Evaluating Data Influence in Meta Learning
Authors: Chenyang Ren
Huanyi Xie
Shu Yang
Meng Ding
Dongrui Liu
Lijie Hu
Di Wang
Venue: ICLR 2026 Poster (2026)
Keywords: Data Attribution
Influence Function
Meta Learning
Primary area: interpretability and explainable AI

Abstract: As one of the most fundamental models, meta learning aims to effectively address few-shot learning challenges. However, it still faces significant issues related to the training data, such as training inefficiencies due to numerous low-contribution tasks in large datasets and substantial noise from incorrect labels. Thus, training data attribution methods are needed for meta learning. However, the dual-layer structure of meta learning complicates the modeling of training data contributions because of the interdependent influence between meta parameters and task-specific parameters, making existing data influence evaluation tools inapplicable or inaccurate. To address these challenges, based on the influence function, we propose a general data attribution evaluation framework for meta learning within the bilevel optimization framework. Our approach introduces task influence functions (task-IF) and instance influence functions (instance-IF) to accurately assess the impact of specific tasks and individual data points in closed forms. This framework comprehensively models data contributions across both the inner and outer training processes, capturing the direct effects of data points on meta parameters as well as their indirect influence through task-specific parameters. We also provide several strategies to enhance computational efficiency and scalability. Experimental results demonstrate the framework's effectiveness in training data evaluation via several downstream tasks.

Published as a conference paper at ICLR 2026
EVALUATING DATA INFLUENCE IN META LEARNING
Chenyang Ren
Ren 1,3,4,, Huanyi Xie Shu Yang Meng Ding", Dongrui Liu Lijie Hu Di Wang 1.
1 Kiin Abdullah University of Science and Technology (KAUST)
Prooabbee Responsible AI and Data Analytics (PRADA) Lab
3 Mohameed bin Zayed University of Artificial Intelligence (MBZUAI)
"Shanghai Jiao Tong University (SJTU)
6 State University of New York at Buffalo
Shanghai Artificial Intelligence Laboratory
ABSTRACT
As one of the most fundamental models, meta learning aims to effectively address few-shot learning challenges. However, it still faces significant issues related the training data, such training inefficiencies due to numerous low-contribution tasks large datasets and substantial noise from incorrect labels. Thus, training data attribution methods are needed for meta learning. However, the dual-layer structure of meta learning complicates the modeling of training data contribu- tions because of the interdependent influence between meta parameters and task- specific parameters, making existing data influence evaluation tools inapplicable inaccurate. To address these challenges, based the influence function, we propose a general data attribution evaluation framework for meta learning within the bilevel optimization framework. Our approach introduces task influence func- tions (task-IF) and instance influence functions (instance-IF) to accurately assess the impact of specific tasks and individual data points in closed forms. This frame- work comprehensively models data contributions across both the inner and outer training processes, capturing the direct effects of data points on meta parameters as well as their indirect influence through task-specific parameters. We also provide several strategies to enhance computational efficiency and scalability. Experimen- tal results demonstrate the framework's 's effectiveness in training data evaluation via several downstream tasks.
INTRODUCTION
Bilevel Optimization (BLO) has received significant attention and has become influential frame- work in various machine learning applications including hyperparameter optimization (Sun et al.,
Okuno et al. 2021), data selection (Borsos et af., 2020 2024). meta learning (Finn et al, and reinforcement learning (Stadie et al. (0220). A general BLO framework consists of two hierarchical optimization levels (outer and inner levels) and can be formulated as the following:
2022
2017
min f(), (A)) s.t. 0* (*) arg min g(A, (",
where the objective and variables of the outer-level problem f are influenced by the inner-level problem g.
Among the various instantiations of BLO, meta learning has gained considerable interest due to its effectiveness in addressing the challenges of few-shot learning (Finn et al., 2017: Jamal & Qi, 2019 Hospedales et al.) 2021] Franceschi et al. 2018; Yang et al. 2021). Meta learning involves two interdependent sets of parameters: meta parameters and task-specific parameters. Its inner level focuses on independently training across multiple few-shot tasks using the meta parameters A supplied by the outer level to derive the task-specific parameters 6* (X). The outer level, in turn, evaluates the performance of the inner level's task-specific parameters trained based on the meta
'The first two authors contributed equally to this work.
Correspondence Lijie Hu (Fmail Address: i jie. hu@mbzzai .ac.ae) and Di Wang (Email Ad- dress: di wang@k aust edu. sa).

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parameter. Specifically, the outer level guides and assesses the training process of the inner level, while the outcomes of the inner level training provide task-specific model support for the outer level.
Recent advancements highlight the applicability of meta learning across various domains, including federated learning (Fallah et al. 2020) and multi-task learning (Wang et al. 202T), underscoring its versatility and potential. However, due to the large number of tasks, the training efficiency of meta learning often suffers, as these tasks do not contribute equally or effectively within extensive meta-datasets. This inefficiency highlights the need discern the contributions of individual tasks to overall model training. Although various task selection methods have been proposed to improve training efficiency (Jamal & Qi 2019 Liu et al.) 2020; Achille et al., 2019), the dual-layer struc- ture complicates the relationship between tasks and meta parameters, making analysis challenging. Additionally, the large datasets used in meta learning often contain some degree of noise, which poses significant issue because model training typically assumes that all labels are accurate, and thus the presence of noisy data can severely hinder model performance (Ding et al., 2024). Fur- thermore, the two-layer bilevel structure complicates the understanding of learning processes and the interpretation of meta parameter predictions for specific tasks, undermining transparency and in high-stakes sectors like healthcare and finance (Kosan et al.) 2023). Thus, it is necessary to develop methods for assessing data quality and attribution of meta learning.
Influence functions have emerged as powerful tool for evaluating training data attribution they efficiently estimate the contributions of data points using gradient information without requiring retraining, in contrast to the Shapley Value. However, it is essential note that influence functions were initially designed for M-estimators (Huber) 1981), which limits their direct applicability to bilevel structures. Due the bilevel structure, data in meta learning can be evaluated two distinct levels: (1) the task level, which includes the training and validation sets for each task used to learn task-specific parameters and assess their effectiveness, and (2) the instance level. which focuses on the specific data points within the train or validation dataset of individual tasks. Each of these levels presents unique challenges for meta learning. First, unlike classical supervised learning models whose objective is single-level optimization problem that is directly influenced by each training data point, the data in meta learning from each task influences the meta parameters directly and indirectly via task-specific parameters. While the influence function approach can be adapted to the meta- meta-parameter by neglecting the inner-level training and indirect influences (as discussed in Section 4.1) this adaptation presents a challenge in accurately quantifying the data's attribution. Secondly, evaluating data is challenging because they do not explicitly appear in the training process of the outer meta parameter; instead, their influence is mediated through task-specific parameters.
To bridge the gap, in this paper, we propose an efficient data evaluation framework for meta learning (under the BLO setting). Specifically, we provide two methods: task-IF for measuring the influence of specific tasks in the meta learning dataset and instance-IF for measuring the influence of specific instances. For the task-level evaluation, our task-IF method emphasizes that the optimal conditions for model training are not only determined by the meta parameters but also incorporate the gradient information from task-specific parameters. At the instance level, we separately analyze data points in the validation and training datasets. We use method similar to the task-IF to estimate data point impact for validation data. For training, we apply a novel two-stage closed-form estimation process across the model's inner and outer levels. First, we estimate the influence of instances on task-specific parameters, then introduce this influence at the outer level to approximate the effect of removing the instance. Our contribution can be summarized as follows:
We conceptualize the data valuation problem in meta learning under the BLO framework, cate- gorizing it into task and instance levels. At the instance level, we propose evaluation methods for training and validation samples. Specifically, we introduce novel task-IF, which allows for precise modeling of task influence in meta learning by effectively capturing influences on both meta parameters and task-specific parameters. Additionally, we address the challenging instance- level evaluation through a two-stage estimation process that comprehensively accounts for both the direct and indirect effects of instances on meta parameters.
