Enhancing Prediction Performance through Influence Measure

Dear Reviewer,

Thank you for your thorough review and feedback. We have provided detailed responses to the points you raised below:

Weakness 1.1: My primary concern with the work presented is the limited scalability analysis of the proposed methods for larger datasets, and possibly the lack of discussion of the computational complexity of the algorithms.

• Below are the computational complexity of each compared method:
  • $*FI*^{*util*}$: $\mathcal{O}(p^3 + nd^3 + ndp)$. Here, $n = \max\\{n_{\text{train}}, n_{\text{test}}\\}$, $d$ is the dimension of the covariates, and $p$ is the dimension of the model parameters. The term $\mathcal{O}(p^3)$ comes from computing the inverse of the Hessian matrix. The $\mathcal{O}(nd^3)$ term is due to inverting $G$, repeated $n$ times. The $\mathcal{O}(ndp)$ term corresponds to computing $\partial_{x}\partial_{\theta}L$ for $n$ times.
  • $*FI*^{*util*}$ with approximation methods: $\mathcal{O}(\alpha ndT\log(\alpha n) + \alpha n(p + d)ks + \alpha nd^3 + \sum_{i=1}^L p_i^3)$. Here, $\alpha \in (0,1)$ is the proportion of data for which we compute $FI$ accurately during subsampling. $T$ is the number of decision trees in the random forest, $k$ is the Truncated SVD parameter, $s$ is the number of iterations for computing each eigenvalue during power iteration, and $\\{p_1, p_2, \dots, p_L\\}$ represent the number of parameters in each layer of an $L$-layer neural network.
  • $*FI*^{*active*}$: $\mathcal{O}(p^3 + n^2d^3 + n^2dp)$. Here, $n = \max\\{n_{\text{label}}, n_{\text{unlabel}}\\}$.
  • $*FI*^{*active*}$ with approximation methods: $\mathcal{O}(\alpha ndT\log(\alpha n) + \alpha n^2(p + d)ks + \alpha n^2d^3 + \sum_{i=1}^L p_i^3)$.
  • IV: $\mathcal{O}(p^3 + np)$.
  • BALD: $\mathcal{O}(np)$.

By employing two approximation techniques, we can significantly reduce the computational gap between our method and the compared methods such as IV. Additionally, we can further optimize $G$ using Truncated-SVD in the future, which will eliminate the $d^3$ term from the computational complexity. When the dataset is large enough, setting $\alpha n$ as a constant allows us to achieve better computational complexity than the IV method. For complex models (e.g., neural networks) where $p$ is sufficiently large, the $p^3$ term dominates the computational complexity of IV. In such scenarios, the computational costs of our method could be lower than IV.

• Apart from that, our approximation method can also reduce the storage complexity of the algorithm. And we provide the storage complexity analysis for the algorithms discussed in our paper:
  • In the computation of FI, the highest storage requirements are for the Hessian matrix and $\partial_x\partial_{\theta}L$, which together occupy a space of $\mathcal{O}(p^2+dp)$.
  • After using the KFSVD approximation method, the space requirement of FI is reduced to $\mathcal{O}(\sum_{i=1}^Lp_i^2+(p+d)k)$.
  • In the computation of IV, it is necessary to store sample information and the Hessian matrix, resulting in a storage complexity of $\mathcal{O}(p^2+d)$.
  • The storage complexity of BALD is $\mathcal{O}(p+d)$.

After applying approximation techniques, our algorithm achieves a storage complexity similar to that of the IV method. In fact, when $p$ is large, our method offers an advantage.

Weakness 1.2: Practical datasets for most real-world applications could have much higher number of samples, as well as features, so it would be important to benchmark with such datasets.

We plan to enhance our experiments as follows:

• In the Data Trimming section, we will incorporate the evaluation of our method on more multi-class classification tasks, such as Students Performance dataset from the UC Irvine Machine Learning Repository, to demonstrate its applicability beyond binary classification.
• In the Active Learning section, we plan to evaluate the effectiveness of our approach in more complex image classification tasks, including CIFAR-100, which presents more challenging classification scenarios, and Caltech101, a widely-used imbalanced dataset. Furthermore, we will investigate the applicability of FI in Natural Language Processing (NLP) by applying BERT to the Microsoft Research Paraphrase Corpus (MRPC), a standard benchmark for text classification.

