MODEL SHAPLEY: Find Your Ideal Parameter Player via One Gradient Backpropagation
Authors: Xu Chu
Xinke Jiang
Rihong Qiu
Jiaran Gao
Junfeng Zhao
Venue: NeurIPS 2025 poster ()
Keywords: Model Shapley; LLM
Primary area: deep_learning

Abstract: Measuring parameter importance is crucial for understanding and optimizing large language models (LLMs). Existing work predominantly focuses on pruning or probing at neuron/feature levels without fully considering the cooperative behaviors of model parameters. In this paper, we introduce a novel approach--Model Shapley to quantify parameter importance based on the Shapley value, a principled method from cooperative game theory that captures both individual and synergistic contributions among parameters, via only one gradient backpropagation. We derive a scalable second-order approximation to compute Shapley values at the parameter level, leveraging blockwise Fisher information for tractability in large-scale settings. Our method enables fine-grained differentiation of parameter importance, facilitating targeted knowledge injection and model compression. Through mini-batch Monte Carlo updates and efficient approximation of the Hessian structure, we achieve robust Shapley-based attribution with only modest computational overhead. Experimental results indicate that this cooperative game perspective enhances interpretability, guides more effective parameter-specific fine-tuning and model compressing, and paves the way for continuous model improvement in various downstream tasks.

MODEL SHAPLEY: Find Your Ideal Parameter Player via One Gradient Backpropagation
Xu Chu 1,2,4 chu_ thu_xu@pku.c edu. en
Xinke Jiang 1.3,4
Rihong Qiu 43,4
xinkejianglstu, pku. edu. cn
r rihonggiu@stu. pku. edu. cn
Jiaran Gao 134
Junfeng 7..0 zhao edu. cn
ji jirangaoostu, pky edu. cn
Key Laboratory of High Confidence Software Technologies, Ministry of Education, Beijing, China
CCenter on Frontiers of Computing Studies, Peking University, Beijing, China
National Engineering Research Center For Software Engineering, Peking University, Beijing, China
"ccooo of Computer Science, Peking University, Beijing, China
$ Big Data Technology Research Center, Nanhu Laboratory, Jiaxing, China
btttps://gthhan com
Abstract
Measuring parameter importance is crucial for understanding and optimizing large language models (LL.Ms). Existing work predominantly focuses pruning probing at neuron/feature levels without fully considering the cooperative behaviors of model parameters. In this paper, introduce novel Aaproach-Monee SHAPLEY to quantify parameter importance based on the Shapley value, a principled method from cooperative game theory that captures both individual and synergistic contributions among parameters, via only one gradient backpropagation. We derive scalable second-order approximation compute Shapley values the parameter level, leveraging blockwise Fisher information for tractability in large-scale settings. Our method enables fine-grained differentiation of parameter importance, facilitating targeted knowledge injection and model compression. Through mini-batch Monte Carlo updates and efficient approximation of the Hessian structure, we achieve robust Shapley-based attribution with only modest computational overhead. Experimental results indicate that this cooperative game perspective enhances interpretability, guides more effective parameter-specife fine-tuning and model compressing, and paves the way for continuous model improvement in various downstream tasks.
Introduction
Many of today's most powerful deep neural networks contain hundreds of millions or even billions of parameters, but not all parameters are created equal. From the perspective of parallel distributed processing (PDP) [1], the emergent behavior of a network arises from collective interactions among numerous parameters, each of which may appear individually unimportant but becomes crucial in synergy with others. Identifying precisely which weights drive performance is therefore essential for tasks such as knowledge injection [2], interpretability [3], and model compression [4]. While heuristic approaches like weight magnitude [5] or first-order sensitivity [6] provide rough estimates of parameter importance, they often neglect the synergy among parameters-ine a weight might appear "unimportant" on its yet become critical when combined with certain others [6].
"Coreesooniigg author.
39th Conference on Neural Information Processing Systems (NeurIPS 2025).