To address the computational intensity and instability of inverse Hessian vector products, which have appeared several times in our method, we introduce several acceleration techniques, such as the EK EKC method and Neumann expansion, to enhance computational efficiency and numerical stability, which allows for scaling the method to large-scale neural networks.
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Our method has potential applications in several real-world scenarios, including the automatic identification and removal of harmful tasks, and model editing, and the improvement of the interpretability of meta parameters. We verify the effectivvnees and efficiency of task-IF and instance-IF through comprehensive experimental evaluation. Experimental results indicate that our framework performs well in data evaluation and editing of meta parameters.
2 RELATED WORK
Meta learning. Meta learning, often described as "learning to learn," has emerged as a pivotal ap- proach for enabling models to rapidly adapt to new tasks by leveraging prior experience (Thrun & Pratt 1998). (Finn et al., 2017) introduced Model-Agnostic Meta-Learning (MAML), framing the meta learning process as a bilevel optimization problem where the inner loop adapts model param- eters to specific tasks, and the outer loop optimizes for performance across tasks. This framework has inspired range of aptimization-baser meta learning methods (Finn & Levine, 2017 Finn Finn al.) 2018 Li et al. 2017 Lee Choi 2018 Grant et al., 2018: Flennerhag et al.) 2019). For example, Li.tal 2017) extended MAML to Meta-SGD, which learns not only the initial parameters but also the learning rates, enhancing adaptability across tasks. Grant et al. (2018) reformulate MAML as method for probabilistic inference in a hierarchical Bayesian model. Finn et al. (2018) proposed Probabilistic MAMI. (PMAML), which incorporates uncertainty into the meta learning process by modeling the task distribution probabilistically. [Fennerhagg et al. (2019) introduced the Warped Gradient Descent, addressing the non-convergence due to bias and limited scale. Moreover, metric- based methods such Matching Networks (Vinyals et 2016) and Prototypical Networks (Snell et al. (017)) focus on learning similarity metrics to classity new tasks effectively. Oreshkin et al. (2018b) advanced Prototypical Networks by proposing task task-specific adaptive scaling mechanism to learn a tesendepn metric.
Influence function. The influence function, initially a staple in robust statistics (Cook) 2000 Cook
& Weisberg 1980), has seen extensive adoption within machine learning since Koh & Liang (2017)
introduced to the field. Its versatility spans various applications, including detecting mislabeled data, interpreting models, addressing model bias, and facilitating machine unlearning tasks. Notable works in machine unlearning encompass unlearning features and labels (Warnecke et al.) 2023),
minimax unlearning (Liu et al. 2024), forgetting subset of image data for training deep neural net- works (Golatkar et al. 2020: 2021), graph unlearning involving nodes, edges, and features. Recent advancements, such as the LISSA method (Agarwal et al.) 2017 Kwon al. 2023) and kNN-based techniques (Guo et al. 2021). have been proposed to enhance computational efficiency. Besides, various studies have applied influence functions to interpret models across different domains, cluding natural language processing (Han et al.) (020) and image classification (Basu et al. 2021). while also addressing biases in classification models (Wang et al. 2019), word embeddings (Brunet
et al. 2019), and finetuned models (Chen et al. 2020).
Despite numerous studies on influence functions, we are the first to introduce this concept into bilevel optimization. In traditional models, the application of influence functions is relatively straightforward; however, in the bilevel optimization framework, becomes significantly more com- plex due to the interactions between the variables across the outer and inner layers. This complexity necessitates a simultaneous consideration the relationships between the outer objective function and the inner constraints. This paper provides more comprehensive perspective quantifying data influence and introduces, for the first time, a stratified approach to quantifying data within bilevel optimization.
3 PRELIMINARIES
Notation. For twice differentiable function L(A,0(A): D). L 0(A); D) denotes the direct gradient (partial derivative) of L w.r.t. À and L(, ((A); D) denotes the direct gradient of L w.r.t. (()). And the total gradient (total derivative) of L(>, 0(A); D) w.r.t. X is calculated as de(A) L(, (A))
D2L(A, (,L(),((A);))) D) = L (()); D) + dA
Influence function. The influence function (Huber 1981) quantifies how estimator relies on the value of each individual point in the sample. Consider neural network i arg ming L(0, D)
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with loss function & and dataset D {24 1-= When individual data point 2m removed from the training set, the retrained optimal retrained model is denoted A- The influence function method provides an efficient way to approximate 0 -_.... without the need of retraining. By up-weighing -m-reete term in the loss function by 6, a series of e-parameterized optimal models 0_ amme will be obtained by 0- Ami = arg ming [L, D) + ()mmm0).. Consider the term
arg ming Σi=1 ((2))) 0)
V D) te. Vi(zmi 1 )mm.e) O,
we perform a Taylor expansion at i and incorporate the optimal gradient condition at 0 Ó_.... and Ò: Σ V((z;;; + e Vi(zm; 0) + H1g (0- -) O, where Ha - - Vil(2;; 0) is the
Hessian matrix. Consequently, the Influence Function is defined as the derivative of the change in parameters of the retrained model due to perturbation with respect to the perturbation: IF(2m) =
db_
-H Ve(zm: 0). By setting c -1 2m is completely removed from the c=0
de
retraining process. Then, 0_z.. can be approximated by linear approximation 0 - IF )- Additionally, for a differentiable evaluation function, such as one used to calculate the total model loss over test set, the change resulting from up-weighting to 2m in the evaluation results can be approximated - (()) (F(2m))
Meta learning. In this work, we consider the meta learning framework formulated as a bilevel optimization problem. Note that the BLO framework encapsulates most of the current objective functions in meta learning, such as MAML (Finn & Levine 2017) and its variants (Finn et al. 2018 Nichol 2018 Rusu al. 2018 Killamsetty et al. (022).
In meta-learning, the inner and outer objectives are typically defined by averaging the training and validation errors across multiple tasks. Given a set of tasks I, we use a meta-training set D = {D; = D U Dirhhez, hiez, where D and D are the training and validation sets for each task The task-specific model parameter (2(A) depends on the meta- meeaa-arr À and the training data, and is obtained by minimizing the inner objective: (A(
arg ming, [, 0g; D(T), where (A, 8;; D.") is the regularized empirical loss on D (with Ez-regularization to prevent overfit- ting). Ly(A, D/") = ExEDY- E(2; 0.) + 211: A|I. And the i-th task related outer loss func- tion is the empirical risk of X and 0g on validation dataset D. defined as (A):: D ((xp)eer ((2; (A)). Noting À here is shared between different tasks, and the outer objec- tive is determined by calculating the average validation error on the parameters we obtained from the inner objective across multiple tasks as (we use fz-regularization to avoid overfitting): X = arg mitt Lomal (A; O(A); = arg min, Lier Lo (2, (2,00((A)) Oc(A); Dral) ) + - |//2".
4 MODELING DATA INFLUENCE IN META LEARNING
It is evident that the datasets utilized for meta learning exhibit a hierarchical structure. Specifically, the whole dataset consists of multiple task-specific datasets. Consequently, when evaluating data influence, we can assess the influence of individual tasks on training outcomes while also conducting a detailed analysis of the impact of specific data points-iee. instances -witheeess-tthnn given task.
Due to the bilevel structure in meta-learning, is essential distinguish whether instances belong to the training or validation set. Consider the k-th task, if the data point we are concerned about is in the validation dataset, then only the meta parameter will be impacted, and we can directly apply the influence function method to evaluate its contribution. The data from task validation set directly influences the update of meta parameters. In contrast, the impact of the task training data on meta parameters is more complex; it indirectly affects the update of meta parameters by influencing the few-shot learning parameters that are relevant to the task. The following sections will provide our general frameworks for task and instance levels' data influence evaluation.