Since the rebuttal phase is short, we will aim to discuss the performance of our algorithm on the mentioned datasets in the final version.

Dear Reviewer,

We truly appreciate the time and effort you invested in reviewing our work, as well as your valuable feedback. In the following, we offer detailed responses to each of the points you raised:

Weakness 1: The evaluation is conducted for binary classification with logistic regression, supplemented by additional experiments using CNN on existing image datasets. However, the analysis is somewhat limited in scope. Although this setup is based on a related ICLR work [Chhabra et al. (2024) / chhabra2024data] and compares against the method named Influence Value (IV) from that published work, I find the dataset evaluation to be relatively insufficient. To thoroughly assess the performance of using FI or related methods, I believe it is essential to test it on a diverse range of datasets with varying characteristics, which would help to confirm its strengths and potentially reveal its limitations.

In response to your recommendation, we intend to supplement our experiments as follows:

• In the Data Trimming section, we will incorporate the evaluation of our method on more multi-class classification tasks, such as Students Performance dataset from the UC Irvine Machine Learning Repository, to demonstrate its applicability beyond binary classification.
• In the Active Learning section, we plan to evaluate the effectiveness of our approach in more complex image classification tasks, including CIFAR-100(which presents more challenging classification scenarios) and Caltech101(a widely-used imbalanced dataset). Furthermore, we will investigate the applicability of FI in Natural Language Processing (NLP) by applying BERT to the Microsoft Research Paraphrase Corpus (MRPC), a standard benchmark for text classification.

Since the rebuttal phase is short, we’ll aim to discuss the performance of our algorithm on the mentioned datasets in the final version.

Weakness 2: Following that, while FI appears to outperform the compared methods (IV and Random for Data Trimming; IV, Random, and Bald for Active Learning), its superiority is not necessarily absolute (at least this is not clearly shown), as performance varies across datasets. Notably, on the CelebA dataset for Data Trimming, Random actually outperforms IV and achieves results comparable to FI. This dataset-specific variation raises concerns about the generalization of the FI's performance across different datasets, leaving questions about how they would perform on other datasets.

We would like to clarify that not all datasets contain a significant number of data points that meaningfully impact the model's performance. As a result, our method can not always show a siginificant improvement over the baseline methods. For instance, in the CelebA dataset you mentioned, no data points can be removed to significantly improve prediction results. In such cases, our method achieves performance comparable to "random," whereas using IV results in a decline in model performance. This demonstrates that our approach is more stable and reliable in practice.

Question 1: What scenarios or conditions are likely to result in FI to underperform?

In active learning experiments, we found that our algorithm is really good at identifying complex samples. However, not all datasets contain such complex sample points. In these relatively simple problem environments, the performance of our method may not differ significantly from that of other approaches.

Dear Reviewer,

We are grateful for your thoughtful review and the time you dedicated to evaluating our work. Please find below our detailed responses to your comments:

Weakness 1: The main weakness of this work is lack of ablation studies to better understand the impact of the contributions of this work on its performance.

We have added ablation studies based on your suggestion, and the results, which clarify the impact of our contributions, are presented below. We will continue to refine these studies in the final version.

Weakness 1.1: For example, the first contribution of the paper is to evaluate the impact of perturbations against the model performance for the test data set. It would have been useful to see how much this slight modification is helping with improving the proposed method's performance for data trimming.

To assess the impact of perturbations on model performance for the test set, we use the Bank dataset as a case study. Specifically, we introduce Gaussian noise to $b$ samples in the training set with the highest and the lowest FI values, respectively, where $b$ takes values from [50, 100, 150, 200, 250, 300]. After introducing the perturbations, we evaluate the model's performance on the test set. The results are presented in Table 1 below. It can be observed that samples with higher FI values are more sensitive to perturbations, as their accuracy decreases more significantly compared to samples with lower FI values.

Table 1: Impact of Perturbations on Model Accuracy.

Weakness 1.2: As another example, authors utilize two approximation techniques to reduce the computational cost of their proposed method, however, there is no experiment providing more insights into the trade offs between the potential reduction in the proposed method's performance vs. the quantified computational cost reduction.

