Revisit, Extend, and Enhance Hessian-Free Influence Functions

## Abstract

Abstract Influence functions serve as crucial tools for assessing sample influence. By employing the first-order Taylor extension, sample influence can be estimated without the need for expensive model retraining. However, applying influence functions directly to deep models presents challenges, primarily due to the non-convex nature of the loss function and the large size of model parameters. This difficulty not only makes computing the inverse of the Hessian matrix costly but also renders it non-existent in some cases. Various approaches, including matrix decomposition, have been explored to expedite and approximate the inversion of the Hessian matrix, with the aim of making influence functions applicable to deep models. In this paper, we revisit a specific, albeit naive, yet effective approximation method known as TracIn, and simplify it further, introducing the name Inner Product (IP). This method substitutes the inverse of the Hessian matrix with an identity matrix. We offer deeper insights into why this straightforward approximation method is effective. Furthermore, we extend its applications beyond measuring model utility to include considerations of fairness and robustness. Finally, we enhance IP through an ensemble strategy. To validate its effectiveness, we conduct experiments on synthetic data and extensive evaluations on noisy label detection, sample selection for large language model fine-tuning, and defense against adversarial attacks.

## 1 Introduction

Data-centric learning is a growing research field that focuses on enhancing machine learning model performance by refining the quality and characteristics of training data []. In contrast to model-centric approaches, which prioritize improving algorithms or optimization techniques without altering the dataset, data-centric learning involves adjusting the dataset itself—through methods like trimming, relabeling, and reweighting—while keeping the learning algorithm fixed. This approach plays a vital role in areas such as model interpretability, selecting training subsets, generating synthetic data, detecting noisy labels, improving active learning, and promoting fairness learning [, ].

Sample influence estimation, as the foundation of data-centric learning, can be generally categorized into two categories []. (a) Retraining-based methods assess the sample influence by retraining the model with and without a specific sample and checking the performance change, which include the classical leave-one-out influence approach [] and Shapley value approaches [, , , ]. (b) Gradient-based methods estimate influence without expensive overheads of retraining, known as influence functions. The seminal work in this category is that of [], which utilizes a Taylor-series approximation and LiSSA optimization [] to compute sample influences. However, the limiting assumption is that the model and loss function are convex. Despite debates on the necessity of convexity [, , , ], challenges persist when directly applying gradient-based methods to large models. The size of model parameters complicates calculations, particularly in obtaining the inverse of the Hessian matrix. Efforts, including matrix decomposition techniques [, , ], aim to expedite and approximate Hessian matrix inversion, therefore enhance the feasibility of influence functions for deep learning models.

Contributions. In this paper, we focus on the influence function category and revisit a specific naive yet aggressive approximation method, TracIn []. This method substitutes the inverse of the Hessian matrix with an identity matrix, representing it as a Hessian-free influence function—the inner product of the gradient of the validation set and a sample to be assessed, which we refer to as Inner Product (IP). We summarize our major contributions as follows:

- •
We simplify the checkpoints of TracIn, define it into IP formulation, and delve into the rationale behind this simple approximation, offering insights into why it performs well in practice.
- •
Expanding our IP framework, we extend its applicability beyond measuring sample influence on model utility to encompass considerations of fairness and robustness.
- •
To improve generalization, we introduce IP Ensemble, an innovative approach that employs dropout mechanisms to simulate a diverse array of models. IP Ensemble consolidates IP scores from these different models, thereby strengthening the method’s overall generalization capabilities.
- •
By assessing the sample influence, model performance can be enhanced by trimming detrimental samples from the training or fine-tuning set, downweighting them, or even relabeling them. Specifically, we validate the effectiveness of IP through synthetic data experiments and conduct extensive real-world evaluations using IP Ensemble. These experiments span various applications, including noisy label correction for vision data, data curation aimed at fine-tuning fairer NLP models, and defense strategies against adaptive evasion adversaries.

## 2 Related Work

In this section, we introduce the literature on influence functions, with a focus on the acceleration of the calculation of the inverse of Hessian matrix, followed by various applications and miscellaneous.

Efficient Influence Estimation. Influence functions serve as crucial tools for estimating the individual valuation of data without requiring model retraining. However, the computation of the inverse of the Hessian matrix poses challenges for large-scale data and models. To address this issue, various approaches have been proposed to simplify or estimate the inverse of the Hessian matrix effectively. A seminal work is that of [], which employs a Taylor-series approximation and LiSSA optimization [] to compute sample influences. Arnoldi [] employs the random projection and simplified Hessian matrix for acceleration. EKFAC [] enhances Kronecker-Factored eigendecomposition for a precise Hessian approximation. More recently, DataInf [] efficiently computes influence even for large models by replacing the inverse Hessian computation with a readily computable closed-form expression, although their framework may suffer from significant theoretical errors. TracInc [], a straightforward yet aggressive approximation, substitutes the inverse of the Hessian matrix with an identity matrix, essentially considering gradients directly as a measure of influence. Beyond the conventional influence function that gauges sample influence on the validation set, self-influence [, ] computes influence using the training set alone. Moving beyond using a single model checkpoint, GEX [] leverages a geometric ensemble of multiple checkpoints to approximate influence functions, alleviating the bilinear constraint and non-linear losses. Moreover, TDA [] also introduces a checkpoint-based segmentation approach, which further enhances the training process by combining the techniques of implicit differentiation and unrolling. This is achieved through the utilization of EKFAC, providing a more efficient and scalable solution for complex models [].

Various Applications of Influence Functions.
With the above efficient approximation, influence functions have diverse applications. One major application is identifying detrimental samples []. The learning performance can be further improved by removing [], relabeling [], or reweighting [] the identified detrimental samples, which has significant implications in various fields such as noisy label detection [], subset selection [], and the identification of the most influential samples [, ]. Other applications encompass few-shot learning [], where influence functions help improve model performance with minimal data, and recommendation systems [, ], enhancing the accuracy and personalization of recommendations. Influence functions are also valuable in selecting data for active learning [], fairness machine learning [, , ], adversarial attack [], graph machine learning [, ], machine unlearning [, ], out-of-distribution generalization [], data privacy [], domain adaptation [], to name a few.

Miscellaneous. Several studies have examined the fragility of influence functions in explaining deep learning model predictions. [] discovers that while influence estimates may not perfectly align with leave-one-out retraining, they approximate the proximal Bregman response function, offering valuable insights for identifying influential or mislabeled examples. [] demonstrates that the effectiveness of influence functions in neural networks varies with network architecture, depth, width, parameterization, and regularization, underscoring their fragility in deep learning due to non-convex loss functions. [] suggests that the instability of current validation procedures, rather than non-convexity or lack of regularization, may be responsible for their unreliability. [] expands influence functions from estimating the effects of removing one point to large groups of training samples; [] enhance influence estimation in large-scale models by concentrating on target parameters and addressing computational instability with a robust inverse-Hessian-vector product approximation;  [] extend traditional influence functions to monitor the impact of pre-training data on fine-tuned model predictions, facilitating the identification of crucial pre-training examples.

## 3 Methods

In this section, we introduce the preliminaries of the influence function, with a focus on the Hessian-free approximation, then elaborate on our extension and upgrade.

Revisit and Simplify. Given a training set $T$$=$$\{z_{i}$$=$$(x_{i},y_{i})\}_{i=1}^{n}$ and a classifier with empirical risk minimization by a convex loss function $\ell$, the optimal parameters of the classifier can be obtained by $\hat{\theta}=\operatorname*{arg\,min}_{\theta\in\Theta}\frac{1}{n}\sum_{i=1}^{
n}\ell(z_{i};\theta)$. To measure the influence of an individual data sample, we can train the model with and without the specific sample and see the performance change. However, the retrain-based approach is expensive for large-scale data and models. To avoid model retraining, influence functions estimate the effect of changing an infinitesimal weight of samples on a validation set $V$$=$$\{z_{j}$$=$$(x_{j},y_{j})\}_{j=1}^{n^{\prime}}$, based on an impact function $f$ evaluating the quantity of interest. Considering the sample impact on model utility, i.e., the loss on the validation set, by removing this sample from the original training set, the sample influence can be estimated as follows []:

$$ $\mathcal{I}^{\textrm{util}}(-z_{i})=\sum\nolimits_{z_{j}\in V}\nabla_{\hat{ \theta}}\ell(z_{j};\hat{\theta})^{\top}\mathbf{H}^{-1}_{\hat{\theta}}\nabla_{ \hat{\theta}}\ell(z_{i};\hat{\theta}),$ (1) $$

where $\mathbf{H}_{\hat{\theta}}$$=$$\sum_{i=1}^{n}\nabla_{\hat{\theta}}^{2}\ell(z_{i};\hat{\theta})$ is the Hessian matrix of the convex $\ell$ loss function.

Influence functions encounter a challenge in direct application to deep models, primarily due to the non-convex nature of the loss function and the considerable size of model parameters. This obstacle not only renders the calculation of the inverse of the Hessian matrix costly but also leads to its non-existence. Various attempts, including matrix decomposition methods [, , ], have been undertaken to expedite and approximate the inversion of the Hessian matrix, aiming to render influence functions viable for deep models. In this paper, we revisit a particular naive yet aggressive approximation method TracIn [] by substituting the inverse of the Hessian matrix with an identity matrix, outlined as follows:

$$ $\mathcal{I}_{\textrm{IP}}^{\textrm{util}}(-z_{i})=\nabla v^{\textrm{util}} \nabla_{\hat{\theta}}\ell(z_{i};\hat{\theta}),\ \textrm{and}\ \nabla v^{ \textrm{util}}=\sum\nolimits_{z_{j}\in V}\nabla_{\hat{\theta}}\ell(z_{j};\hat{ \theta})^{\top}.$ (2) $$

Figure: Figure 1: Illustration of $\nabla v^{\textrm{util}}\mathbf{H}^{-1}_{\hat{\theta}}$ and $\nabla v^{\textrm{util}}$.
Refer to caption: x1.png

The above equation is the inner product of $\nabla v^{\textrm{util}}$ and the sample gradient $\nabla_{\hat{\theta}}\ell(z_{i};\hat{\theta})$; therefore, we call this method Inner Product (IP). Note that TracIn incorporates multiple checkpoints to record model parameters throughout the optimization process, whereas IP only takes into account the final or converged model. We believe that sample influence should be assessed using a fixed model; simply summing sample influences at different stages fails to capture the dynamics of model optimization. The converged model can significantly diverge from its earlier stages, making the sample influences derived from initial checkpoints less accurate.

