Revisit, Extend, and Enhance Hessian-Free Influence Functions

We would like to thank the reviewer for the deep analysis of our work, and the time, effort, and consideration. Below, we answer the questions raised by the reviewer:

Influence in IP
Thank you for the insightful interpretation on influence function and TracIn. We fully agree! From the formulation, our IP is similar to influence function and TracIn. Here we can provide another intuition of our method. If the training samples are similar to validation samples, we expect the model trained with such samples can also perform well on the validation set. Note that the similarity should be calculated not in the original feature space, where this space is orthogonal to the learning model. Therefore, the cosine similarity in the gradient space measures how the training sample is similar to the validation set in the point of the model, i.e., the gradient space. This approach circumvents the instability of Hessian inverse computations while providing an intuitive and computationally efficient alternative.

Order consistency
Thank you for the careful thought. In this paper, we do not claim the order consistency always holds. Moreover, in our synthetic datasets, we empirically demonstrated the linear case might have the order consistency, but the non-linear case not.

We agree with Reviewer HFTU that order consistency might not be a good interpretation. We will remove this part.

Remove the inverse of Hessian
We agree with Reviewer KR1G that the inverse of Hessian matrix contains information. However, computing the inverse of the Hessian matrix in non-convex scenarios is notoriously challenging and unstable, particularly when the Hessian matrix contains near-zero or negative eigenvalues, leading to numerical instability. This instability can introduce additional noise, undermining the reliability of influence computation. The cost of approximating the inverse of Hessian matrix is even higher than directly removing it. From our empirical evaluation, our IP does not suffer from too much performance sacrifice but has simple formulation. We will revise the manuscript to articulate these points more clearly.

Ensemble
Your understanding is correct. Our ensemble method generates diverse models by applying dropout to the final model checkpoint, avoiding the traditional approach of selecting different checkpoints from the training process. The rationale behind this design is as follows:

Traditional methods often rely on selecting multiple checkpoints during training to generate diverse models. However, these checkpoints are typically derived from non-converged model states, which may contain significant optimization noise, thereby affecting the accuracy of influence function estimation. In contrast, our method is based on the final converged model and leverages the stochasticity of dropout to simulate uncertainty in model parameters, creating diverse models efficiently. This approach mitigates the noise introduced by non-converged states while avoiding the high computational cost of retraining. Experimental results demonstrate that our method achieves robust influence estimation and significantly improves final model performance (see Figure 3 and Figure 4). In the revised manuscript, we will provide a more formal definition of the ensemble procedure and include additional experimental analyses to illustrate the applicability and potential limitations of this method.

Improvement very limited
Our major contribution lies in that IP is a simplified version of TracIn and works well in empirical verification. We agree with Reviewer KR1G that the inverse of Hessian matrix contains information, but approximating it might bring higher cost than removing it. Our IP without this term can achieve better and comparable performance, where we consider this simplicity and effectiveness to be a key strength of our approach. If we can achieve similar performance with a simpler tool, there would be no reason to add unnecessary components to the existing complex tool.

Survey
This survey paper is a key reference for our work. In fact, it first caught our attention two years ago when the arXiv version was made available. We are pleased to hear that the paper has now been accepted by Machine Learning. We would like to kindly inquire about which section Reviewer KR1G suggests we revisit or re-learn. Thank you.

We would like to thank the reviewer for the deep analysis of our work, and the time, effort, and consideration. Below, we answer the questions raised by the reviewer:

Order consistency
Thank you for the careful thought. In this paper, we do not claim the order consistency always holds. Moreover, in our synthetic datasets, we empirically demonstrated the linear case might have the order consistency, but the non-linear case not.

We agree with Reviewer HFTU that order consistency might not be a good interpretation. We will remove this part. But here we can provide another intuition of our method. If the training samples are similar to validation samples, we expect the model trained with such samples can also perform well on the validation set. Note that the similarity should be calculated not in the original feature space, where this space is orthogonal to the learning model. Therefore, the cosine similarity in the gradient space measures how the training sample is similar to the validation set in the point of the model, i.e., the gradient space. This approach circumvents the instability of Hessian inverse computations while providing an intuitive and computationally efficient alternative.

Conditions under which the angle alpha is small
See the above response.

Correlation to the inverse Hessian
In non-convex scenarios, the inverse of the Hessian matrix is notoriously difficult and unstable to compute. The indefinite nature of the Hessian makes its inverse highly sensitive to small perturbations. It is impossible to calculate the approximation error or correlation to the inverse of Hessian matrix for complex data. Instead, we employ the downstream tasks (i.e., model performance without identified detrimental samples) to compare different algorithms.

On Leave-one-out training
Thank you for your thoughtful suggestion. However, we respectfully and frankly believe that leave-one-out training may not be an effective evaluation tool. From a computational perspective, leave-one-out training is prohibitively expensive, especially for large datasets and complex models, as it requires multiple retraining cycles. Additionally, in non-linear cases, particularly with deep learning, removing a single training sample may not have a significant impact, especially when considering the inherent randomness in the training process. In other words, we conducted experiments using ResNet on CIFAR-10, comparing performance with the complete training set versus with one sample removed. After running these experiments multiple times, we found that the average performance between the two settings did not show statistical significance.