4.1 EVALUATING THE INFLUENCE OF TASKS
Consider the k-th task, the information related to this task includes De = D U Ded We will leverage influence function evaluate how Dg will affect the model given by meta learning. When
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remove the k-th task from meta learning, the k-th inner training task will be completely elimi- nated. This is reflected in the outer layer, where Lo (2, (A);;; D will disappear. Therefore, we can ignore the specifics of the k-th inner training and focus solely on the outer level. The retrained meta parameter is defined as x arg min, Liehe Lo (A, (A);; Deai) + |A]2.
direct approach. If we directly adopt the same idea of the original influence function and consider Da data point, we then may up-weight the loss for the k-tth task Lo (x, (x.0x (A); Daal) by a small perturbation с and replace the x by the minimizer of the e-parameterized model X Ignoring the dependence between the task-specific parameters and the meta parameters, can intuitively derive an influence function by using a similar Taylor expansion of the partial derivative w.r.t. À of the outer objective function as
IF(Dx) (6 + Hireia OyLo (A*, (x())):
(2)
where dideea is the Hessian matrix directly with respect to A, defined as
HA, direct EosaLo (A", 0,((";;
(3)
REI
Our method. However, this method is flawed as overlooks an important aspect: (;(A) is de- pendent X, and the experimental results in Table [ further demonstrate the inaccuracy of this method. Mathematically. in the original IF definition, since there no inner level function, we have LL D) = 0 and (1. However, as in bilevel optimization the outer objective function Lyoua depen- dents both À and (A) for X we do not have the direct gradient L (X",, 0(x*) D) and its corresponding ① Thus, we cannot use similar Taylor expansion to ①I and get (
To address the above issue, the key observation that while the direct gradient x is non-zero, its corresponding total gradient of the loss function is zero, i.e., DTmaaa (X",,(A>))))>> ((*)); D) = 0. The following result provides a result on the computation of the total gradient.
Theorem 4.1. The total gradient of the i-th task related outer loss Lo (A, (A);; Diya) with respect to X can be written as: DxLo (>, (A); = θrLo (A); + d(.(2) dA
0o Lo (2, 0z(A); Dral). The term d0,(2) can calculated by dA
(A,(A) -x Li (A)D,(A): H
dd
where Hi,n is the i-th inner-level Hessian matrix, defined Hi, 0o, 0o L1 (>, (,): D.T).
Based on the discussion above, it is clear that the direct Hessian matrix in (3) is also inadequate for the influence function in the bilevel setting. Therefore, we propose an expression for the total Hessian matrix that incorporates the total derivative.
Definition 4.2. The total Hessian matrix of the outer total loss Loaa ()); with respect to defined as Hoy = EiEz LD (A):
Based on the total gradient and the total Hessian matrix we proposed, we can up weight the (),,, 8g: Dial) term at the outer level and derive its influence function.
Theorem 4.3. (task-IF) Define the task-IF of the k-th task in the meta parameter X
task-IF(Dx: X", ,x((A") = (8- + Hx, DxLo (X*, (0x((")); Dral).
Then after the removal of the k-th task, the retrained meta parameter X- can be estimated by X_x = x task-IF(DDs (D;;; X² (x*)).
Remark 4.4. Note that the task influence function depends on the total Hessian matrix. For simplic- ity, we denote Lo (A)),(A); D.al) as Lg) then it can be written as
0,0,D' + dd,(A) yd
dPB,(A) dx²
(dB,(a)) dd
HotTotal
de, OxL' +
Do. Li
+
do 00,00.13 L'
TET.
Due to the complex definition of do:(2) derivative, presenting significant computation challenges. To address these complexities, we will
dA as discussed in
44 d
4x2 requires a three-order partial
introduce several accelerated algorithms in Section 5.1
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4.2 EVALUATING THE INFLUENCE OF INDIVIDUAL INSTANCE
There are tWO instances in the meta learning dataset: instances in the training dataset and instances in the validation dataset. The validation instance appears the outer level and will only influence the meta parameter without directly impacting the task-specific parameter. The training instances appear at the inner level and impact the task-specific parameter, thereby influencing the meta pa- rameter indirectly. Therefore, we must delve into the inner training process and conduct two-stage evaluation, In word, the evaluation of validation instances is similar the task evaluation problem, while evaluating training instances is more challenging and requires new techniques.
Validation data influence. We first focus on the evaluation of influence for the validation data è (ž, ū) of the k-th task. Since the validation data was not used in the inner level loss, we can directly follow the idea of task-IF to the outer loss function. The only difference is that we up-weigh a term related to data point ī in the k-thh task {(2; (A)) rather than the task term Lo 0g (A); Diat). Then similar to Theorem 4.3] we obtain the following theorem.
Theorem 4.5 (Instance-IF for Validation Data). The influence function of the validation data (x())) - (6 I + H Ve O2 (X*)) After the re- moval of 2. the retrained optimal meta parameter X== can be estimated by X= = x instance-IF(2;X1, X,,x,,00 (Xx)))
(z. 5) is instance-IF(2;X (0x((xx
Training data influence. We evaluate the influence of training data first. We will employ a two- phase analysis to analyze the impact of an individual data point \ = (œ, ÿ) in the k-th task. Initially, we utilize the IF to assess the data point's effect on its corresponding task-related parameter 02 (A)- Since the meta parameter, À is passed from the outer level and remains fixed during each inner-level training period. Thus, we can use the classical influence function directly to LI Ozi D..)
Theorem 4.6. The influence function for the inner level (inner-IF) of 2 (ã. ū) from the training set is inner-IF(2; (x(())) H Vo, [(2; 0x (X))), where HK.m 0o 0o Lx(X,, Ox(x') D is the kith inner-level Hessian matrix. After the removal of 2, denote 0g (X*) as its corresponding retrained task-parameter 0% (X*) = arg ming, Lr(X", 02, ())) Then it can be estimated by og (X*) xe (x*) (iter-IF(;;0(X11))
Now we aim to approximate the influence of replacing 0₃ by 0 in the outer level. That is, how will the optimal meta parameter X" change under the removal of ž. Note that the outer level loss only depends on the validation data and remains unchanged after the removal of ž. This challenges applying up-weighting to the influence function effectively as the change of the outer loss is im- plicit. The result, which is shown in Theorem 4.7 shows the change in the outer-level loss function resulting from the removal of I.
Theorem 4.7. the k-th task related parameter changes from Og to O5, we can estimate the related loss function difference Lo ((>, 0g (A)) ; Deal)) - Lo OxA); Dral) by P(A, (2); which is T inner-)F($,0((A)).
defined as Vo To (>, (A); Dgal)"
Now derive the influence function. Motivated by the above result, instead of weighittiing the loss function term, we focus on up-weighting the data influence term P(), 0g (A); 2). We add this term into the outer level loss function and up-weight it by a small which varies from 0 to 1. Then the outer level loss function becomes x= = arg min ΣiET LToca ((); P)CC On (). When < = 1, the influence of 2 is fully removed by adding this term into the loss function.
Proposition 4.8 (Instance-II for Training Data). Let ( O, we can obtain the influence function of the outer level as instance-IF X", OxxA*)) = - -H DDP((*, (1(x*,0x(X) (x()²);; 2). After the removal of ž, the optimal meta parameter X can be estimated by X_g RJ X instance-IF(Ea X 02 0x(X)))
5 COMPUTATION ACCELERATION AND PRACTICAL APPLICATIONS
5.1 COMPUTATION ACCELERATION
Although the results in Section are closed-form solutions for evaluating data influence, they involve extensive calculations of the inverse Hessian-vector product (iHVP), which are coupled with other

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Table I: Performance comparison of different methods.