In the following, we link the IP to influence functions and provide our insights of IP on why such a naive approximation works well in practice. Let $\nabla v^{\textrm{util}}$$=$$\sum\nolimits_{z_{j}\in V}\nabla_{\hat{\theta}}\ell(z_{j};\hat{\theta})^{\top}$, then influence functions in Eqs. (1) and (2) can be reformulated as $\nabla v^{\textrm{util}}\mathbf{H}^{-1}_{\hat{\theta}}\nabla_{\hat{\theta}}
\ell(z_{i};\hat{\theta})$ and $\nabla v^{\textrm{util}}\nabla_{\hat{\theta}}\ell(z_{i};\hat{\theta})$. We illustrate the directions of $\nabla v^{\textrm{util}}\mathbf{H}^{-1}_{\hat{\theta}}$ and $\nabla v^{\textrm{util}}$ in Figure 1, with the influence scores roughly representing the angle between the sample gradient and $\nabla v^{\textrm{util}}\mathbf{H}^{-1}_{\hat{\theta}}$ or $\nabla v^{\textrm{util}}$. Remarkably, although influence scores by Eqs. (1) and (2) may differ, they exhibit order-consistency in several scenarios. For instance, if two samples $z_{i}$ and $z_{i^{\prime}}$ have gradients in Region II or IV, then $\mathcal{I}^{\textrm{util}}(-z_{i})\gtrless\mathcal{I}^{\textrm{util}}(-z_{i^{
\prime}})$ implies $\mathcal{I}_{\textrm{IP}}^{\textrm{util}}(-z_{i})\gtrless\mathcal{I}_{\textrm{
IP}}^{\textrm{util}}(-z_{i^{\prime}})$. However, this order-consistency does not hold in Regions I and III. Fortunately, samples in these regions are jointly recognized by $\mathcal{I}^{\textrm{util}}$ and $\mathcal{I}^{\textrm{util}}_{\textrm{IP}}$ as either beneficial or detrimental, resulting in minimal practical differences. A full analysis is in Appendix A.

In general, IP emerges as a straightforward, efficient, and remarkably effective alternative to influence functions. In non-convex scenarios, approximations of the inverse Hessian matrix may introduce significant errors, rendering $\nabla v^{\textrm{util}}\mathbf{H}^{-1}_{\hat{\theta}}$ ineffective, even incorrect. We substantiate this observation empirically in Section 4. Conversely, while $\nabla v^{\textrm{util}}$ in IP may not precisely align with the optimal direction for enhancing model performance, it remains a reliable indicator for distinguishing between detrimental and beneficial samples, even for non-convex models. Intuitively, if a sample gradient aligns with the gradient from the validation set, it suggests that incorporating this sample contributes to enhancing the model’s utility. Similarly, the negative samples works oppositely.

Extension. Beyond measuring the sample influence on model utility, we extend IP to assess the sample influence on fairness and robustness by modifying
the impact function $f$.

Specifically, we can instantiate the impact function $f$ by group fairness [], such as demographic parity (DP) to measure influence on fairness []. Consider a binary sensitive attribute defined as $g\in\{0,1\}$ and let $\hat{y}$ denote the predicted class probabilities. The fairness metric DP is defined as: $f^{\textrm{DP-fair}}(\hat{\theta},V)$$=$$\big{|}\mathbb{E}_{V}[\hat{y}|g$$=$$1]$$-$$\mathbb{E}_{V}[\hat{y}|g$$=$$0]\big{|}$. Within the above IP framework, we can calculate the training sample influence on fairness as follows:

$$ $\mathcal{I}^{\textrm{DP-fair}}_{\textrm{IP}}(-z_{i})=\nabla v^{\textrm{fair}} \nabla_{\hat{\theta}}\ell(z_{i};\hat{\theta}),\ \textrm{and}\ \nabla v^{ \textrm{fair}}=\nabla_{\hat{\theta}}f^{\textrm{DP-fair}}(\hat{\theta},V)^{\top }_{\hat{\theta}}.$ (3) $$

Similarly, we can also measure the sample influence on adversarial robustness within the IP framework. To achieve this, we follow [] and consider a white-box adversary [] specific to linear models, which can be easily extended to other models and settings. To craft an adversarial sample, we take each sample $z_{j}=(x_{j},y_{j})$ in the validation set $V$ and only perturb $x^{\prime}_{j}=x_{j}-\gamma\frac{\hat{\theta}^{\top}x_{j}+b}{\hat{\theta}^{
\top}\hat{\theta}}\hat{\theta}$ and keep $y_{j}$ unchanged, where $\hat{\theta}\in\mathbb{R}^{d}$ are the linear model coefficients, $b\in\mathbb{R}$ is the intercept, and $\gamma>1$ controls the amount of perturbation added. In this manner, we can obtain an adversarial validation set $V^{\prime}$ which consists of $z^{\prime}_{j}=(x^{\prime}_{j},y_{j})$ for each sample $z_{j}$ of $V$. Now, we can compute adversarial robustness influence for each training sample as follows:

$$ $\mathcal{I}^{\textrm{robust}}_{\textrm{IP}}(-z_{i})=\nabla v^{\textrm{robust}} \nabla_{\hat{\theta}}\ell(z_{i};\hat{\theta}),\ \textrm{and}\ \nabla v^{ \textrm{robust}}=\sum\nolimits_{z^{\prime}_{j}\in V^{\prime}}\nabla_{\hat{ \theta}}\ell(z^{\prime}_{j};\hat{\theta})^{\top}.$ (4) $$

Figure: Figure 2: Illustrating our IP on two synthetic datasets and convex/non-convex models. A-C illustrate a 2D linearly separable synthetic dataset with a subset of detrimental samples bearing incorrect labels, trained using a Logistic Regression model, and D-F demonstrate the similar analysis on a non-linear synthetic half-moon dataset using a Multilayer Perceptron neural network. Specifically, A and D depict training sets with two classes, where detrimental samples are marked with $\times$ and regular samples with $\circ$. B and E show test sets. C and F present influence scores and IP scores by Eqs. (1) and (2), respectively. In the linear case, there is a clear correlation between influence scores and inner product scores, the detrimental samples have both negative influence scores and IP scores. However, in the non-linear case, the influence scores of detrimental samples appear intermixed; but the detrimental samples can be identified via IP.
Refer to caption: x2.png

Enhancement. The simplicity of IP offers opportunities to enhance the generalization of influence functions. In convex cases, the model parameter $\hat{\theta}$ is both optimal and unique. However, in non-convex scenarios, the presence of local minima introduces instability and non-uniqueness into the solution. Typically, ensemble strategies are employed to bolster model generalization [, ]. Yet, while employing different models can enhance performance, it also escalates the costs associated with model training and complicates the calculation of influence functions. This arises from the variability in model parameters, necessitating multiple computations of the inverse of each individual Hessian matrix. The introduction of Hessian-free IP circumvents this issue, eliminating the need for costly calculations of Hessian matrices and their inverses. Drawing inspiration from dropout mechanisms, diverse models can be swiftly generated without necessitating model retraining. By computing sample gradients from various models, we propose IP Ensemble that amalgamates IP scores from distinct models. Experiments detailed in Section 5 illustrate the superior performance of the inner product ensemble over other influence function-based methods.

## 4 Correctness Verification on Synthetic Data

Here we verify the correctness of our Inner Product (IP) as an influence score surrogate on two synthetic datasets with convex/non-convex models. First, we use Logistic Regression on a linear dataset to demonstrate the effectiveness of IP in a convex optimization scenario. Second, we use a

Figure: Figure 3: Directions of gradients of the validation set and detrimental samples in Figure 2. In the linear case, $\alpha$ is $0.66\degree$, and we draw a larger angle for better visualization. For the non-linear case, we use the dimension reduction [] for visualization, and $\alpha$ is $94.36\degree$.
Refer to caption: x3.png

Multi-Layer Perceptron on a half-moon dataset to validate the performance of IP in a non-convex optimization scenario.(^1^11The detailed characteristics of datasets and models used in this paper can be found in Appendix B.) In Figure 2, A and B illustrate the training and test sets of a linearly separable dataset, consisting of 150 and 100 samples, respectively. Notably, the training set contains 10 manually generated noisy samples with incorrect labels. Similarly, D-F illustrate a non-linear separable half-moons dataset, consisting of 200 training samples including 20 noisy samples with incorrect labels, and 100 test samples. C and F display the influence score and inner product score for each training sample, calculated according to Eqs. (1) and (2), respectively. Additionally, fairness and robustness analysis is in Appendix C.1.