Label perturbation detection
Our research theme focuses on goal-oriented data-centric learning, aiming to improve model performance by distinguishing between beneficial and detrimental samples. This is fundamentally different from noisy label detection, as these concepts address separate issues. Note that a sample with a correct label might still be a detrimental sample. Thus, the requested metric does not align with our research focus.

U(0, 0.01)
We will formally define the IP Ensemble process below and will include a clearer explanation in future revisions. This process involves the following steps:

1. Generating Diverse Models:
   For each ensemble model, a dropout rate $p_k$ is randomly sampled from a uniform distribution $U(0, 0.01)$. The dropout is then applied to the model parameters $\theta$ to generate diverse models:
   $\theta^{(k)} = \text{Dropout}(p_k, \theta), \quad k = 1, 2, \dots, K.$

2. Computing IP Scores for Each Model:
   For each diverse model $\theta^{(k)}$, the IP score of a training sample $z_i$ is computed as:
   $I_{\text{IP}}^{(k)}(-z_i) = \nabla_{v_{\text{util}}}^{(k)} \cdot \nabla_{\theta^{(k)}} \ell(z_i; \theta^{(k)}),$
   where
   $\nabla_{v_{\text{util}}}^{(k)} = \sum_{z_j \in V} \nabla_{\theta^{(k)}} \ell(z_j; \theta^{(k)})^\top$
   is the aggregated gradient of the validation set $V$ based on the $k$-th model.

3. Ensembling the IP Scores:
   The final IP Ensemble score for $z_i$ is obtained by averaging the IP scores from all $K$ models:
   $I_{\text{IP-Ensemble}}(-z_i) = \frac{1}{K} \sum_{k=1}^K I_{\text{IP}}^{(k)}(-z_i).$

Thank you for your acknowledgment on reading our response.

We would like to thank the reviewer for the deep analysis of our work, and the time, effort, and consideration. Below, we answer the questions raised by the reviewer:

Benefits of this simplification
Our simplified approach calculates influence values based solely on the final converged model. The rationale is that only a converged model has stable parameters that can accurately reflect the true influence of samples on the model's predictions. As highlighted in [1], during non-converged states, parameter fluctuations can lead to unstable influence values, thereby reducing the reliability of the analysis. Our experimental results further support this conclusion. The IP Ensemble method consistently outperforms TracIn across nearly all scenarios presented in our paper.

Intuition
We have validated the effectiveness of removing the Hessian inverse not only on synthetic datasets but also on real-world datasets. These experiments demonstrate that influence value evaluation remains accurate and reliable even without the computation of the Hessian inverse. Here we can provide another intuition of our method. If the training samples are similar to validation samples, we expect the model trained with such samples can also perform well on the validation set. Note that the similarity should be calculated not in the original feature space, where this space is orthogonal to the learning model. Therefore, after the removal of the Hessian inverse, the cosine similarity measures how the training sample is similar to the validation set in the point of the model, i.e., the gradient space. This approach circumvents the instability of Hessian inverse computations while providing an intuitive and computationally efficient alternative.

Fairness or adversarial robustness
We follow [2] to use influence function to measure the sample influence towards fairness and robustness. For instance, under different loss functions, influence functions can quantify the contribution of samples to model decisions, thereby demonstrating their role in fairness or adversarial robustness tasks. To make this point clear, we will address this more thoroughly in future revisions.

Ensemble process
In the current version, we provide a simplified description of the ensemble process. However, the process itself is relatively direct, involving the aggregation of influence scores from different models to improve robustness and stability. We will formally define this process below and will include a clearer explanation in future revisions. This process involves the following steps:

1. Generating Diverse Models:
   For each ensemble model, a dropout rate $p_k$ is randomly sampled from a uniform distribution $U(0, 0.01)$. The dropout is then applied to the model parameters $\theta$ to generate diverse models:
   $\theta^{(k)} = \text{Dropout}(p_k, \theta), \quad k = 1, 2, \dots, K.$

2. Computing IP Scores for Each Model:
   For each diverse model $\theta^{(k)}$, the IP score of a training sample $z_i$ is computed as:
   $I_{\text{IP}}^{(k)}(-z_i) = \nabla_{v_{\text{util}}}^{(k)} \cdot \nabla_{\theta^{(k)}} \ell(z_i; \theta^{(k)}),$
   where
   $\nabla_{v_{\text{util}}}^{(k)} = \sum_{z_j \in V} \nabla_{\theta^{(k)}} \ell(z_j; \theta^{(k)})^\top$
   is the aggregated gradient of the validation set $V$ based on the $k$-th model.