Method
cmnigle
MINI--mogret
Accurscy
RT (scomd))
Accuracy
RT (scones)
Accuracy
RT (reend))
Aecurey
RT (seeaed)
Retrais
315.9435.45
2011
341.2422.18
692.5667.01
033092200000
362.02200.01
Takk
0.5432 10 EC00
7-26 0.05
0.5347.10000 110224
2.57 0.94
0,2216
10.033
10.000005
[13906 L00015
18A11000
IKFAC t0
0.872.18 L803
611 01100110 1003
510.001 20
0.188 1U000
1931-42.000 24
103151 100001
688 48841113031
TRAK
0.9078.111 1891
1022
04180011.0011
1001000000
0.2922110 16,220
1154.17
[.1282 100000
27th 14.12.441
Retraan Traingg?
0,785220.008 0090
251.76:1.21
015402200148
252.9021.7
0.3253201834
392.3420.40
035-0)30005
361, 1975
lasane di Thaising)
0,288628 0037
4,4920022
06123441122
4,50=0002
02254331833
7,17:008
113645
7 7,227740082)
Retraia
0080
200,8:27622
0e10
(ilidaion)
0.775020.000
0547662019244
5.71:031
027672000850
7,4:0085
0.3098500005
7.244:20.0
computational steps and also necessitate the computation of third-order partial derivatives. Thus, developing acceleration techniques is essential to enhance the scalability of our methods. We now present approximation methods designed to accelerate iHVP and third-order partial derivatives.
iHVP acceleration via EK-FAC. Due the specific method for calculating the derivative of task- related parameter 0; (A) in 4 and the iHVP calculation in Theorem [466, our algorithm involves the computation of iHVP at several different stages. To expedite this computation and eliminate the need to store the Hessian matrix explicitly, we employ the EK-FAC method (Kwon et al. 2023 Additional implementation details can be found in the Appendix.
Total Hessian matrix approximation. As mentioned in Remark 4.4 computing the total Hessian matrix for large-scale neural networks presents significant challenges, primarily due to the require- ment of calculating the third-order partial derivatives of the loss function with respect the param- eters. Given that the information conveyed by third-order derivatives is relatively limited compared to that of first and second-order derivatives, we propose approximation for the total Hessian.
Theorem 5.1. To approximate total Hessian Hx,
replace the H' term in total Hessian
(see 4.4, defined as H' d²0,(2) dx²
On. Li + +(1000 (20((2)) -J, 0o Li by Eiezz 110 I,
where I is the identity matrix.
Generally, compared to the original term H', reflects the extent to which each parameter influences the loss function and captures significant gradient information by focusing on the main influencing factors via ignoring the second-order derivative terms. This enables the practical simplification of complex mathematical expressions while preserving essential information. However, EK-FAC is no longer applicable for computing the iHVP for the total Hessian proposed in this paper. This is because EK-FAC is based on the assumption that the parameters between different layers are independent. In contrast, the definition of the total Hessian in 4.4) needs interactions between different layers. Therefore, EK-FAC may be insufficiently accurate here. To compute the iHVP precisely, efficiently, and stably, we propose the following accelerated method:
iHVP acceleration via EK-FAC. Due to the specific method for calculating the derivative of task- related parameter (A) in (4) and the iHVP calculation in Theorem 4.6. our algorithm involves the computation of iHVP at several different stages. To expedite this computation and eliminate the need to store the Hessian matrix explicitly, we employ the EK-FAC method (Kwon et al. 2023). Additional implementation details can be found in the Appendix.
Neumann series approximation method. For vector
express H Total through the Neumann series: X V + Σt (T- H v. By truncating this series at order J, we derive the following approximation: Hz v + (I- xmal) . +.... (I Teaa)) " .. is important to note that we do not specifically accelerate the inversion of the Hessian matrix; rather, we directly optimize the iHVP procedure. As a result, throughout the entire computation process, there is no need to simultaneously store the large Hessian matrix, which significantly reduces the memory requirements of our method.
5.2 APPLICATIONS OF INFLUENCE FUNCTIONS
In this section, we present application methods for downstream tasks based on the previously derived influence functions for tasks and instances.
Model editing by data removal. We can use IFs update the model under the removal of certain data or tasks. Specifically, the model after removing the k-th task can be estimated as x task-IF((D))) (D2;; X", (A*)). Additionally, the model for removing instance E from the k-th task
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can be estimated as X instance-IF( ( X" (x{(*)).
Data evaluation By appropriately selecting the model evaluation function, we can define influence scores(ISs), which measure the influence of specific tasks or instances can then be applied to task selection in order to enhance meta learning performance or to adversarial attacks assess model robustness.
tasks. These scores
Definition 5.2 (Evaluation function for meta learning). Given new task specific dataset Dnew Der U new the few-shot learned parameter 0' for the task is obtained by minimizing the loss new
(X",, 0; Diew) We can evaluate the performance of the meta parameter X based on the loss of O' on the validation dataset, expressed as ExEDuat new [(2;0)
Based on this metric, we propose a calculation method for the task and instance Influence Score, defined as IS = EnEDY! Vo.((2; 8'FF. A detailed theoretical derivation can be found in Appendix.
EXPERIMENTS
In this section, we demonstrate our main experimental results on utility, efficiency, effectiveness, and the ability to identify harmful data.
6.1 EXPERIMENTAL SETTINGS
Dataset and evaluation metric. We utilize datasets: omniglot (Lake et al. 2015 MNIST (LeCun et al., 1998). MINI-Imagener (Vinyals et al. 2016) and FC100 (Oreshkin et al. (018a). We report two metries: Aceuracy and runtime (RT). Accuracy the proportion of correct predictions, and Runtime (seconds) measures the time to update the model. See Appendix Doo details.
Baselines. At the task level, besides the ground truth-retraining, we compare direct IF and Task- IF with three baseline data attribution evaluation methods: EKFAC IF (Grosse et al. 2023), TARK (Park et al. 2023), and TracIN (Pruthi et al., 2020). Direct IF, as described in Section 4.1 is a direct implementation of (2) and a generalization of the original IF. At the instance level, we use retraining as the ground truth. Retrain after removing selected instance from the training validation dataset for a task-specific dataset. Additionally, we conduct experiments using instance-IF ing) and Instance-IF (Validation) directly implement the theoretical results from Proposition[4. (.. and Theorem 4.5) respectively. Detailed algorithms for all data evaluation methods are provided in the Appendix.
Vaaidation) involves retraining the model
Instance-IF (Train-
6.2 EVALUATION OF UTILITY AND EDITING EFFICIENCY
TF and direct-IF can be utilized for retraining surrogates and data attribution. To assess their utility and editing efficiency, we compare our methods with other baselines from the perspectives of accuracy and runtime. Specifically, we employ various data evaluation techniques to identify and remove 40% of harmful data from the training dataset. Subsequently, the model is retrained to evaluate its test accuracy. Notably, task-IF and direct-IF can also be used to update the model directly, eliminating the need for retraining. Our results in Table highhlitt the superiority of task- IF over baselines, while highlighting the critical trade-off between computational efficiency and accuracy. On the Omniglot dataset, task-IF achieved accuracy comparable retraining (0.7804 vs. 0.7908) while reducing runtime from 316.74 to 65.69. Similarly, on MINI-ImageNe task-IF cut runtime from 682.56 74.63 with a modest accuracy trade-off (0.3071 vs. 0.2668). These findings demonstrate task-IF's ability to save 75%-80% of runtime while maintaining competitive accuracy. While direct-IF achieves the fastest runtime among all approaches, it shows significant degradation in accuracy across all datasets. This is due to the fact that it ignores the dependence between the task-specific parameters and the meta parameters. As we mentioned in Theorem [ task-IF leverages the total gradient and Hessian to fill the gap, which will also slightly increase the computation time due to their addition terms.