In the linear case depicted in Figure 2C, the inner product score serves as a reliable surrogate to distinguish detrimental samples from beneficial ones. It exhibits an almost perfect correlation and order-consistency with the influence score. Specifically, detrimental samples yield negative scores for both inner product and influence, while other samples typically show positive or nearly zero values. Figure 3 illustrates the relationship between directions among $\nabla v^{\textrm{util}}$, $\nabla v^{\textrm{util}}\mathbf{H}^{-1}_{\hat{\theta}}$, and the gradients of detrimental samples. It is worth noting that the angle between $\nabla v^{\textrm{util}}\mathbf{H}^{-1}_{\hat{\theta}}$ and $\nabla v^{\textrm{util}}$ is only $0.66\degree$, indicating that the identity matrix is a simple and effective surrogate of $\mathbf{H}^{-1}_{\hat{\theta}}$ in the linear case.

However, the limitations of influence scores become apparent in the context of non-convex models, as illustrated in Figure 2F. Here, the influence scores of detrimental samples are intermingled with those of normal ones, where $\nabla v^{\textrm{util}}\mathbf{H}^{-1}_{\hat{\theta}}$ is not the optimal direction due to the inaccuracies in approximating the Hessian matrix. Fortunately, the IP score effectively isolates detrimental samples from inliers. Even for this non-linear dataset, $\nabla v^{\textrm{util}}$ remains useful for discerning detrimental samples from beneficial ones, as all detrimental samples exhibit obtuse angles with $\nabla v^{\textrm{util}}$.

**Table 1: Accuracy results of influence function-based methods on the CIFAR10N, CIFAR-100N and Animal-10N datasets with 5% identified detrimental samples removed**
| Methods / Datasets | CIFAR-10N-a | CIFAR-10N-r | CIFAR-10N-w | CIFAR-100N | Animal-10N | Avg. |
| --- | --- | --- | --- | --- | --- | --- |
| Cross Entropy | 91.62 | 90.25 | 85.66 | 56.41 | 80.54 | 80.90 |
| LiSSA [] | 92.13 $\pm$ 0.29 | 90.98 $\pm$ 0.16 | 85.97 $\pm$ 0.47 | 59.24 $\pm$ 0.39 | 81.93 $\pm$ 0.14 | 82.05 |
| TracIn [] | 90.48 $\pm$ 0.12 | 88.09 $\pm$ 0.24 | 85.18 $\pm$ 1.05 | 56.47 $\pm$ 1.87 | 80.12 $\pm$ 0.57 | 80.07 |
| EKFAC [] | 91.76 $\pm$ 0.23 | 90.47 $\pm$ 0.10 | 83.25 $\pm$ 0.38 | 59.91 $\pm$ 0.90 | 80.89 $\pm$ 0.54 | 81.26 |
| DataInf [] | 91.88 $\pm$ 0.39 | 90.79 $\pm$ 0.21 | 86.22 $\pm$ 0.13 | 58.40 $\pm$ 0.22 | 81.60 $\pm$ 0.23 | 81.78 |
| Self-TracIn [] | 92.03 $\pm$ 0.09 | 90.43 $\pm$ 0.24 | 86.00 $\pm$ 0.18 | 61.99 $\pm$ 0.29 | 81.82 $\pm$ 0.34 | 82.45 |
| Self-LiSSA [] | 91.91 $\pm$ 0.17 | 90.66 $\pm$ 0.35 | 85.73 $\pm$ 0.41 | 61.56 $\pm$ 0.56 | 81.23 $\pm$ 0.24 | 82.22 |
| TDA [] | 91.95 $\pm$ 0.19 | 89.87 $\pm$ 0.32 | 84.02 $\pm$ 0.41 | 58.91 $\pm$ 0.48 | 80.57 $\pm$ 0.25 | 81.06 |
| GEX [] | 91.81 $\pm$ 0.27 | 90.68 $\pm$ 0.39 | 85.64 $\pm$ 0.20 | 58.47 $\pm$ 0.48 | 80.78 $\pm$ 0.58 | 81.49 |
| IP (Ours) | 92.42 $\pm$ 0.17 | 90.82 $\pm$ 0.08 | 86.31 $\pm$ 0.35 | 60.59 $\pm$ 0.20 | 81.19 $\pm$ 0.22 | 82.27 |
| IP Ensemble (Ours) | 92.26 $\pm$ 0.19 | 91.28 $\pm$ 0.29 | 86.50 $\pm$ 0.35 | 62.25 $\pm$ 0.54 | 82.35 $\pm$ 0.55 | 82.93 |

## 5 Noisy Label Correction for Vision Datasets

In this section, we demonstrate the effectiveness of our IP Ensemble in identifying detrimental samples on noisy vision datasets. Specifically, we choose three benchmark datasets CIFAR-10N [], CIFAR-100N [], and Animal-10N datasets [] in the noisy label learning area. CIFAR-10N encompasses three distinct noise settings: aggregate, random, and worst, denoted as “-a,” “-r,” and “-w,” respectively. “a” means that labels are derived via majority voting among three annotators, with ties being resolved randomly, “r” adopts the label provided by the first annotator, while “w” selects the label from the least reliable annotator.

For competitive methods, we choose the following influence function-based methods. TracIn [] replaces the Hessian matrix with the identity matrix and considers checkpoints during the training process; LiSSA [] and EKFAC [] employ implicit Hessian-vector products and Kronecker-Factored curvature to efficiently approximate the inverse of the Hessian matrix; DataInf [] swaps the order of the matrix multiple for obtaining a closed-form expression; Self-TracIn [] and Self-LiSSA [] are the self-expression versions of TracIn and LiSSA, where $\nabla v^{\textrm{util}}$ is replaced with $\nabla_{\hat{\theta}}\ell(z_{i};\hat{\theta})$ and only the last checkpoint, i.e., the converged model parameters, is used in Self-TracIn. GEX [] utilizes ensemble methods based on checkpoints from extra stochastic gradient descent on the converged model. TDA [] focuses checkpoints during the training process, ensembles the influence via EKFAC []. Our IP Ensemble method constitutes an ensemble version of IP with $\mathcal{U}(0,0.01)$ dropout applied on model parameters and an ensemble size of 5.

To reduce randomness, we train a ResNet-34 network [] once on each dataset to establish a baseline model. Subsequently, based on the same model, we employ the aforementioned influence function-based methods to identify 5% of detrimental samples. Following this, we conduct five retraining iterations of the ResNet-34 network, each time removing the identified detrimental samples from the training set. We report the average accuracy and standard deviation of the above influence function-based methods across these five retrainings in Table 1. In general, these influence function-based methods are effective in identifying detrimental samples. Upon retraining the ResNet-34 model without these identified samples, nearly every result obtained by these methods outperforms the vanilla ResNet-34 trained on the entire dataset, except for EKFAC on CIFAR-10N-w. Notably, our IP Ensemble consistently outperforms other baseline methods across various noise conditions and datasets. Particularly noteworthy is the performance of our IP Ensemble in the most challenging scenario, CIFAR-100N, achieving the highest recorded accuracy of 62.25% on the test set, surpassing the vanilla cross-entropy accuracy of 56.41%. Moreover, TDA and TracIn only achieve moderate performance because they introduce multiple checkpoints throughout the training process. These approaches might dilute the effectiveness of identifying truly detrimental samples for the converged model, as the influence of each checkpoint may vary. Additionally, compared to IP, IP Ensemble delivers further performance gains, highlighting the benefits of the ensemble strategy for enhancing model generalization. Moreover, the average accuracy of the IP Ensemble on the test set reaches its peak at 82.93%, surpassing both the vanilla cross-entropy performance of 80.90% and the second-best accuracy of Self-TracIn at 82.45%. More experimental results and analysis on different percentages of removed samples, parameter analysis on different dropout rates, ensemble sizes, and base model architectures can be found in Appendix C.2- C.4, respectively.

Figure: Figure 4: Speed improvement factors of IP over other baseline methods.
Refer to caption: x4.png

In addition, we also present the running times of these influence function-based methods. Despite some baselines having linear time complexity, there is significant divergence in their real execution times. Given that our IP exhibits exceptional speed and similar execution times, we consider them as the baseline and compute the speed improvement factors over other baseline methods, as depicted in Figure 4. For ensemble methods including TracIN, TDA, GEX, and our IP Ensemble, parallel computation can be applied to accelerate the running time if enough resources are allowed; if calculated serially, the time grows linearly with the ensemble size. We do not report their running time in Figure 4. With the exception of Self-TracIn, our IP runs over 100 times faster than LiSSA, EKFAC, DataInf, and Self-LiSSA. Notably, on Animal-10N, IP is over 800 times faster than EKFAC. It is worth noting that Self-TracIn and Self-LiSSA are slower than their standard versions due to the fixed $\nabla v^{\textrm{util}}$ when calculating the influence of each sample. The simplicity and efficiency of IP make it a promising tool for analyzing sample influence on deep models. The time complexities and execution time of these methods can be found in Appendix C.5.

Figure: Figure 5: Accuracy and fair score of different influence function-based methods on fine-tuning RTE, COLA, and QNLI datasets. The X-axis denotes accuracy, and the Y-axis for fair score is inverted. The brown crossing denotes the performance of using all the samples for fine-tuning the RoBERTa model as the baseline model. Building upon this, we plot a horizontal and a vertical line in each figure and divide the space by fairness and utility results into four regions. The green area in the top right corner signifies a model that is both fairer and more accurate compared to the baselines.
Refer to caption: x5.png

## 6 Data Curation Towards Fine-Tuning of Fairer NLP Models

In this section, we further demonstrate the efficacy of our IP method in gauging the impact of individual samples on fairness within the realm of curating suitable data samples for fine-tuning language models. Beyond mere utility, fairness has emerged as an indispensable attribute for machine learning models to mitigate inadvertent discrimination.