3. Ensembling the IP Scores:
   The final IP Ensemble score for $z_i$ is obtained by averaging the IP scores from all $K$ models:
   $I_{\text{IP-Ensemble}}(-z_i) = \frac{1}{K} \sum_{k=1}^K I_{\text{IP}}^{(k)}(-z_i).$

This approach not only avoids the computational cost of retraining but also improves the robustness of influence estimation by leveraging model diversity introduced through dropout.

Approximation errors
Computing the inverse of the Hessian matrix in non-convex scenarios is notoriously challenging and unstable. The indefinite nature of the Hessian makes its inverse highly sensitive to small perturbations. Note that it is likely possible that the inverse of Hessian matrix does not exist without extra regularization to ensure its validity. Even with such regularization, approximation errors in the Hessian inverse can significantly impact the influence function results.

In Section 4, we employ a synthetic dataset to validate the effectiveness of removing the Hessian inverse in a controlled experimental environment. For real-world data and models, the Hessian matrix is of high dimension, which brings in the difficulty in computation and visualization. Therefore, it is impossible to calculate the approximation error of the inverse of Hessian matrix for complex data. Instead, we employ the downstream tasks (i.e., model performance without identified detrimental samples) to compare different algorithms.

We would like to thank the reviewer for the deep analysis of our work, and the time, effort, and consideration. Below, we answer the questions raised by the reviewer:

Gap between the contribution claimed in this paper and actual literature. We believe Reviewer Y3UC understood our method. Clearly, we do not claim that we propose TracIn; instead, our major contribution lies in that IP is a simplified version of TracIn and works well in empirical verification. We suspect that Reviewer Y3UC may view our method as too simple. However, we consider this simplicity to be a key strength of our approach. If we can achieve similar performance with a simpler tool, there would be no reason to add unnecessary components to the existing complex tool.

Thanks for providing these references. For [1], it primarily focuses on measuring the similarity between samples, rather than directly addressing data valuation. We will cite it in our paper. For [2], we have already cited the relevant works, discussed their advantages and limitations in our paper (Line 573), and compare it in the experiments. For [3], we understand there are several existing studies that employ influence functions to estimate sample impact on fairness and robustness. Here we follow [4] to extend our IP method for fairness and robustness as well, where we briefly mentioned this in the contribution part.

Unfair Experiments. We solicit Reviewer Y3UC for more evidence, which helps for our informative response, rather than just opinion. Here we follow the setting of [4] in a fair comparison. Again, our contribution lies in the simplicity, and our IP achieves better or comparable performance.

Dropout trick. We agree that the dropout trick can also be applied to other baseline methods. We would like to note that it is impractical to apply an ensemble strategy for complicated methods.

Figure 2 (D). We tested the synthetic data with uniformly noisy labels, which still produced an ideal decision boundary. To illustrate the negative impact of noisy samples on the decision boundary, we introduced two 'clustered' noisy samples. In other words, we simulate a scenario where noisy samples can affect model performance, where sample influence finds a place to apply.

For the negative performance of the inverse of Hessian, it is indeed likely and very likely. We believe Reviewer Y3UC would agree that in non-convex scenarios, the inverse of the Hessian matrix is notoriously difficult and unstable to compute. The indefinite nature of the Hessian makes its inverse highly sensitive to small perturbations. Additionally, the inverse may not even exist, demonstrating that negative performance is certainly possible.

TRAK. We report the performance of TRAK on CIFAR-10 and CIFAR-100 in the table below. In general, TRAK employs the random project for acceleration and suffers from the performance drop compared with the unaccelerated version.

Unclear claims. Apologies for any confusion. We would like to clarify that the influence of samples varies during the model optimization process. Some samples may be beneficial in the early stages but become detrimental later on. Specifically, when model parameters change significantly, the influence of a sample in the early stages does not accurately reflect its true impact on the converged model. Simply summing the sample influences at different checkpoints, as done in TracIn, can lead to a compromise that results in inferior performance. We have verified this through our experiments and will make this point clearer in the paper.

Similar performance between each method. Since all methods aim to estimate from the formulation of influence functions, the similar performance is expected. That is the reason why our simple method is more valuable.

Thanks for pointing out LDS. Respectfully, we do not use this metric for two reasons. 1) We believe data-centric learning should be goal-oriented, which should be evaluated in the task context; 2) LDS is time-consuming and impractical for large models, as the original authors suggest 100-500 models are needed to calculate LDS.

Revisit. We always strive to provide accurate expressions, and it was an oversight on our part to use an imprecise term. If you recall our response in the previous round, revisit is not one of our contributions, let alone something that undermines our work. Our primary contribution lies in presenting IP as a simple yet effective tool for influence estimation, which performs well in empirical validation.

LDS. The calculation of LDS requires multiple simulations then and fits the correlation. Such a way returns indeterministic results and the linear correlation will be affected by outliers. Personally and humbly (maybe biased), such a metric is inappropriate and impractical (slow running time) as a golden metric for evaluation.