reviewer comment (Reproducibility):
**Details of the approach**: The exact procedure of the trimming portion is not quite clear to me. I think the authors miss discussing retraining. I assume that the authors are referring to a model retrained after a subset of examples are deleted? So in Fig. 2, x axis==zero is a model trained on all data points? Then you trim a percent of the training samples and then retrain a model on the new dataset? If yes, is it the original model that is used for deciding which samples to trim or is the model changing?

Author response:
* **Details of the approach**: We apologize for any lack of clarity in presenting the approach.

    - > I assume that the authors are referring to a model retrained after a subset of examples are deleted?

        Yes, the reviewer is correct. 
    
    - > So in Fig. 2, x axis==zero is a model trained on all data points?

        Yes, in Figure 2, x-axis = 0 the model is trained on all data samples.

    - > Then you trim a percent of the training samples and then retrain a model on the new dataset?

        Yes, the reviewer is correct, this is precisely what we do.

    - > If yes, is it the original model that is used for deciding which samples to trim or is the model changing?

        The original unchanged model is used to decide which samples to trim and then we sequentially trim the samples to the desired percentage. 
    

    Thank you for these details. We will make the above points clearer in the text.
    
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Reviewer comment (Novelty):
**What is the motivation behind the cart regression procedure?**: As it stands it seems the cart procedure takes as input $(x_i, y_i)$, and the predicts some influence score per example? More details could be useful here. Are the samples used in the training of the cart model a subset of the original training set for which the influence was estimated? Since we know that the influence score measures the effect of up(down)weighting the training sample, alone, we also know that the label should not have any effect on predictive quality of the tree. What is the point of then concatenating the label? It seems like the goal here is to estimate the effect of a feature on the performance metric of interest. I take this judging from Figure 1 where the authors plot performance metric vs features that is colored by influence. If the goal is really to determine the effect of a feature on the performance metric of interest, then how to do that is already in section 2.2 of the original Koh and Liang paper. If the goal is not to measure the effect of the feature on the influence score, then I am not sure I understand the point of this section. Another point here is that in the rest of the paper, the trimming-based approach is really what the authors use and not the cart procedure. If this is the case, I don't think we can that as a contribution of this work. I am asking all these questions as a way to better understand the motivation and goal of fitting the tree to predict the estimated influence score.

Author response:
* **What is the motivation behind the CART regression procedure?**: Thank you for these insightful questions. We would first like to illustrate our motivation/goal and then address these questions one by one. Our goal is to discover a subset of the training samples that benefit the learning model (via influence functions to trim the samples that are a detriment to the model) and interpret where the benefits come from in the feature space (via CART regression). 

    - > Are the samples used in the training of the CART model a subset of the original training set for which the influence was estimated?

        Yes, the reviewer is correct, we use an 80-20 split with the original samples (concatenated with their labels) for training the CART model that will then be used to predict the influence value. 

    - > Since we know that the influence score measures the effect of up(down)weighting the training sample, alone, we also know that the label should not have any effect on predictive quality of the tree. What is the point of then concatenating the label?

        The label implicitly contributes to the computation of the influence function. Since the influence function estimation utilizes the gradient of the loss term $l$, which takes in as input both $x_i$ (training sample) and $y_i$ (class label), the label also contributes to the influence computation. This is evident in Eqs. (1-3) in our paper and Eq. (2) in Koh and Liang's paper, where they use $z_i$ which is a tuple = $(x_i, y_i)$. This is why we concatenate the label for influence estimation.

Reviewer comment (Novelty):
**Related Work**: There is some related work that this paper should be aware of. I list them here: Kong et. al., Resolving Training Biases via Influence-based Data Relabeling, Adebayo et. al. Quantifying and mitigating the impact of label errors on model disparity metrics, Richardson et. al. Add-Remove-or-Relabel: Practitioner-Friendly Bias Mitigation via Influential Fairness, (concurrent) Understanding Unfairness via Training Concept Influence, Sattigerri et. al. Fair infinitesimal jackknife: Mitigating the influence of biased training data points without refitting. All of these papers have a trimming and/or relabelling scheme in them. I am not claiming that this work is not novel/important. I think the insights here are quite useful actually, but it would be helpful for the authors to acknowledge these works, and contextualize their contributions in light of these papers.

Author response:
* **Related Work**: Thank you for bringing these papers to our notice, it is highly appreciated and will strengthen the discussion in our paper. We will cite these and include discussion on them in the main text. We contextualize them with regards to our approaches below:

    * A major difference between our work and [3-7] is that none of these works focus on feature-level interpretability approaches that can allow practitioners to understand and interpret what feature ranges benefit the model or lead to its detriment. There are other differences as well, which we discuss individually below. 

    * [3,4,5]: In [3], authors focus specifically on relabeling schemes for harmful training samples using influence functions. While thematically related to improving model performance using influence, the approach taken in our work contrasts with theirs in terms of overall motivation and generality of methods. Our data selection methods and feature-level interpretations via influence are extremely general and simple, and can hence be applied to fairness, accuracy, robustness, and many scenarios such as distribution shift, adversarial attacks, active learning, etc. The authors in [4,5] extend the ideas proposed in [3] to fairness metrics, with [5] also allowing for addition and removal of samples specific to the profile of the practitioner. However, it is important to note that based on [3] and [4], different labeling functions need to be proposed for different functions of interest (fairness, utility, etc.) whereas with our approach this issue does not arise due to its simplicity and generality. 
    * [6]: This work specifically studies fairness of samples by generating counterfactual samples and then measuring their influence. While this work can aid understanding of the fairness of training samples, its motivation is different from our work, which seeks to extend and apply influence functions for model performance gains across multiple metrics of interest and for feature-level interpretability.
    * [7]: In this work authors propose a downweighting scheme for training samples to study improvements in fairness for a number of fairness metrics. In spirit, this work is close to ours on influence-based data selection. However, the approaches proposed are more complex and specific to the study of fairness, whereas we opt for simpler and general approaches that can be widely applied in many settings, even beyond traditional classification.

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Reviewer comment (Experiments):
**Tabular Data**: I don't see this as an important weakness; however, most of this work is demonstrated on tabular data. It will be tricky to extend the feature analysis portion, as done in Figure 1 for example, to say images or text.

Author response:
* **Tabular Data**: We understand the reviewer's concern. We would like to point out that while 3 of the 5 datasets we consider are natively tabular datasets, the remaining 2 are vision/image (CelebA) and text/NLP datasets (Jigsaw Toxicity). Our approaches can be used with image and text datasets by using a feature extraction step. That is, feature extraction from the raw data needs to be undertaken as a preprocessing step using embeddings from a deep learning model or some other approach. This is how we conduct experiments on CelebA and Jigsaw Toxicity datasets. For example, we use the Mini-LM transformer model to get embeddings for the Jigsaw Toxicity dataset and then conduct experiments on it as if it were a tabular dataset. We would also like to mention that past work has found that pixel-level interpretation of images is not akin to human interpretation/reasoning, and hence, prototype-based explanations are generally preferred [8]. Hence, for raw image data, using a feature extraction step for preprocessing is a viable option for improved interpretation.

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