In this experiment, we employ three datasets—RTE, COLA, and QLIN—from the GLUE repository [] to fine-tune the widely-used language model RoBERTa []. Our focus is on group fairness, necessitating that machine learning models treat samples within various predefined subgroups comparably. To assess fairness, we adopt the methodology outlined in [], which involves perturbing the demographic information within each sample and scrutinizing whether the model yields identical predictions for the original sample $x$ and its corresponding perturbed counterpart $\tilde{x}$. This evaluation metric "fair score" is defined as $|\mathcal{C}(x)-\mathcal{C}(\tilde{x})|$ over all the samples in the test set, normalized by the size of the test set, where $\mathcal{C}(\cdot)$ is the model predictor. Note that the fair score is a negative metric, so smaller values are preferable. Utility and fairness serve as distinct perspectives for evaluating the performance of a model. Thus, in our fine-tuning experiments, we consider both utility and fairness. Employing the same influence function-based methods and our IP Ensemble as in the previous section, we conduct a comparative analysis. For each method, we calculate the influence on both utility and fairness within the fine-tuning set. Subsequently, We identify and remove the 5% most detrimental samples in terms of utility and fairness before fine-tuning the RoBERTa model to optimize performance across both accuracy and fairness metrics.

The results of this experiment are presented in Figure 5, where the Y-axis for fairness is inverted. The brown crossing denotes the performance of using all the samples for model fine-tuning as the base model. Building upon this, we plot a horizontal and a vertical line in each figure and divide the space by fairness and utility results into four regions. The green area in the top right corner signifies a model that is both fairer and more accurate compared to the baseline model. Results within this green region can be considered Pareto improvements, enhancing both utility and fairness simultaneously. It is evident most results are located in the green area, indicating the existence of detrimental samples, and
not all the samples are helpful to the model performance. This also implies that influence function methods are effective to identifying the detrimental samples, even for non-convex deep models. However, it is important to note that some competitive methods yield a trade-off or even Pareto deterioration. For instance, LiSSA demonstrates a better fair score but worse accuracy compared to the base model on RTE; conversely, it exhibits better accuracy but worse fairness on QLIN. EKFAC shows similar performance on COLA and QLIN. Self-LiSSA demonstrates Pareto deterioration on COLA. We hypothesize that the approximation of the inverse of the Hessian matrix may suffer from large errors, leading to heavily divergent influence estimations from the true values. Our IP Ensemble consistently achieves Pareto improvements across all three datasets, yielding the best results compared to other influence-based methods.

Figure: Figure 6: Performance variations across various influence function-based methods over 10 distinct attacks on Bank, CelebA, and JigsawToxicity. The dashed gray line presents the pre-attack performance, while the brown line denotes the average accuracy of post-attack over 10 runs.
Refer to caption: x6.png

## 7 Defense Against Adaptive Evasion Adversaries

In this section, we demonstrate how the influence-based approach can effectively fortify defenses against an adaptive adversary [, ] that performs evasion attacks on the learning model. In this scenario, the attacker randomly selects a subset of test samples to launch the evasion attack. We defend by proactively trimming the training set by a predetermined amount, although we lack specific knowledge about which samples are adversarial during inference.

In this experiment, we utilize a Logistic Regression model with three datasets: Bank [], CelebA [], and JigsawToxicity []. Following the protocol in Section 3, we consider a white-box adversary [] to craft adversarial samples. For each sample $(x_{j},y_{j})$ in the validation set $\mathcal{V}$, we perturb it by changing the feature $x^{\prime}_{j}=x_{j}-\gamma\frac{\hat{\theta}^{\top}x_{j}+b}{\hat{\theta}^{
\top}\hat{\theta}}\hat{\theta}$ and keeping $y_{j}$ unchanged. The attacker perturbs between 5% and 25% of the test set samples at random. By quantifying the impact of samples on model robustness, we trim 5% detrimental samples in the training set through influence functions. The boxplot depicted in Figure 6 demonstrates the performance variations across various influence function-based methods over 10 distinct attacks. Since the Logistic Regression model is convex, TDA simplifies to EKFAC, and TracIn and GEX simplify to IP; therefore, we will omit their duplicate results in this figure and the following table. In general, influence function-based methods are highly effective against adaptive evasion attackers, showcasing superior resilience particularly post-attack, notably evident in scenarios involving Bank and JigsawToxicity. Among these influence-based methods, our IP Ensemble demonstrates competitive efficacy with the best or the second best on all the datasets.

**Table 2: Defense performance of various influence function-based methods under the relabeling and reweighting strategies on Bank, CelebA, and JigsawToxicity datasets**
| Defense Strategy | Relabeling | Reweighting |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- |
| Bank | CelebA | JigsawToxicity | Avg | Bank | CelebA | JigsawToxicity | Avg |  |
| Pre | 80.31 | 85.26 | 73.58 | 79.72 | 80.31 | 85.26 | 73.58 | 79.72 |
| Post | 70.79 $\pm$ 3.71 | 75.33 $\pm$ 3.04 | 66.18 $\pm$ 3.52 | 70.77 | 70.79 $\pm$ 3.71 | 75.33 $\pm$ 3.04 | 66.18 $\pm$ 3.52 | 70.77 |
| LiSSA [] | 86.07 $\pm$ 0.39 | 68.68 $\pm$ 1.39 | 73.30 $\pm$ 2.96 | 76.02 | 78.69 $\pm$ 3.56 | 73.23 $\pm$ 3.70 | 70.13 $\pm$ 1.00 | 74.02 |
| EKFAC [] | 87.38 $\pm$ 0.89 | 77.36 $\pm$ 1.94 | 70.04 $\pm$ 0.96 | 78.26 | 83.50 $\pm$ 5.90 | 75.11 $\pm$ 1.73 | 70.05 $\pm$ 0.76 | 76.22 |
| DataInf [] | 87.46 $\pm$ 2.44 | 77.36 $\pm$ 1.85 | 70.39 $\pm$ 0.84 | 78.40 | 85.99 $\pm$ 2.49 | 74.74 $\pm$ 1.68 | 70.82 $\pm$ 0.35 | 77.18 |
| Self-TracIn [] | 86.12 $\pm$ 1.10 | 75.58 $\pm$ 2.97 | 68.31 $\pm$ 2.73 | 76.67 | 85.22 $\pm$ 1.72 | 73.12 $\pm$ 3.71 | 71.37 $\pm$ 1.81 | 76.57 |
| Self-LiSSA [] | 78.97 $\pm$ 3.40 | 75.55 $\pm$ 2.99 | 69.18 $\pm$ 1.76 | 74.57 | 85.16 $\pm$ 1.74 | 73.17 $\pm$ 3.72 | 71.41 $\pm$ 1.86 | 76.58 |
| IP (Ours) | 87.45 $\pm$ 0.27 | 77.28 $\pm$ 1.87 | 70.43 $\pm$ 0.87 | 78.38 | 86.17 $\pm$ 2.04 | 75.10 $\pm$ 1.71 | 70.82 $\pm$ 0.36 | 77.36 |
| IP Ensemble (Ours) | 87.45 $\pm$ 0.32 | 77.30 $\pm$ 1.89 | 70.51 $\pm$ 1.00 | 78.42 | 86.44 $\pm$ 2.58 | 75.10 $\pm$ 1.73 | 70.67 $\pm$ 0.27 | 77.40 |

In addition to the trimming strategy, we continue to explore relabeling and reweighting strategies to tackle detrimental samples. The relabeling strategy [] changes the identified detrimental samples from their original classes into another class, since the datasets we used here are binary classes, we directly flip their labels. The reweighting strategy [] takes the influence score of each sample as the exponential weight with a softmax normalization, and then trains a model with weighted samples. We report the defense performance of various influence function-based methods and our IP Ensemble under the relabeling and reweighting strategies in Table 2. Under the relabeling strategy, DataInf achieves the best results on Bank and CelebA with 87.46% and 77.36% accuracy, respectively. However, LiSSA is not stable as other methods on CelebA with only 68.68%, which is even worse than the performance of post attack. Our IP ensemble method continues to achieve competitive performance, with accuracy scores of 87.45%, 77.30%, and 70.51% on three datasets with the second-best performance among all influence functions-based methods. For the reweighting strategy,
the influence function-based methods are still effective on Bank and JigsawToxicity, which outperform the post-attack by a large margin, but achieve similar performance with post attack on CelebA. Notably, when considering the average accuracy of both the relabeling and reweighting strategies, our IP Ensemble method performs the best.

## 8 Conclusion

In this paper, we revisited, extended, and enhanced the TracIn method into the Inner Product (IP) formulation. By proving the exhibition of order-consistency in several scenarios between IP and influence function, we demonstrated substituting the inverse of the Hessian matrix with an identity matrix offers a practical and computationally efficient solution to estimating sample influence. Based on that, we extended our IP to measure the sample influence on fairness and robustness. Continually, we enhanced the generalization of IP by introducing an ensemble strategy. We verified the correctness of IP on synthetic datasets and extensive evaluations on noisy label detection, data curation for fair NLP model fine-tuning, and defense against adaptive adversarial attacks. Overall, our IP Ensemble highlighted the potential of simple, yet powerful, approximations in influence estimation.

## Appendix

## Appendix A Rationality of Inner Product

In Figure 1, we visualize the direction of $\nabla v^{\textrm{util}}\mathbf{H}^{-1}_{\hat{\theta}}$ and $\nabla v^{\textrm{util}}$, with the influence scores roughly representing the angle between the sample gradient and $\nabla v^{\textrm{util}}\mathbf{H}^{-1}_{\hat{\theta}}$ or $\nabla v^{\textrm{util}}$, where $\mathbf{H}^{-1}_{\hat{\theta}}$ induces a rotation of $\alpha$ between these two directions. For a complete analysis, we further split Region II and IV into Sub-Regions as shown in Figure 7 assuming $\alpha<45\degree$. Given gradients of two samples $z_{i}$ and $z_{i^{\prime}}$, we analyze all possible scenarios as follows.