For the instance-level evaluation, observe that instance-IF demonstrates promising results, par- ticularly in the scenario where we aim to evaluate the attribution of training samples. The approach maintains relatively stable performance between training and validation cases, with accuracy scores
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of 0.7686 and 0.7790, respectively, omniglot, while achieving remarkable runtime efficiency (4.49 and 5.53 seconds). This sight difference in model accuracy between IF method and retraining may be attributed different factors. For instance, theoretical framework relies the stabil- ity of influence estimation across training, which may vary depending on the model's convergence behavior. Moreover, the effectiveness of instance-level influence computation might be affected by the inherent complexity and feature distribution of different datasets, explaining the varying perfor- mance gaps observed between MNIST and MINI-ImmgeNNe..
6.3 RESULTS OF HARMFUL DATA REMOVAL
We also influence functions to assess how help- ful or harmful each task is. To validate this, we create harmful tasks by flipping labels in 80% of tasks and 40% of MNIST samples. Figure | re- ports test accuracy and the fraction of mislabeled tasks detected as we vary the proportion of train- ing data checked. "IS" removes tasks by influence score; "Random" removes tasks at random. The left plot shows that our IS-based approach consistently outperforms random removal across all checking ratios. When examining 25% of the training data, the IS-based method achieves a lest accuracy of approximately 62.5%, while random removal yields only 55% accuracy. This significant performance gap of 7.5 percentage points indicates that method effectively identifies and removes truly harmful tasks rather than arbitrary ones. The right plot further validates our approach's is effectiveness in identifying mislabeled tasks. The IS-based method demonstrates remarkably efficient detection, identifying nearly 100% of mislabeled tasks after checking only 40% of the training data. In contrast, the random removal baseline shows linear relationship between the fraction checked and identified, achieving only about 45% detection rate even after examining 95% of the training data. This substantial difference in detection efficiency (approximately 2.2x better) underscores our method's 's strong capability in targeting harmful tasks.
Figure I: Harmful tasks removal experiment.
6.4 EFFECTIVENESS OF IF
Here, we aim to show that our proposed influence functions can indeed be used attribute data. To in- vestigate the effectiveness, we first calculate the in- fluence scores for all tasks or instances in the train- ing set based on our proposed task-IF and instance- IF. Using these scores, we iteratively remove tasks or instances in descending order of influence score and subsequently retrain the model examine its robust- ness. As shown in Figure removing high- high-influence tasks or instances significantly reduces test accuracy. At the task level, accuracy drops from about 0.62 0.58, while at the instance level, it falls from roughly 0.40 0.25. In comparison, randomly removing data has a much smaller impact, highlighting the ef- fectiveness of targeted removal based on influence scores. These results suggest that the influence score calculation effectively identifies critical data points, with high-influene tasks or instances playing a piv- otal role in maintaining model performance. This analysis demonstrates the effectiveness of influ-
Harber Tastonev
Marber Tasta Samroved
Figure 2: MNIST Data Attribution (up: Task, below: Instance).
ence scores in pinpointing valuable data. Additional visualizations are provided in Appendix B
7 CONCLUSION
This paper presents data attribution evaluation methods for meta learning, addressing challenges like training inefficiencies and noise from mislabeled data. By introducing task-IF and instance- offer a comprehensive approach for evaluating data across tasks and instances. Our framework

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captures both direct and indirect impacts of data points meta parameters, improving efficiency and scalability.
ACKNOWLEDGMENT
Di Wang. Huanyi Xie. Chenyang Ren, and Shu Yang are supported in part by the funding FCC/1/5940-20-05, FCC/1/5940-06-02, and King Abdul- lah University of Science and Technology (KAUST) Center of Excellence for Generative AI. under award number 5940 and a gift from Google. Lijie Hu is supported by the funding BF0100 from Mohamed bin Zayed University of Artificial Intelligence (MBZUAI). For computer time, this research used lbex managed by the Supercomputing Core Laboratory at King Abdullah University of Science & Technology (KAUST) in Thuwal, Saudi Arabia.
BAS/1/1689-01 (1110/901--00..G//711225524
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URL htttt:tt//xxiiii http://..wrrra.....e// 04692
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A LLM USAGE
We used LLMs to refine grammar and improve language fluency. The authors reviewed and edited all LLM-generated content and assume full responsibility for the final text.
B OMITTED PROOFS
B.1 EVALUATING THE INFLUENCE OF TASKS
Theorem B.1. The total gradient of the i-th task related outer loss Lo (A);; Dral) with respect Ả can be written as:
DxLo (A,0;(A); D!
d0,(A)
Lo (A, 0, (A); Dr +
0b, Lo (A)):(A);D))
dA
The term d0.(2) dA can be calculated by
dA,(A) dd 0x0n L1 (A) H
(5)
where Hiji is the i-th inner-level Hessian matrix, defined Hi, 0b,0o, I., (A, (A);;
Proof. From the Chain Rule, the total gradient of Lo (A): ( denoted as
DyLo (A, 0,(A); Dyal) is calculated by:
DxLo (A, O,(); D.
do,(A) dA
=rxLo (A)),(A); Dr
0g, Lo (A)
We will begin to derive the calculation method for d0,(2)
dA term. From the optimal condition of the
i-th inner level: 0,L, (A, (A): Dr = 0. Then take derivative respect to À
0, (2,0 dO,(A)
L1,0 D/)) =0
dA
Then from Implicit Function Theorem, we can calculate
d0,(X) dA term as:
d0,(A) -0,00t.1(,,,1))): (.00
dx
Noting here L, (A); D.") is the Hessian matrix of the i-th inner level loss w.r.t (())), denote as H,,in. Then, the final expression of dR_(A)
dA is:
(a)(2) - 0x L, (2,0(A);;; High
Yp
Definition B.2. The total Hessian matrix of the outer total loss L (A, () with respect is defined as:
H Total
Σ (D.DALO (2, (N)
iEI.
Theorem B.3. (task-IF) Define the task-IF of the k-th task the meta parameter X* as
(ask-IF(Ds;: x0,00 (xx(A*))
(8. Hx,rona) DALo (x*,0x('') Deat).
Then after the removal of the k-tth task, the retrained meta parameter X²x be estimated by
X R X* para-IF(Dss .0, Ox*)))
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Proof. The outer objective is determined by caleulating the average validation the task- specific parameters we obtained from the inner objective across multiple tasks as:
X arg min
Σ
Lo (A: + |Al?
(6)
IEI
We upweight the term in the total loss function (6) related to task & by and obtain
de -
arg min Σ
8c D( +e e. Lo(A, 0x: Drat) + -
1112
(7)
IET
Because of the minimization condition, we have
DALo Oc(ee) D.al) DALo (de, Ox(e): Daal) + - =0
4EI
where DALo (de, (de); Di is the total derivative with respect À defined in Definition B.1 which is
DrLo (ee, (Ac)); Di = OxLo (Ae, (.Ae): + d0,(2e) dr.
OgLo (ee, (;
where d0,()e) is obtained by -x,0
dd.
d0,(Ae) dA
Then, for i I, perform Taylor expansion of DxLo (Ae Oc(cc): Daal) around the point (X",0; (X*)), then we have
DxLo (Ae, O.(,)
(de - --))
(9)
Substituting Equation
into g and repeating the same process for all the tasks, have:
8. DALo (x (A");; Diol) + DyLo (A,, (r,O((A*)) Doal)
KeI
(10)
8.1 ED.DALO(X"000'))
(A. - x²) =0 =.
4EI
The high order terms have been dropped in above equation. From Equation 6 derive an optimal condition for X as:
EDALO 0,()); + X = =
TET.
Then the first term in Equation
10)
equals By simple algebraic manipulation, we have
(. D Draa) DALo (x*,0((())) Dwel).