To address the trade-off between computational cost and performance, we have added ablation experiments on two datasets . As shown in the table, the approximate method is over three times faster than the exact computation, with minimal impact on performance. It even achieves better results on the CelebA dataset. Additionally, as the dataset size increases, the proportion of time spent on training and prediction with the random forest decreases, making the speedup even more significant.

Table 2: Comparison of Approximate and Exact Methods on Different Datasets

Question 1: Based on equation 1 it looks like the perturbations applied to the training instances is by adding noise to them versus changing the training data's weight/importance. Can authors explain the rationale behind picking this type of perturbation over changing training instances' weights

To offer a new perspective on assessing the impact of individual samples on model performance, we decided to directly add perturbations to the sample points, rather than using traditional influence functions. While adjusting sample weights reflects the impact of removing a point from the dataset, adding noise is more closely aligned with adversarial attack scenarios. If a sample significantly affects model performance after perturbation, it indicates that the sample could cause a notable performance drop when exposed to attacks or measurement errors. Thus, we believe it may offer useful insights in adversarial attack scenarios. We will consider exploring this possibility in future research.

Dear Reviewer,

We sincerely appreciate the time and effort you have put into reviewing our work, as well as your valuable comments. Below, we provide a detailed response to each of the points you raised:

Weakness 1: The paper claims "Our metric directly assesses the effect of small perturbations in the training samples on the model's performance on the test set", whether this leaks information about the test samples during the training process, which needs to be clarified by the authors.

We apologize for any confusion caused by our text. Our objective is to measure the impact of small perturbations in the training set on the model's performance, and we want to emphasize that no data leakage occurs. In our FI computation, the dataset is divided into three parts: training, validation, and test sets. Importantly, we use only the training and validation sets during the computation, not the test set. Specifically, we analyze how small changes in the training samples affect the model's performance on the validation set. The test set is exclusively used to evaluate the overall performance of the proposed method, ensuring that no leakage takes place. We will clarify this point in the final version to prevent further misunderstandings.

Weakness 2: The parameter settings in the experiments need further clarification.

For the Data Trimming experiments, we mainly followed the setups of Chhabra et al. (2024)[1] with the same data sources, and we used similar models and parameter scales. The specific parameters can be found in Table 3 of our Appendix D.1.

For the Active Learning experiments, we primarily followed the setups from previous works, with some minor adjustments:

• In the Tabular Datasets experiments, we based our design on the approach used in Chhabra et al. (2024)[1], where parameter settings are available in their open-source code. The experimental plots in their Appendix G guided our choices, and we used similar parameter scales. You can find the specific parameters in Table 4 of our Appendix D.2.
• For the Image Classification experiments, we followed the design from Andreas Kirsch et al. (2023)[2], matching the parameter scale used in their Appendix E. The detailed experimental setup is provided in Table 5 of our Appendix D.2.

Regarding the network architectures, we made the following decisions to ensure a rigorous evaluation:

• For MNIST, we used a custom-designed simple CNN, as shown in Table 6 of the Appendix D.2.
• For CIFAR10, we implemented the simplest version of the ConvMixer architecture [3].

[1] Chhabra, A., Li, P., Mohapatra, P., & Liu, H. (2024). "What data benefits my classifier?" Enhancing model performance and interpretability through influence-based data selection. In Proceedings of the Twelfth International Conference on Learning Representations (ICLR 2024), Vienna, Austria. https://openreview.net/forum?id=HE9eUQlAvo

[2] Kirsch, A., Farquhar, S., Atighehchian, P., Jesson, A., Branchaud-Charron, F., & Gal, Y. (2023). Stochastic batch acquisition: A simple baseline for deep active learning. Transactions on Machine Learning Research, 2023. https://openreview.net/forum?id=vcHwQyNBjW

[3] https://github.com/locuslab/convmixer

Dear Reviewer,

Thank you for your thoughtful review. We provide detailed responses to the points you raised below:

Weakness 1: I wonder if it is possible to provide the complexity analysis in terms of both memory and speed for the proposed method and the state-of-art strategies for examples in the paper.