- •
(I, I). In the scenario where both $z_{i}$ and $z_{i^{\prime}}$ have gradients in Region I, $\mathcal{I}^{\textrm{util}}(-z_{i})\gtrless\mathcal{I}^{\textrm{util}}(-z_{i^{
\prime}})$ implies $\mathcal{I}_{\textrm{IP}}^{\textrm{util}}(-z_{i^{\prime}})\gtrless\mathcal{I}_
{\textrm{IP}}^{\textrm{util}}(-z_{i})$, which is the opposite of order-consistency. While these samples are jointly recognized by $\mathcal{I}^{\textrm{util}}$ and $\mathcal{I}^{\textrm{util}}_{\textrm{IP}}$ as detrimental, there is little practical difference.
- •
(I, II). In the scenario where $z_{i}$ has gradients in Region I and $z_{i^{\prime}}$ has gradients in Region II, the order-consistency partially holds, depending on which Sub-Region the gradients of $z_{i^{\prime}}$ is in. For Sub-Region $\mathrm{II^{b}}$ and $\mathrm{II^{c}}$, we have $\mathcal{I}^{\textrm{util}}(-z_{i})>\mathcal{I}^{\textrm{util}}(-z_{i^{\prime}})$ and $\mathcal{I}_{\textrm{IP}}^{\textrm{util}}(-z_{i})>\mathcal{I}_{\textrm{IP}}^{
\textrm{util}}(-z_{i^{\prime}})$, and the order-consistency holds. For Sub-Region $\mathrm{II^{a}}$, however, $\mathcal{I}^{\textrm{util}}(-z_{i})\gtrless\mathcal{I}^{\textrm{util}}(-z_{i^{
\prime}})$ and $\mathcal{I}_{\textrm{IP}}^{\textrm{util}}(-z_{i})>\mathcal{I}_{\textrm{IP}}^{
\textrm{util}}(-z_{i^{\prime}})$, thus the order-consistency partially holds. While samples in Sub-Region $\mathrm{II^{a}}$ are jointly recognized by $\mathcal{I}^{\textrm{util}}$ and $\mathcal{I}^{\textrm{util}}_{\textrm{IP}}$ as detrimental, there is little practical difference.
- •
(I, III). In the scenario where $z_{i}$ has gradients in Region I and $z_{i^{\prime}}$ has gradients in Region III, $\mathcal{I}^{\textrm{util}}(-z_{i})>\mathcal{I}^{\textrm{util}}(-z_{i^{\prime}})$ and $\mathcal{I}_{\textrm{IP}}^{\textrm{util}}(-z_{i})>\mathcal{I}_{\textrm{IP}}^{
\textrm{util}}(-z_{i^{\prime}})$, therefore the order-consistency holds.
- •
(II, IV). In the scenario where $z_{i}$ has gradients in Region II and $z_{i^{\prime}}$ displays gradients in Region IV, the concept of order-consistency is partially maintained.
- •
(II, II). In the scenario where both $z_{i}$ and $z_{i^{\prime}}$ have gradients in Region II, $\mathcal{I}^{\textrm{util}}(-z_{i})\gtrless\mathcal{I}^{\textrm{util}}(-z_{i^{
\prime}})$ implies $\mathcal{I}_{\textrm{IP}}^{\textrm{util}}(-z_{i})\gtrless\mathcal{I}_{\textrm{
IP}}^{\textrm{util}}(-z_{i^{\prime}})$, the order-consistency holds.

Based on the symmetry in samples ($z_{i}$ and $z_{i^{\prime}}$) and regions (I and III, II and IV), we can get all order-consistencies in Table 3. The bold parts are discussed above, and the rest parts are from the symmetry. In all scenarios where order-consistency does not hold, samples are jointly recognized by both $\mathcal{I}^{\textrm{util}}$ and $\mathcal{I}^{\textrm{util}}_{\textrm{IP}}$ as either beneficial or detrimental, resulting in minimal practical difference.

Figure: Table 3: Order-consistency in different scenarios
Refer to caption: x7.png

## Appendix B Detailed Information on Datasets and Model Training

We describe dataset details, model training, and other information used in the main paper, below.

### B.1 Datasets

We cover our generated synthetic datasets in Section 4, vision datasets with noisy labels in Section 5, text datasets in Section 6, and datasets for robustness in Section 7.

#### B.1.1 Synthetic Datasets

We generate two synthetic datasets to validate the correctness of our IP method in measuring the convex and non-convex models’ utility, fairness, and robustness. Specifically, we generate a linear dataset using the scikit-learn [] library’s make_blobs function, which consists of 150 training samples and 100 test samples. The second dataset is the non-linear half moons dataset so that we can train a Multi-Layer Perception network with two hidden layers with ReLU activations. The training set has 250 samples and the test set has 100 samples, and the dataset is generated using the scikit-learn library’s make_moons function. Here we manually flip the labels of 10 samples from each class to add noise to the data.

#### B.1.2 Vision Datasets

Both the CIFAR-10N and CIFAR-100N datasets [] consist of the same input images that make up the CIFAR-10 (10 classes) and CIFAR-100 (100 classes) datasets [], respectively. Each input is a 32x32 RGB image with a dimension of (3,32,32). However, for CIFAR-10N and CIFAR-100N, the labels are noisy, as they contain real-world human annotation errors collected using 3 annotators on Amazon Mechanical Turk. As these datasets are based on human-annotated noise, they model noisy real-world datasets more realistically, compared to synthetic data alternatives. The training set for both datasets contains 50,000 image-label pairs, and the test set contains 10,000 image-label pairs that are free from noise. For CIFAR-10N we utilize three noise settings for experiments in the paper– (1) Worst is the dataset version with the highest noise rate (40.21%) as the worst possible annotation label for the image is chosen, (2) Aggregate is the least noisy dataset (9.03%) as labels are chosen via majority voting amongst the annotations, and (3) Random has intermediate noise (17.23%) and consists of picking one of the annotators’ labels. We use the first annotator for the random labels. Due to the substantial number of labeling classes in CIFAR-100N, only a single noisy setting is available, with an overall noise rate of 40.20%.

#### B.1.3 NLP Datasets

For the three GLUE datasets—RTE, CoLA, and QNLI[], we perturbe the validation set as follows.
The Recognizing Textual Entailment (RTE) dataset consists of sentence pairs labeled as entailment or not entailment. It is derived from a series of annual textual entailment challenges. The dataset includes 2,490 training examples and 277 validation examples. Since the test set does not have labels, we split the validation set into two parts, with one half used as the validation set for computing influence, and the other half used as the test set. Subsequently, we generated perturbed datasets for both the validation set and the test set using a seq2seq model [].
The Corpus of Linguistic Acceptability (CoLA) dataset consists of sentences labeled as grammatically acceptable or unacceptable. This dataset is derived from publications in linguistics and includes 8,551 training examples and 1,043 validation examples. As with RTE, the test set for CoLA does not have labels, so we split the validation set into two parts, using one half as the validation set for computing influence, and the other half as the test set. Perturbed datasets were generated using a seq2seq model [] for both the validation set and the test set.
The Question Natural Language Inference (QNLI) dataset is a large-scale corpus for question answering, consisting of question-sentence pairs from the Stanford Question Answering dataset. The task is to determine whether the context sentence contains the answer to the question. It includes 104,743 training examples and 5,463 validation examples. Similar to the other datasets, the test set does not have labels, so we split the validation set into two parts, using one half as the validation set for computing influence, and the other half as the test set. Perturbed datasets were also generated using a seq2seq model [] for both the validation set and the test set.
These datasets represent diverse natural language understanding tasks, and using them helps evaluate the fairness and utility of the fine-tuned RoBERTa model.

#### B.1.4 Datasets for Defencing Attacks

We utilize three datasets: Bank, CelebA, and JigsawToxicity, to evaluate the defense against adaptive evasion adversaries.
The Bank dataset [] consists of features extracted from direct marketing campaigns of a Portuguese banking institution. The goal is to predict whether a client will subscribe to a term deposit. The dataset includes 18,292 training examples, 6,098 validation examples, and 6,098 test examples. The feature dimension of the dataset is 51. Both the validation and test sets have labels.
The CelebA dataset [] is a large-scale face attributes dataset with more than 200,000 celebrity images, each annotated with 40 binary attributes. For this experiment, we focus on a subset of these images. The dataset is split into 62,497 training examples, 20,833 validation examples, and 20,833 test examples. The feature dimension of the dataset is 39. Both the validation and test sets have labels.
The JigsawToxicity dataset [] contains a large number of comments from Wikipedia labeled for toxicity. The goal is to predict the toxicity level of a given comment. The dataset includes 18,000 training examples, 6,000 validation examples, and 6,000 test examples. The feature dimension of the dataset is 385.
These datasets cover a range of tasks from marketing prediction and face attribute recognition to toxicity detection, providing a robust evaluation of the model’s defense mechanisms against adaptive adversaries.

### B.2 Models and Methods

We now describe the models and the methods used in our experiments throughout the main paper. First, we describe the ResNet-34 [] architecture used as the base model for the noisy vision datasets, then the RoBERTa [] NLP transformer model. We also describe implementation details and parameter values for the label correction baselines used in Sections 7 and the influence-based baselines used throughout the paper.

#### B.2.1 MLP

In our experiments as described in Section 4, we utilized a three-layer Multi-Layer Perceptron (MLP) classifier. This MLP is structured with three fully connected layers, each followed by a ReLU activation function. Specifically, the input layer maps the input features to 32 neurons, the hidden layer maintains this dimensionality, and the output layer is a single neuron producing the final prediction. The final layer’s output is passed through a sigmoid activation function, generating a probability score for prediction.