A - x- --- - 8
8 I+
iEI
Revising the definition of the total Hessian matrix HA.o in Definition B.2 be rewritten as:
the above equation
>; X = -e (8 HoTT -1 DxLo (X* (A"); Dr
(11)
Letting O, we define the task-IF
task-IF(k)) = dd
= - (5 I HA, DALo (x*))) Dral).
de c=
When -1, 2-1 A approximation for X=E;
first-order approximation estimate X_1
Ma x X para-II((k).
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Remark B.4. Note that the task influence function depends the total Hessian matrix. For simplic- ity, we denote Lo (A): as Lo. then it can be written as
@xOAL' +2. (A) BEZ
Hotal Total
Σ
du. Dx
dd
(A)
(d dd
0g do, 1
0g L' +
dx
Proof. From Definition B.2
the total Hessian matrix Hx,Tonl is defined as:
Hotal
Σ
(DADALo (A)
SEI
Then, we can expand total Hessian matrix Hotal as:
H
-E
LO (A, D(aa)
iEI
d0,(A)
ΣDA DA
(OaLo (A,0:(A))
-b, Lo (A, (,))(2)))
d)
OxOxLi + d0,(A)
Σ iEZ
02
d2
d(O((A)
Og L'o + (c0,(A)) (Ax))
-0,,00, 8g do 00.
dx2
B.2 EVALUATING THE INFLUENCE OF INDIVIDUAL INSTANCE
Theorem B.5 (Instance-IF for Validation Data). The influence function of the validation data ă (ū,9) is
X ,0x(x")))
- I T VE(2;
After the removal of ž, the retrained optimal meta parameter X can be estimated by
X= ~ X* instance-IF(2;X) (
Proof. Consider evaluating the influence of the data ž (#. () in Deal in parameter X, the loss
function and parameter after the removal is defined as follows:
x arg min X
Σ
0:(A): ((2; Og (A))
(12)
SEI
To estimate XWg. firstly, up- up-weight the term Lo related to ă by e and obtain the outer level loss
function
Σ Lo(A, - ((2;(((((())) 11
NET
Then set of new parameters d, are defined by
A = arg min Σ (A)) + (();0)(A))) + 5 |A12
(13)
The from the minimizing condition in 13 we can obtain:
Σ
(, c (x((); +8 A =0
1ET
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Perform Taylor expansion at X X²,
D 01:(A);;; + ( + -Ae -x)
IET
(14)
(A, (A, - -*) x² ~0
REI
From the definition of X we can find it the minimizer the original outer level loss function
LiET (x, (A); Deal) + I|lll? therefore, the first term in Equation (14) equals 0. Besides,
Hx, Total Eiez DoD (X* (,"())); Drat) is the total Hessian matrix of the outer level loss
function. Then we have
x* - (8 HotT Vxl(2;
Let e + O, we can obtain the definition the instance-level influence function for validation data as follows:
11, (x(A*)) = = (6. I Ha (x*))) instance- (F; X",
When -1 the influence of (z, 17) is fully removed. We can use this instance-IF(2; X", (A*)) to estimate x when data ž is removed with A fixed as
\ ~ ((; 0c(A)))
Theorem B.6. The inftuence function for the inner level (inner-1F) of = (ă, V) from the training sel is
inner-IF(2; 0s (A*)) = - - kijn Vo, ((2;0(((
where Hk.i 0o, 0o. (x(X", (x*));; DI") is the k--th inner-level Hessian matrix. After
the removal of 2. denote 0% (X*) as its corresponding retrained kask-parameter 0g (X*)
arg ming; Lr(X", O;; {)). Then it can be estimated by
0% (X*) 0&(A') inner-F((;00A(x))
Proof. Consider evaluating the influence of the data Σ (ū, 7) in Dral in parameter On (A), the loss function and parameter after the removal is defined as follows:
0% (A) arg min L, (A, 02 D (k2
Revising the definition of 0 (A):
(x( arg min min LA, LA, Σ (2)00x) + 2 10 - All2.
0
ZED
Firstly, we-weight ((;; 02) in I, by e, resulting in the modified loss function. Then, a set of new parameters 0z (A))(A) are defined by:
(A) = arg min RED
((2;0x) + 2 2100 - AlI2 + (.e((())))) = L1, + ((22 0x).
(15)
The from the minimizing condition in 15). we can obtain:
Vol. (A);; D(T) + (ol();
Perform Taylor expansion at (k,c(A) (A),
(ALi DKC) e Vol(2; (A)) VAL 0z. D(") (0k.e 0x()))
From the definition of (x) find it as the minimizer of L, 0g (2), A), therefore, the first term in above equation equals 0. Then we have
(A) 0,(A)
Hoo. vol(z,,;
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where Hø.; V. (()....)
Let e - 0, we can obtain the definition of the inner-level influence function as follows:
(A),e(A)
Va, E0, ū; 0x))
de
When -1, the influence of (z:, g) is fully removed. estimate 0r (A) when data (zr y) is removed with fixed as
We can use this inner-IF(ar, y; (A)) to
8 (A) = (A()) inner-IF(2:, y: (.)) & ((())
Theorem B.7. If the k-th task related parameter changes Gy 0%. we can estimate the related loss function difference
Lo ((,, (2)) Dg Lo (A) DEa)
by P(), (N); which is defined
VaaLo (A, (A);; DE)) inmer-IF(2;0g(A)
Proof. The loss function difference can estimated by the first-order Taylor expansion as:
Lo ((2, 0% (A)) Dg Lo (A, (A); Dgat)
VoLo (A, On (A); Deal)" (0% (A) (s()A)
VoLo (2,0 (A); D inner-IF(2;0x(A)
The second equation is obtained by Theorem B.6 Then the proof is completed.
Proposition B.8 (Instance-IF for Training Data). Let Ç -> O, we can obtain the influence function of the outer level as follows:
Instonce-I((;:X10.0
- DxP(X*, ;
After the removal of 2, the optimal meta parameter X be estimated by
X= ~ X + instance-IF(2; X (x())).
Proof. In the following, we will derive the influence of removing the meta parameter X², Follow the settings in the Preliminary part, we expand the form of the outer level loss function as:
Σ TET
Lo (>, (A):
6 (10.(2)) 2112.
Σ
Σ ((2;8((A)
TE ZEDrai
If the data ? we want to remove is in the training dataset of the k-th task, then 0x (A) will change to 0% (A). And we can estimate the loss function difference with the help of Theorem B.7 as follows:
Lo (>, O (A); Deat) Lo (A,
se (A);
Then P(A, 02 ()); the actual change the outer-level loss function resulting from the remove of Σ. We add this term into the total loss function and up- up-weight it by small from 0 to 1. When
< = 1, the influence of remove is fully add into the loss function.
Σ Lo (A); + + P(A, 0s ()) SET
And set of parameter X is obtained by the minimization process as
x
arg min
Σ Lo (A,0((A): + P(A, (2):
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Published as a conference paper at ICLR 2026
Naturally have the following optimal condition:
DALo (A2) D."l) + P O.
IEI
Perform Taylor expand X²,
DALO (x",) C (x("):;
DADALO (X', D( (Az- 1²)
Noting here X is the minimizer for the original loss function Eier Lo (2, (xA); Dial), therefore the first term equal O, then we have
- =
Tosal DxP(A", (x(());
where Toul Eiez ez Lo OA"); Dral). Let ( + 0, we can obtain the definition of the instance level influence function as follows:
dyp
instance-IF($; X", (x()))
--H X,tall D2P(A", Bx(A)))
dc c=0
When (= 1, the influence of removing ğ is fully added into the outer level loss function. We can use this instance-IF(;; 02 (X*)) to estimate the meta parameter X when data is removed as follows:
x A instance-IF(3;X x*, (2,,,
B.3 APPLICATIONS OF INFLUENCE FUNCTIONS
iHVP Acceleration via EK-FAC We utilize EK-FAC to perform an approximation for Ha,n and then compute the inversion efficiently. Revising the definition of Hijin:
H = OgdoT, (X*, ((("));DTT)
The EK-FAC method relies on two approximations of the Fisher information matrix Gg. which is equivalent to Hi when the loss function L in L Ox()); Dfr) is specified as cross-entropy. Firstly, assume that the derivatives of the weights different layers are uncorrelated, which implies that H.,. has block-diagonal structure.