• Below are the computational complexity of each compared method:
  • $*FI*^{*util*}$: $\mathcal{O}(p^3 + nd^3 + ndp)$. Here, $n = \max\\{n_{\text{train}}, n_{\text{test}}\\}$, $d$ is the dimension of the covariates, and $p$ is the dimension of the model parameters. The term $\mathcal{O}(p^3)$ comes from computing the inverse of the Hessian matrix. The $\mathcal{O}(nd^3)$ term is due to inverting $G$, repeated $n$ times. The $\mathcal{O}(ndp)$ term corresponds to computing $\partial_{x}\partial_{\theta}L$ for $n$ times.
  • $*FI*^{*util*}$ with approximation methods: $\mathcal{O}(\alpha ndT\log(\alpha n) + \alpha n(p + d)ks + \alpha nd^3 + \sum_{i=1}^L p_i^3)$. Here, $\alpha \in (0,1)$ is the proportion of data for which we compute $FI$ accurately during subsampling. $T$ is the number of decision trees in the random forest, $k$ is the Truncated SVD parameter, $s$ is the number of iterations for computing each eigenvalue during power iteration, and $\\{p_1, p_2, \dots, p_L\\}$ represent the number of parameters in each layer of an $L$-layer neural network.
  • $*FI*^{*active*}$: $\mathcal{O}(p^3 + n^2d^3 + n^2dp)$. Here, $n = \max\\{n_{\text{label}}, n_{\text{unlabel}}\\}$.
  • $*FI*^{*active*}$ with approximation methods: $\mathcal{O}(\alpha ndT\log(\alpha n) + \alpha n^2(p + d)ks + \alpha n^2d^3 + \sum_{i=1}^L p_i^3)$.
  • IV: $\mathcal{O}(p^3 + np)$.
  • BALD: $\mathcal{O}(np)$.

By employing two approximation techniques, we can significantly reduce the computational gap between our method and the compared methods such as IV. Additionally, we can further optimize $G$ using Truncated-SVD in the future, which will eliminate the $d^3$ term from the computational complexity. When the dataset is large enough, setting $\alpha n$ as a constant allows us to achieve better computational complexity than the IV method. For complex models (e.g., neural networks) where $p$ is sufficiently large, the $p^3$ term dominates the computational complexity of IV. In such scenarios, the computational costs of our method could be lower than IV.

• Apart from that, our approximation method can also reduce the storage complexity of the algorithm. And we provide the storage complexity analysis for the algorithms discussed in our paper:
  • In the computation of FI, the highest storage requirements are for the Hessian matrix and $\partial_x\partial_{\theta}L$, which together occupy a space of $\mathcal{O}(p^2+dp)$.
  • After using the KFSVD approximation method, the space requirement of FI is reduced to $\mathcal{O}(\sum_{i=1}^Lp_i^2+(p+d)k)$.
  • In the computation of IV, it is necessary to store sample information and the Hessian matrix, resulting in a storage complexity of $\mathcal{O}(p^2+d)$.
  • The storage complexity of BALD is $\mathcal{O}(p+d)$.

After applying approximation techniques, our algorithm achieves a storage complexity similar to that of the IV method. In fact, when $p$ is large, our method offers an advantage.

Weakness 2: I am a bit confused about what robustness means in the paper. Is it about how the model would behave after removing or perturbing any data? Or its about after doing these procedures, the prediction performance is consistently better than not doing it? If it is the latter case, improving prediction performance should be a better way of describing it.

In our paper, "robustness" refers to the ability of our algorithm to enhance the model's prediction performance. We will revise the text in the final version according to your suggestions.

Weakness 2.1: Moreover, since we are talking about robustness, should the standard deviation of prediction accuracies in Table 2 also be included? also error bars in figure 2 and 8.

• We have revised Table 1 and Table 2 in our paper to include the standard deviation of prediction accuracies, and we are now presenting the updated tables with this additional information.

Table 1: Comparison of two methods on linear model

Table 2: Comparison of two methods on nonlinear model

• Regarding your concern about the error bars in Figures 2 and 8, we would like to clarify that, given the datasets, both our method and the approach of Chhabra et al[^1] involve deterministic selection of data points. As a result, error bars are only applicable to Random.