#### B.2.2 ResNet-34

The ResNet-34 model in Section 5 was proposed in [] and is a 34-layer convolutional neural network pretrained on the ImageNet-1K dataset at resolution 224 $\times$ 224. The pretrained model block is fine-tuned on the CIFAR-10N/CIFAR-100N training set experiments with default parameters– minibatch size (128), optimizer (SGD), initial learning rate (0.1), momentum (0.9), weight decay (0.0005), and
number of epochs (100), for all experiments.

#### B.2.3 RoBERTa

We use the Roberta-base model from Huggingface(^2^22[https://huggingface.co/docs/transformers/model_doc/roberta](https://huggingface.co/docs/transformers/model_doc/roberta)) in Section 6. The learning rate is 0.00001 and the batch sizes of RTE, CoLA, and QNLI are 64, 16, and 32. The model is fine-tuned over 10 epochs. The loss function used is a negative log-likelihood as the datasets are all for binary classification.

#### B.2.4 Logistic Regression

The Logistic Regression model in Section 7 is implemented using the sklearn library. The model uses L2 regularization with and a maximum iteration limit of 2,048 [].

#### B.2.5 Influence-Based Baselines

In our experiments, we utilize the following influence function-based methods as baselines: TracIn [] replaces the Hessian matrix with the identity matrix; LiSSA [] and EKFAC [] use implicit Hessian-vector products and Kronecker-Factored curvature to efficiently approximate the inverse of the Hessian matrix; DataInf [] swaps the order of matrix multiplication to obtain a closed-form expression; Self-TracIn [] and Self-LiSSA [] are self-expression versions of TracIn and LiSSA, where $\nabla v^{\textrm{util}}$ is replaced with $\nabla_{\hat{\theta}}\ell(z_{i};\hat{\theta})$. GEX [] utilizes ensemble methods based on checkpoints from extra stochastic gradient descent on the converged model. TDA [] focuses checkpoints during the training process, ensembles the influence via EKFAC [].
Our IP closely resembles TracIn and GEX, with the added application of dropout with $\mathcal{U}(0,0.01)$ on the model parameters. TracIn, GEX, IP, and IP Ensemble are all Hessian-free and they are ensemble versions, except for IP. Here, we present their calculations to highlight the differences between them as follows:

$$ $\begin{split}\mathcal{I}^{\textrm{util}}_{\textup{IP}}(-z_{i},\hat{\theta})&= \sum\nolimits_{z_{j}\in V}\nabla_{\hat{\theta}}\ell(z_{j};\hat{\theta})^{\top} \nabla_{\hat{\theta}}\ell(z_{i};\hat{\theta}),\\ \mathcal{I}_{\textrm{TracIn}}^{\textrm{util}}(-z_{i},\Theta_{\textrm{TracIn}}) &=\frac{1}{T}\sum\nolimits_{\hat{\theta}_{t}\in\Theta_{\textrm{TracIn}}} \mathcal{I}^{\textrm{util}}_{\textup{IP}}(-z_{i},\hat{\theta}_{t}),\\ \mathcal{I}_{\textrm{GEX}}^{\textrm{util}}(-z_{i},\Theta_{\textrm{GEX}})&= \frac{1}{T}\sum\nolimits_{\hat{\theta}_{t}\in\Theta_{\textrm{GEX}}}\mathcal{I} ^{\textrm{util}}_{\textup{IP}}(-z_{i},\hat{\theta}_{t}),\\ \mathcal{I}_{\textrm{IP Ensemble}}^{\textrm{util}}(-z_{i},\Theta_{\textrm{IP Ensemble}})&=\frac{1}{T}\sum\nolimits_{\hat{\theta}_{t}\in\Theta_{\textrm{IP Ensemble}}}\mathcal{I}^{\textrm{util}}_{\textup{IP}}(-z_{i},\hat{\theta}_{t}), \\ \end{split}$ (5) $$

where $\hat{\theta}$ in IP is the converged model parameters; $\Theta_{\textrm{TracIn}}$ denotes the set of model parameters from the saved checkpoints during model training; $\Theta_{\textrm{GEX}}$ is obtained by training the converged model for several extra iterations; IP Ensemble gets $\Theta_{\textrm{IP Ensemble}}$ by applying dropout mechanisms on the model; $T$ is the ensemble size. The advantages of IP Ensemble are that it does not require recording checkpoints compared to TracIn, and it does not need extra model training processes compared to GEX. We argue that the early checkpoints might not be helpful to analyze the sample influence and the extra iterations on the converged model make little effects.

Figure: Figure 8: Illustrating our inner product approach on measuring fairness and robustness. A and D illustrate the same 2D linearly separable synthetic dataset in Figure 2 trained using a Logistic Regression model for binary classification, where the solid and dashed point boundaries denote the majority and minority subgroups. B and E represent the validation set. C and F show the estimated influence on fairness and robustness by the traditional influence function and our IP.
Refer to caption: x8.png

## Appendix C Additional Experimental Results and Analysis

### C.1 IP for measuring fairness and robustness

Similar to Figure 2, we conduct the experiments based on Eqs. (3) and (4) to demonstrate the effectiveness of our IP in measuring the sample influence on fairness and robustness. Figure 8 shows the relationship between IP and the traditional influence function, indicate that IP is a good surrogate of the traditional influence function in measuring model’s fairness and robustness as well.

### C.2 Performance of IP Ensemble with different rates of removed samples

We also explore different rates of removed samples on CIFAR-10N-a.
Table 4 shows the performance of IP Ensemble with different rates of removed samples. Except for not removing, there is no significant difference in remaining results.

**Table 4: Performance of IP Ensemble with different rates of removed samples on CIFAR-10N-a**
| Method | Ensemble Size | Remove Rate | ACC |
| --- | --- | --- | --- |
| Cross Entropy | 5 | 0 | 91.62 |
| IP Ensemble (Ours) | 5 | 0.025 | 92.29 $\pm$ 0.16 |
| IP Ensemble (Ours) | 5 | 0.050 | 92.26 $\pm$ 0.19 |
| IP Ensemble (Ours) | 5 | 0.075 | 92.50 $\pm$ 0.13 |
| IP Ensemble (Ours) | 5 | 0.100 | 92.37 $\pm$ 0.15 |

### C.3 Parameter Analysis on Dropout Rate and Ensemble Size

Table 5 shows the performance of different dropout rates and ensemble sizes of IP Ensemble. As can be observed, our IP Ensemble is not sensitive to dropout rate, and its performance increases with a large ensemble size, demonstrating the effectiveness of the ensemble strategy. It is also worth noting that although our IP Ensemble runs fast, it might take a longer time to calculate the sample gradient.

**Table 5: Performance of different dropout rate and ensemble size of IP Ensemble on CIFAR-10N-a, CIFAR-10N-r, CIFAR-10N-w, and CIFAR-100N**
| Ensemble Size | Dropout Rate | CIFAR-10N-a | CIFAR-10N-r | CIFAR-10N-w | CIFAR-100N |
| --- | --- | --- | --- | --- | --- |
| 5 | 0.01 | 92.26 $\pm$ 0.19 | 91.28 $\pm$ 0.29 | 86.50 $\pm$ 0.35 | 62.25 $\pm$ 0.54 |
| 5 | 0.1 | 92.26 $\pm$ 0.19 | 91.28 $\pm$ 0.29 | 86.50 $\pm$ 0.35 | 62.25 $\pm$ 0.54 |
| 5 | 0.5 | 92.26 $\pm$ 0.19 | 91.28 $\pm$ 0.29 | 86.50 $\pm$ 0.35 | 62.25 $\pm$ 0.54 |
| 1 | 0.01 | 92.42 $\pm$ 0.17 | 90.82 $\pm$ 0.08 | 86.31 $\pm$ 0.35 | 60.59 $\pm$ 0.20 |
| 5 | 0.01 | 92.26 $\pm$ 0.19 | 91.28 $\pm$ 0.29 | 86.50 $\pm$ 0.35 | 62.25 $\pm$ 0.54 |
| 10 | 0.01 | 92.58 $\pm$ 0.04 | 91.32 $\pm$ 0.29 | 86.89 $\pm$ 0.37 | 61.82 $\pm$ 0.61 |
| 15 | 0.01 | 92.27 $\pm$ 0.09 | 91.27 $\pm$ 0.28 | 86.47 $\pm$ 0.41 | 61.59 $\pm$ 0.34 |
| 20 | 0.01 | 92.41 $\pm$ 0.15 | 91.33 $\pm$ 0.26 | 86.65 $\pm$ 0.16 | 62.25 $\pm$ 0.11 |

### C.4 Performance on ViT and MLP-Mixer

To verify the effectiveness of our IP ensemble across different network architectures, we conducte experiments using ViT [] and MLP-Mixer [] on various vision datasets in Table 6. We use a batch size of 512 for all experiments in this section. The models are trained on CIFAR-10N and CIFAR-100N datasets for 100 epochs, and on Animal-10N for 400 epochs. The learning rate was set to $1\times 10^{-4}$ for ViT and $1\times 10^{-3}$ for MLP-Mixer.

Our IP Ensemble consistently demonstrates superior performance compared to the vanilla models and other baseline methods on different base models. Specifically, when trained on ViT, IP Ensemble achieves an average accuracy of 70.06%, outperforming the vanilla ViT’s average accuracy of 68.87%. This improvement is observed across all datasets, with notable gains on CIFAR-100N (51.01% versus 48.53%) and Animal-10N (77.20% versus 76.20%). Additionally, compared to baseline methods, our IP Ensemble achieves the best performance with ViT or MLP-Mixer based models, indicating the effectiveness of our methods on different base models.