Suppose model fo can be denoted by fo(ar) fo, .... fo, (x). We fold the bias into the weights and vectorize the parameters in the I-th layer into vector B E Rdi, di E N is the number of 1-th layer parameters.
Then Go can be replaced by (GI (G()))))),,))))) where
G()
Denote hi, the output and pre-activated output of 1-th layer. Define Ω = Ci(() -
Di 14-1 (xx) h- (xx)"
- V.VVV. C. Then Gi(O) can be approximated by G0 8411 Tr.
Furthermore, in order to accelerate transpose operation and introduce the damping term, perform eigenvalue decomposition of matrix Ω4-1 and I and obtain the corresponding decomposition results QuARQT and Qr Then the inverse of (0) can be obtained by
(1) (Gi(0) Adaa) -Q Qr.) (Aq., & Ar, + Aela.) (Qa.,, @ Qr.)²
(16)
Besides, George et (2018) proposed a new method that corrects the error in equation I6 which sets the i-th diagonal element An,-, @ Ar, as A = n EE= ((or, & Qr.) V (),
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Published as a conference paper at ICLR 2026
Definition B.9 (Evaluation Function for Meta Learning). Given new task specific dataset Dnew Diew U Deei the few-shot learned parameter for the task is obtained by minimizing the loss Lr(X² 00 Deww) We can evaluate the performance of the meta parameter X based on the loss of 6' on the validation dataset, expressed as EeEDret nec {(2;0')
Based on this metric, propose calculation method for the task and instance Influence Score, defined as EeED Vz) IF.
Proof. Assume we consider the contribution of set of data D, define the initial optimal meta parameter as X", and retrained optimal meta parameter after removing the data D A We define the evaluation metric as the test loss difference:
Σ (=; ((r))
Σ (z;0'(1_
Dest
DED new
We can estimate this difference by Taylor expansion as follows:
Σ
(();0(x²))
Σ (X=
# Dval
a Duaa
Σ
Vol(=;0((x)) (f' 0' (x(x))
sE
d0' (X*)
vo((z;((<>))
-'-x
dx*
To make method more efficient, neglect the influence in d0'(A*) and directly estimate the loss difference by
Vol(z;;(()))) (A-
A= -== can be estimated by our proposed IF. Then we define the Influence Score(IS) as IS EeE Duat (x+)) IF, and use it to evaluate the influence of the data D.
in which -
C FULL ALGORITHMS
Algorithm Task-IF
1: Input:
Data: Training Dataset D = (D U Deal), task & to be evaluated.
Parameter: The learned meta parameter A", the task-related parameters 0, for E I.
2: Subsample a task subset Is. For the detailed methods, there are multiple choices, including random sampling, difficulty-brrt
sampling, uncertainty-basel sampling. 3: Compute the total gradient by Algorithm
4: Use Neumann Series Approximation method to compute the task-IF by Algorithm
5: Approximate the model retained when the k-th task is removed: X X task-IF
6: Return: task-IF, X
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Published as a conference paper at ICLR 2026
Algorithm 2 Direct IF
1:
Input:
Data:
Training Dataset D ( "Duat,, task & be evaluated.
The learned meta-parameter X the task-related parameters 0; for E I. Subsample a task subset Zş. For the detailed methods, there are multiple choices, including random sampling, difficulty-brat based sampling, uncertainty-bast -bssed sampling.
Parameter:
2:
3: Compute the partial gradient of the k-th outer loss function as
T = OxLo(X" 0x: Dial)
4: Compute the direct-IF(k)) based T
Σ IEI
0x00Lo((X,,0011 Diaa) + +0-1) TT
direct-IF =
5: Approximate the model retained when the k-th task is removed: X X" direct-IF.
6: Return: direct-IF(k), X=
Algorithm 3 Total Gradient Computation for Task
1: Input:
Data:
Training Dataset D Usez (Dir Drat), task & to evaluated.
Parameter: the learned meta- paraameareeerr Xt, the task-related parameters 0; for E Z.
2:
Use EK-FAC to compute the middle gradient M as
M₁ = 0z0gLs (x-, 0n; D 0gOg L (X (x"00;:()) OgLo(A", 00; Deal).
3: Compute the total derivative of the k-th outer loss function with respect to X as T
T OxLo(>" On; Deal) M2
4: Return: Total Gradient the k-th task Ti.
Algorithm 4 Neumann Expansion Based Total Hessian Computation
1: Input:
Data: Training Dataset D = ( U Deal), task & to be evaluated. Parameter:
The learned meta-parameter > the task-related parameters 0, for i E I.
Task Gradient: T
2: j
T t To
3: if [| 1,-1l11 > then
4:
for i E do
5:
The total Hessian H' related to the i-th task defined as
020 To (X*, 05 Di + 110,To (X*, 0;; DDat) 11 T O2 L1 (X*,0, 0; D(T) 0gdg L, (X*, D(T) LodxLo (;Do))
where / is the identity matrix.
6: end for
7: Use EK-FAC to compute the Hessian product S, 2 Eiez H 1y.
8: Compute l5+1 by
1,+1 I, 5 I + S₁ +
9: j+ 1
end if
11: Return: IF = 15.
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Published as a conference paper at ICLR 2026
Algorithm 5 Instance-level IF for Validation Data
1:
Input:
Data: Training Dataset D = Usez (D) U UDDa), data (ž, (]) to be evaluated, the corresponding
task k,
Parameter: The learned meta-parameter X* the task-related parameters 0, for i E Z.
2: Use EK-FAC to compute the middle gradient M
M = 010gL1 (x*, 02s D Ogd, - L (x-00;:D 0zl(ā, 0x).
3: Compute the total derivative of the k-th outer loss function with respect to X as T:
(, M.
4: Take T the gradient to be the input, use Neumann Series Approximation method to com- pute the instance-IF for validation data by Algorithm
5: Approximate the model retained when the k-th task is removed: X X insttanceII
6: Return: instance-IF, X
Algorithm Instance-IF IF for Training Data
Input:
Data: Training Dataset D Usez (DF U Dyal), data (æ, to be evaluated, the corresponding
task k;
Parameter: The learned meta-parameter X", the task-related parameters 0; for i E I.
2: Compute inner-IF for the k-th task as:
inner-IF -000 L, (x: Ogl(0, 02)
3: Compute the outer influence term Ps
PB = To (X*, 0z; Dral)" inner-IF.
4: Take T' as the gradient to be the input, use Neumann Series Approximation method to com- pute the instance-IF for validation data by Algorithm
5: Approximate the model retained when the data is removed: X X + instance-IF
6: Return: instance-IF, X
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Published as a conference paper at ICLR 2026
D EXPERIMENT DETAILS
Dataset. Omniglot features 1,623 handwritten characters from 50 alphabets. MNIST is a handwrit- len digit dataset with 60,000 training examples and 10,000 examples. Mini-ImageNet contains 50,000 training images and 10,000 test images across 100 classes. FC100 is a few-shot learning benchmark with 60,000 CIFAR- 100 images, divided into 64 training, 16 validation, and 20 test classes for systematic cross cross-category evaluation.
Implementation Details. Our experiments used an Intel Xeon CPU and GTX 1080 ti GPU. For task-level evaluation, we train the MAML model, randomly remove some tasks, and then use direct IF and task-IF update the model. For instance level, we randomly remove 40% sample for 20 tasks. During the training phase, we update the model for 5000 iterations and randomly sample 10,000 tasks for the omniglot and MINI-Imagenet datasets and 252 tasks for the MNIST dataset. We repeat the experiments 5 times with different random seeds.