**Table 6: Accuracy results of influence function-based methods with ViT and MLP-mixer as based models on CIFAR10N-a, CIFAR-100N, and Animal-10N with 5% detrimental samples removed.**
| Methods / Datasets | CIFAR-10N-a | CIFAR-100N | Animal-10N | Avg. |
| --- | --- | --- | --- | --- |
| ViT | 81.87 | 48.53 | 76.20 | 68.87 |
| LiSSA [] | 81.83 $\pm$ 0.19 | 48.47 $\pm$ 0.24 | 76.90 $\pm$ 0.35 | 69.07 |
| TracIn [] | 81.59 $\pm$ 0.21 | 48.18 $\pm$ 0.28 | 76.21 $\pm$ 0.58 | 68.66 |
| EKFAC [] | 80.79 $\pm$ 0.56 | 48.90 $\pm$ 0.30 | 76.85 $\pm$ 0.30 | 68.85 |
| DataInf [] | 81.68 $\pm$ 0.21 | 49.00 $\pm$ 0.22 | 76.70 $\pm$ 0.27 | 69.13 |
| Self-TracIn [] | 81.88 $\pm$ 0.12 | 49.25 $\pm$ 0.21 | 76.95 $\pm$ 0.25 | 69.36 |
| Self-LiSSA [] | 82.89 $\pm$ 0.12 | 49.30 $\pm$ 0.18 | 76.80 $\pm$ 0.28 | 69.66 |
| TDA [] | 81.89 $\pm$ 0.28 | 49.05 $\pm$ 0.54 | 76.44 $\pm$ 0.32 | 69.12 |
| GEX [] | 82.99 $\pm$ 0.24 | 49.78 $\pm$ 0.20 | 76.88 $\pm$ 0.29 | 69.88 |
| IP (Ours) | 81.93 $\pm$ 0.22 | 49.12 $\pm$ 0.20 | 76.80 $\pm$ 0.22 | 69.28 |
| IP Ensemble (Ours) | 81.96 $\pm$ 0.13 | 51.01 $\pm$ 0.17 | 77.20 $\pm$ 0.24 | 70.06 |
| MLP-Mixer | 74.42 | 35.63 | 72.69 | 60.91 |
| LiSSA [] | 75.35 $\pm$ 0.47 | 36.50 $\pm$ 0.21 | 72.89 $\pm$ 0.28 | 61.58 |
| TracIn [] | 74.43 $\pm$ 0.28 | 35.21 $\pm$ 0.35 | 72.71 $\pm$ 0.43 | 60.78 |
| EKFAC [] | 75.49 $\pm$ 0.28 | 36.34 $\pm$ 0.24 | 73.12 $\pm$ 0.19 | 61.65 |
| DataInf [] | 75.10 $\pm$ 0.45 | 35.78 $\pm$ 0.29 | 73.17 $\pm$ 0.21 | 61.35 |
| Self-TracIn [] | 75.88 $\pm$ 0.21 | 36.72 $\pm$ 0.19 | 73.81 $\pm$ 0.54 | 62.14 |
| Self-LiSSA [] | 75.41 $\pm$ 0.38 | 36.05 $\pm$ 0.27 | 72.56 $\pm$ 0.34 | 61.34 |
| TDA [] | 74.89 $\pm$ 0.31 | 36.84 $\pm$ 0.38 | 71.88 $\pm$ 0.43 | 61.20 |
| GEX [] | 75.57 $\pm$ 0.27 | 36.85 $\pm$ 0.25 | 73.89 $\pm$ 0.43 | 62.10 |
| IP (Ours) | 75.04 $\pm$ 0.23 | 36.15 $\pm$ 0.18 | 73.47 $\pm$ 0.20 | 61.55 |
| IP Ensemble (Ours) | 75.77 $\pm$ 0.19 | 37.12 $\pm$ 0.18 | 73.80 $\pm$ 0.11 | 62.23 |

### C.5 Time Complexity and Execution Time of Various Influence Function-based Methods on Vision Datasets

Table 7 shows the time complexity of various influence-based methods. Except for the vanilla one, all other methods have linear time complexity in terms of the sample size. However, they have large divergence in real execution time.

We omit the specific timing details for the sample gradient, which are readily available during the base model’s training phase. Besides, utilizing vmap in Pytorch, we can efficiently compute gradients in parallel. For example, in Section 5, it only takes 61 seconds and 4 seconds to calculate $\nabla v^{\textrm{util}}$ and $\nabla_{\hat{\theta}}\ell(z_{i};\hat{\theta})$ on CIFAR-10N dataset, respectively.

**Table 7: Computational complexity of influence-function-based methods ($n$ is #training samples and $p$ is #model parameters, $k$ is #checkpoints or #ensemble size with $k$$=$$1$ here. "-" denotes no runs.)**
| Method | Type | Time Complexity | CIFAR-10N | CIFAR-100N | Animal-10N |
| --- | --- | --- | --- | --- | --- |
| Exact by Eq. (1) | Hessian-based | $\mathcal{O}(np^{3})$ | - | - | - |
| LiSSA [] | Hessian-based | $\mathcal{O}(np)$ | 7.67 | 34.59 | 7.46 |
| TracIn [] | Hessian-free | $\mathcal{O}(npk)$ | 0.03 | 0.28 | 0.03 |
| EKFAC [] | Hessian-based | $\mathcal{O}(np^{2})$ | 22.54 | 192.58 | 23.89 |
| DataInf [] | Hessian-based | $\mathcal{O}(np)$ | 11.50 | 45.29 | 10.89 |
| Self-TracIn [] | Self-influence | $\mathcal{O}(npk)$ | 0.15 | 1.41 | 0.15 |
| Self-LiSSA [] | Self-influence | $\mathcal{O}(np)$ | 14.57 | 54.39 | 14.40 |
| GEX [] | Hessian-free | $\mathcal{O}(npk)$ | 0.03 | 0.28 | 0.03 |
| TDA [] | Hessian-based | $\mathcal{O}(np^{2}k)$ | 23.98 | 193.79 | 24.77 |
| IP Ensemble (Ours) | Hessian-free | $\mathcal{O}(npk)$ | 0.03 | 0.28 | 0.03 |

## Appendix D Broader Impact and Limitations

Our work aims to address issues that currently hinder the applicability of influence estimation in deep learning models. By enabling influence estimation for deep models, practitioners can assess whether training samples are beneficial or detrimental to performance. As we demonstrate through extensive experiments across multiple problem settings, our proposed outlier gradient analysis approach outperforms existing baselines and can augment model performance by trimming detrimental samples. As a result, our work paves the way for significant positive societal impact, especially with the increased adoption of larger and deeper neural networks such as LLMs.

However, despite our research demonstrating superior performance and potential benefits, there are several limitations to consider. First, our method assumes a level of model and data homogeneity that might not hold in highly heterogeneous datasets or more complex model architectures.
Furthermore, there is the consideration of robustness and fairness across different domains and types of data; our current evaluations, while extensive, may not cover all possible scenarios or edge cases, potentially limiting the generalizability of our findings.
Finally, ethical considerations around data trimming and sample selection must be carefully managed to avoid unintended biases or negative impacts on model performance for underrepresented groups. As influence estimation becomes more integrated into model training and evaluation pipelines, ongoing research will be essential to ensure that these techniques are applied responsibly and equitably.

Due to the extensive computational demands of our experiments (see details in Appendix E), which require multiple retraining cycles to evaluate model stability and robustness, it becomes impractical for us to conduct comparative experiments on very large datasets, e.g., ImageNet. The limited computational resources at our disposal make it challenging to perform such large-scale retraining efforts, especially given the significant training time and GPU requirements. As a result, we focus our evaluation on middle-size yet representative benchmarks that allow for more feasible experimentation while maintaining the rigor of our comparative analyses.

## Appendix E Code and Reproducibility

Experiments were run on Ubuntu 20.04.6 with NVIDIA RTX 4090 GPUs (24GB VRAM), CUDA 12.3, and driver 545.23.08. The code will be released upon request.

## Peer Reviews

confidence: 5
contribution: 1
initial_rating: 3
presentation: 3
questions: - Could you elaborate on this “We believe that sample influence should be assessed using a fixed model; simply summing sample influences at different stages fails to capture the dynamics of model optimization.” more? It seems hard to understand.
- Noisy label detection and other downstream tasks that do not have a very large difference between each method. Maybe some counterfactual prediction evaluation metrics (e.g., LDS) are more suitable?
soundness: 2
strengths: - The paper is well written, and the reader can understand the main idea of the paper quickly in a short time.
- Efficiency has become a very important topic for TDA(training data attribution).
summary: This paper proposed a simplified method (IP) for training data attribution. The method equipped a hessian-free method to avoid the large computational cost on the Hessian matrix and its inverse. The paper also proposes some tricks (e.g., ensemble) and more applications (e.g., change the target function to robustness/fairness). Some empirical studies are also proposed to show the efficiency improvement and performance.
weaknesses: - There is a large gap between the contribution claimed in this paper and actual literature. The major problem lies in the first and second contribution bullet point in section 1 (line 59 - line 62)
  - The Inner Product (IP) proposed by this paper (as a simplified version of TracIN) has long been proposed [1] and used in a large number of papers[2]. 
  - Replacement of the loss gradient to some other metrics to fairness and robustness is also something tried for influence function or related methods[3].
- The experiment is not fully fair and may intentionally make other baseline methods to have lower results. Furthermore, the improvement is kind of trivial. 
  - The dropout trick can also be used on other methods as well, what are their performance?
  - Some related gradient-based SOTA methods (e.g., TRAK) are ignored. Furthermore, random projection has been a very widely used method for efficiency, but it is also omitted in the paper.
 - The manually added noisy data for Figure 2 (D) seems to be cherry-picked. The negative performance of H^-1 seems to be very unlikely.


Overall the contribution over-claiming is kind of the major problem for this paper, and that’s the main reason why I give a 3 point assessment. 


[1] Charpiat, G., Girard, N., Felardos, L., & Tarabalka, Y. (2019). Input similarity from the neural network perspective. Advances in Neural Information Processing Systems, 32.

[2] Kwon, Y., Wu, E., Wu, K., & Zou, J. (2023). Datainf: Efficiently estimating data influence in lora-tuned llms and diffusion models. arXiv preprint arXiv:2310.00902.

[3] Richardson, B., Sattigeri, P., Wei, D., Ramamurthy, K. N., Varshney, K., Dhurandhar, A., & Gilbert, J. E. (2023, June). Add-Remove-or-Relabel: Practitioner-Friendly Bias Mitigation via Influential Fairness. In Proceedings of the 2023 ACM Conference on Fairness, Accountability, and Transparency (pp. 736-752).

---

confidence: 4
contribution: 1
initial_rating: 3
presentation: 2
questions: What is the precise role and interpretation of the influence function? How does the inverse Hessian factor into the formula?
soundness: 1
strengths: The paper includes a lot of experiments and provides statistical ranges for the reported results.
summary: The paper proposes an approach for measuring influence of samples in trained models by defining it as the inner product between the gradient of a specific sample and the sum of gradients from all other samples. By substituting this gradient sum with a fairness measure, the method claims to capture the influence of a sample on the model's fairness. When this gradient sum is replaced with gradients of adversarially perturbed samples, the method is intended to reflect the impact on robustness. Additionally, the paper suggests enhancing the proposed measurement through an ensemble of models. Experiments were conducted for applications such as detecting noisy labels, curating fairness-sensitive samples, and improving resilience against adversarial attacks.
weaknesses: However, the paper lacks a clear problem statement and positioning of the method within existing approaches. The formula provided to describe the method is the TracIn formula, with the only difference being the reduction of calculations to the final trained model. This raises several questions: (i) the explanation of the benefits of this simplification is vague and could be questioned, (ii) the intuition behind this approximation rests on a comparison with a method involving the Hessian’s inverse and assumes cosine similarity between vectors without a clear justification, and (iii) there is no explanation of what this formula intuitively means in contexts of fairness or adversarial robustness. Furthermore, the paper does not formally define the process for generating ensembles to accumulate the influence scores.

A precise definition of the influence function and an explanation of the role and significance of the inverse Hessian in this context would improve the method’s description.

Section 4, which highlights differences between this approach and an influence method that incorporates the Hessian, presents results on synthetic data and a simple multi-layer perceptron. It shows that Hessian based method fails due to approximation errors. However, it does not clearly demonstrate or discuss the approximation errors, and it is unclear why a setup without approximation errors cannot be tested on more complex, non-linearly separable data. Figure 3 lacks sufficient explanation, diminishing clarity.

In subsequent experiments where the ensemble inner product method shows improvements, only the original method appears beneficial in terms of time complexity. Moreover, as the ensemble size increases, it closely approximates TracIn with multiple measurements, making the specific contributions of this paper unclear.

Section 6 presents improvements in fairness, yet the benchmarks primarily focus on accuracy, limiting the validity of the comparison. Finally, the approach appears to offer limited benefits in adversarial defense.

Minor Issues: The related work section includes descriptions of studies that do not effectively contextualize the current paper’s contributions.

---

confidence: 3
contribution: 2
initial_rating: 3
presentation: 4
questions: What does U(0, 0.01) on line 318 mean in the context of model parameter drop out? Are model parameters set to zero with a chance of 1%, with or without replacement?
soundness: 2
strengths: 1. The proposed method is particularly simple, easy to implement and efficient to compute.
2. The experimental evaluations presented in the paper are fairly thorough and rigorous, with appropriate repeat experiments to establish confidence intervals. 
3. The extension of influence estimation to algorithmic fairness metrics is interesting.
summary: This paper proposes a variation of TracIN called IP, where the ensembling over checkpoints is abandoned in favor of the converged model. 
As this particular variation has appeared several times as baseline in ablation experiments in prior works (e.g. in both TracIN and the original influence function paper), the authors attempt to revisit this formulation with new theoretical and experimental insights.
Some intuition is provided for why this simple setup might work through an argument for similarity to the inverse hessian influence function. Further extensions of IP in the application of fairness, and a more powerful ensemble version of IP, are also presented. Experimentally, the proposed method is competitive with prior art in the benchmarked tasks in terms of downstream performance.

+++++++++++++++++++++++++++++++++++++++++++++++++++++

I have read the author's rebuttal. While I agree with some of the points made by the authors, I'm opting to maintain my original review score as the contribution of the paper and the experimental evidence for impact is still unclear
weaknesses: 1. The order consistency argument for why IP is a good approximation to inverse hessian influence is quite weak. In figure 1, data points with vectors in regions I and III would not satisfy order consistency. The authors argue that since IP and IF both rate such points as beneficial/detrimental, the order doesn’t matter. This is clearly not true, as many applications of influence estimation involve determining set membership at the extreme ends of the influence spectrum (e.g. removing x% detrimental examples based on the most negative influence values or curating x% most beneficial examples for fine tuning based on the highest influence values). Changes to ordering in these regions are in fact more significant than changes to ordering within regions II and IV, which would be assigned neutral influence scores.
2. Furthermore, the conditions under which the angle alpha is small has not been discussed. 
3. While the benchmarks on downstream utility are quite extensive, other quantitative metrics relating to similarity of predictions, such as correlation to the (inverse hessian) influence function, leave-one-out training, and precision-recall curves of label perturbation detection, are missing.

---

confidence: 5
contribution: 1
initial_rating: 3
presentation: 3
questions: The paper clearly cannot be accepted in its current form, but I think a few results on synthetic data in Section 4 look interesting. The authors could explore the following question: under what circumstance (on data and/or model), the directions of $H^{-1}v$ and $v$ are close.
soundness: 1
strengths: - The sanity checks on synthetic data yield some interesting findings, e.g., for non-linear models, IP can separate corrupted samples from normal samples whereas influence function cannot
- The experiments are comprehensive and cover quite a few applications (nevertheless, I think the paper is mostly following Chhabra et al. 2024)
summary: The authors introduce Inner Product (IP), a metric that simplifies the classic influence function by dropping the inverse Hessian term to assess the influence of training samples. They validate IP through sanity checks on synthetic data and demonstrate its superior performance across various applications, including noisy label detection, fairness enhancement, and defense against evasion attacks.
weaknesses: If IP is indeed as useful as the authors were suggesting in this paper, this would be a breakthrough for the field of influence function (and more broadly, data attribution), since the (inverse) Hessian term is indeed the main bottleneck of applying influence functions to large-scale models. Unfortunately, after reading the paper, I am not convinced that IP is a principled and useful metric. 

- The authors confused the relationship between influence function and TracIn, and incorrectly viewed TracIn as an approximation of influence function. Influence function is a first-order approximation of leave-one-out, which measures influence w.r.t. changes to the training set. On the other hand, TracIn is a first-order approximation of the change of loss, which measures the change in model parameters as a function of time. They are fundamentally different metrics since they have completely definition of "influence". The authors seemed to suggest that IP can be derived from both influence function (by replacing the inverse Hessian with identity), and TracIn (by only considering the last iteration). Regardless of whether such simplifications make sense or not, this makes IP hard to interpret --- the basic question is, what notion of "influence" is IP adopting, and what ground truth is it approximating?

- The analysis at the beginning of P4 is completely wrong, as there is no way that IP can preserve order even in specific regions. For instance, in Figure 1, one can easily construct two gradient vectors in Region II as follows: the first vector is orthogonal to the blue vector and has small norm, the second vector forms an acute angle with the blue vector and has large norm. By adjusting the norm, we can reverse the order. The main issue is: removing the inverse Hessian must lose information, and there is no analysis on why such information could potentially be useless. 

- The ensemble procedure is unclear and confusing. The procedure is not formally explained; I assume that the dropout is only applied to the final model checkpoint, since the authors mentioned "diverse models can be swiftly generated without necessitating model retraining". What is the rationale behind this approach? In the literature, people do multiple retraining and ensemble to reduce the inherent noise in training and influence function estimation. It is unclear to me that applying multiple dropout to the final model checkpoint would enjoy the same benefit. 

- Finally, regarding the experimental results, the improvement compared to existing approaches is very limited. In table 1, all methods look similar to me after taking into account the variance.  

There are also some minor points:

- I think the authors should read [1] carefully to develop a better understanding of the literature

- Some algorithms which have similar design (e.g., ensemble) are missing, for instance, TRAK [2]

- Some claims lack evidence, for instance, "We believe that sample influence should be assessed using a fixed model; simply summing sample influences at different stages fails to capture the dynamics of model optimization" --- it seems to me that the essence of TracIn is exactly being dynamic

References

[1] Hammoudeh, Zayd, and Daniel Lowd. "Training data influence analysis and estimation: A survey." Machine Learning 113.5 (2024): 2351-2403.

[2] Park, Sung Min, et al. "Trak: Attributing model behavior at scale." arXiv preprint arXiv:2303.14186 (2023).

Meta-review: The paper proposes a simplified version of influence functions for assessing sample importance in deep learning.
The main concerns that remained after feedback phase remained along the lack of novelty of the paper's primary claimed contribution - the Inner Product (IP) method, and additionally on the correctness of the analysis (specifically on information loss when dropping the Hessian).

We hope the detailed feedback helps to strengthen the paper for a future occasion.

Decision: Reject