E ADDITIONAL VISUALIZATION
F LIMITATION
In this work, we employ influence function to efficiently approximate the performance change that would occur if particular data instance task removed and the model were retrained (leave- one-out retraining). It is important to note that our method still fundamentally relies on estimating retraining outcomes rather than performing actual retraining, As a result, while we provides highly efficient and effective approximation in practice, it may not capture all nuances of true retraining, and potential discrepancies may exist between the estimated and actual data values. However, exper- imental results show that the influence function-baa approximations closely reflect the outcomes of actual retraining, making our data attribution method both practical and effective in real-workd meta-learning applications.
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Published as a conference paper at ICLR 2026
Mini-ImageNNe Dataset
(a) Influence score: 0.000878
(b) Influence score: 0.000879
(c) Influence score: 0. 00005554
(d) Influence score: 0,000577
Omniglot Dataset
e
73
(e) Influence score: 0.000413
(i) Influence score: 0,000424
A
(g) Influence score: 0.000507
(h) Influence score: 0,000511
MNIST Dataset
Influence 0.000492
ii) Influence 0.000498
8
(k) Influence score: 0.000537
(I) Influence score: 0.000557
24


Reviews:
[{"review_id": "m4QEZaMWlN", "reviewer": "Reviewer_u4Gn", "summary": "This paper proposes a general, closed-form framework to evaluate training-data influence in meta-learning, introducing two complementary measures, task-IF and instance-IF. They quantify how entire tasks and individual examples affect meta parameters directly and indirectly through task-specific parameters. It formalizes meta-learning as a BLO problem, derives total gradients and a total Hessian tailored to the bilevel coupling, and shows how to estimate parameter changes from up-weighting or removing data without full retraining. To make the method practical at scale, the authors add acceleration strategies for inverse Hessian vector products and approximate the total Hessian, then demonstrate utility on standard few-shot benchmarks for tasks like harmful-data identification and editing efficiency.", "questions": "- Please provide an ablation isolating the contribution of each ingredient so we can see where the gains come from and whether any component is redundant.\n\n- How do you choose the truncation depth J in practice? Reporting accuracy and runtime vs J would help.\n\n- Can you quantify the error or give guidance on when EK-FAC is acceptable vs when the Neumann route is mandatory?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "- The paper cleanly formulates meta-learning as bilevel optimization and introduces complementary task- and instance-level influence measures tailored to that structure.\n\n- It contributes practical scalability via acceleration techniques, making the closed-form influence derivations computationally feasible.\n\n- The experimental setup is comprehensive.", "weaknesses": "- The paper lacks a true ablation study isolating the contribution of its key components.\n\n- Accuracy gains over strong baselines are modest and sometimes below retraining, with the main advantage being runtime.\n\n- Reproducibility is constrained by the absence of a code repository or data link in the submission.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "2D60kPuWYH", "reviewer": "Reviewer_P6Eo", "summary": "The paper studies how to quantify the contribution of individual tasks and samples in meta-learning.It extends the classical influence function to the bilevel setting of meta-learning, where both task-specific and meta-level parameters interact.\n\nIt introduces two estimators: Task-IF, which measures the influence of entire tasks, and Instance-IF, which aims to measure the effect of individual samples inside tasks.\n\nExperiments on few-shot benchmarks show that their method can efficiently approximate the results of full retraining while identifying harmful or noisy data.\n\nIn essence, the paper attempts to make data attribution feasible for meta-learning by deriving tractable influence estimators.", "questions": "1. Can you analyze the range of cumulative errors, or compare Task-IF and Instance-IF predictions of \\Delta\\lambda to the true change obtained by removing a task/sample and retraining?\n\n2. In Proposition 4.8, the calculation of the Instance-IF for training data relies on the term D_\\lambda P(\\lambda^*, \\theta_k(\\lambda^*); \\tilde{z}). Given that P is a function of the inner-level Hessian inverse H_{k,in}^{-1}, which itself depends on \\lambda, the full derivative D_\\lambda P would involve third-order derivatives (the derivative of H_{k,in}^{-1} with respect to \\lambda). Could you clarify whether your implementation includes these third-order derivative terms? Did you use the same approximation strategy as Task-IF to ignore these higher-order derivatives?\n\n3. How sensitive are your influence estimates to the conditioning of H_{in}? Are the performances of Task-IF and Instance-IF stable across different types of tasks?", "limitations": "", "rating": 4, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "1. Introduced influence functions into meta-learning and adapted them to the bilevel optimization structure. \n2. Clarified why directly applying standard influence functions in bilevel optimization is incorrect and derived the corrected Task-IF formulation that properly accounts for inner–outer dependencies.\n3. Designed two types of influence functions, Task-IF and Instance-IF, enabling multi-granularity data evaluation.", "weaknesses": "1. The proposed method for training instance influence, which models the outer-loss change as a simple additive proxy P, implicitly relies on the sufficiency of local, linear approximations. This two-stage linear approximation (first to compute P, then to compute its influence) may fail to capture the complex, non-linear effects of removing a data point, especially in optimization landscapes with high curvature or when the removal of a point induces a significant shift in the task-specific parameters.\n\n2. The influence functions proposed in the paper are derived reasonably, but multiple approximations may lead to uncontrollable cumulative errors. The influence function is based on a first-order Taylor approximation; to avoid third-order derivatives, the total hessian matrix is simplified. Additionally, the computation of the inverse Hessian-vector products relies on iterative approximations. The paper does not provide an analysis of these cumulative errors.\n\n3. The experiments lack diversity in tasks, as the paper only evaluates few-shot image classification tasks. However, the loss landscapes of different types of tasks can vary significantly, which may affect the performance of Hessian-based influence functions, the core method of this paper.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "pUusj8yXrs", "reviewer": "Reviewer_nWff", "summary": "This paper introduces a framework for evaluating the influence of training data in the context of MAML, which is formulated as a bilevel optimization problem. The authors propose task influence functions (task-IF) and instance influence functions (instance-IF). These methods are designed to accurately assess the impact of entire tasks and individual data points on the meta-training process. The paper provides a theoretical foundation for these methods, including the use of total derivatives to capture the interdependent nature of meta and task-specific parameters. To address the computational complexity, the authors propose acceleration techniques such as EK-FAC and Neumann series approximation. The effectiveness of the proposed framework is demonstrated through experiments on several benchmark datasets, showing its utility in tasks like identifying and removing harmful data.", "questions": "Please refer to Questions", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. Novelty: The paper is the first to formally and introduce influence functions into the bilevel optimization framework for meta-learning, to thoroughly include task influence and data(training/validation) influence. \n\n2. Comprehensive and Well-Structured Framework: The proposed framework is comprehensive, addressing data influence at both the task and instance levels, and for both training and validation data. The distinction between these different levels of influence is well-motivated and clearly articulated.", "weaknesses": "1. The scope: The proposed IF evaluation method is only applicable for second-order MAML, which is only one specific algorithm among numerous  meta-learning algorithms. A narrowed title and scope, e.g. \"Evaluating data influence in MAML by IF\" would be more proper and accurate.\n\n2. Experimental and practical concern: The burden calculating IF is concerned, though some approximation and acceleration techniques have been applied. It seems there are much more simple and intuitive baselines in the application scenario of IF. For example, for retraining, a simple baseline is taking few gradient-ascend steps on the data to remove (both task/instance (train/valid)) are applicable), but has not been compared. For identifying harmful data, a simple baseline is evaluating  the loss scalar, but also has not been compared.\n\n3. Related work: it seems that [1] has included task-if of MAML, which weakens the novelty and contribution of this work, but not has been cited or discussed.\n\n[1] Mitsuka Y, Golestan S, Sufiyan Z, et al. TLXML: Task-Level Explanation of Meta-Learning via Influence Functions. arXiv preprint arXiv:2501.14271, 2025.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)