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To address this shortcoming, we introduce Model Shapley: cooperative thee-theoreetc framework for quantifying each parameter's contribution in a way that naturally accounts for interactions. Specif- ically, we treat each parameter as a "player" in game and measure its Shapley value [7] [ average marginal improvement delivers over all possible subsets of other parameters. The Shapley value known for satisfying desirable fairness axioms, but computing it exactly requires evaluating an expo- nential number of parameter subsets [8]. This has historically limited its use for parameter attribution.
MODEL SHAPLEY overcomes this barrier through a second-order, closed-form approximation that exploits local curvature of the loss (via the Hessian or Fisher information). With a single newwad-bbccweer pass and a blockwise Fisher- based Hessian approximation [9], we estimate every parameter's 's Shapley value scale. Further, we introduce a Monte Carlo incremental scheme that updates these importance scores online, accommodating large datasets or streaming scenarios without repeatedly retraining or pruning.
We demonstrate in extensive experiments that MODEL SHAPLEY significantly outperforms standard heuristics in three key applications:
Knowledge Injection. By protecting regularizing high-Shapley parameters, we reduce catastrophic forgetting [2], since these weights carry more critical knowledge from previous tasks.
Interpretability. Our framework supplies parameter-level attribution, revealing how specific weights or neurons most heavily influence the network's 's outputs, complementing traditional input-based explanations [3].
Quantization/Coprresiii Shapley-aware importance scores guide pruning or mixed-precision quantization [4] more effectively than magnitude or naive second-order methods, preserving accuracy even under aggressive compression.
Contributions. (1) We formulate the parameter- parameter-importance problem using the Shapley value, offering principled way to capture cross-parameter interactions-aligennna with the PDP view that cooperative effects among parameters are central to network function. (2) We derive a second-order, closed-form approximation (and incremental extension) for computing these values in single pass, scalable to modern architectures. (3) We thoroughly validate MODEL SHAPLEY on knowledge injection, interpretability, and compression, demonstrating its superior performance compared to simpler heuristics. Overall, this game-theoretic lens provides a fresh perspective on how to analyze and optimize deep neural networks at the parameter level.
2
Related Work
2.1 Shapley Value
The Shapley value (SV), initially introduced by Shapley [7]. provides a principled framework for fair payoff allocation among players in cooperative games. Building upon this theoretical foundation, SV has been adopted across variety of domains over the decades. [10] extended SV into probabilistic set- tings, while [1 [11] explored interaction indices among players. In data-driven fields, SV has increasingly been leveraged for model interpretability and data valuation. [3] popularized feature-level explanations via SHAP, [12] introduced Data Shapley to quantify the contribution of individual training examples. To address the exponential complexity of SV computation, several works have proposed approximation algorithms [13, 8]. i.e., [14] approximate data SV in one training run. Recent comprehensive surveys, such [15], consolidate SV's applications throughout the entire data analytics lifecycle, identifying major challenges related to efficiency, approximation, privacy, and interpretability. Beyond data and feature levels, SV has also been applied at the model level. Some works utilize Shapley value explain the information flow mechanisms inside neural networks [13, 16]. while others investigate model-level interactions, for example federated learning [17] and in traditional machine learning models such as random forests [18, 19]. However, despite these advances, existing studies predominantly focus on ap- plying SV at the data sample or feature level. or at the model level for inter-model comparisons. Due to the intrinsic combinatorial explosion, only few prior work has attempted to extend Shapley Value to the level of neural network internals [20, 21 aiming to compute cooperative game values for neurons. Nevertheless, these works rely on a multi-armed bandit and Monte Carlo approximation with signif- icant computational overhead, and the random permutation of neurons can harm model performance, introducing estimation bias. This gap motivates exploration of neuron-level Shapley analytics.

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2.2 Parameter Importance Quantification in Deep Learning
In recent years, with the growing interest in understanding the internal mechanisms of neuron networks, Parameter Importance Quantification [22, 23, 5] has become a crucial research direction. Analo- gous to the functional partitioning observed in the human cortex [24, 25, 26, whhre different regions are responsible for different cognitive tasks-LLMs may also develop "functional partitions" during training, with certain structural units (e.g.. attention heads [28, 29, 30J, feed-forward networks (FFNs) [31, 32, 33), individual neurons [34, 35, 36, 3, (.3) specializing in specific tasks storing particular knowledge. Parameter importance quantification typically involves clarifying the target capabilities (c.g., robustness [35], knowledge storage [29, 34]), preparing appropriate probing datasets, and determining the granularity of analysis (c.g.. attention head [39, 29I, neuron [37, 40], or layer [31]). Existing localization methods can be broadly categorized into two groups: O methods with forward signals, which locate functional units based on activation statistics (e.g. LAPE [37], Locate-then: unlearn [40]), activation patching (e.g., Function Vector [39], FVG [41]), integrated gradients [42]; @ and methods involving backward signals, which either trace gradient trajectories (e.g.. Recurrent- KIF [43], TaSL [35], OBS [5], OBD [38], GPTQ [4], DFAMS [44]. LightGNTK [45]) or combine forward and backward propagation signals (e.g.. Parenting (36]) to dynamically discover functional modules. However, these works only consider the intrinsic importance of parameter units for specific tasks, while the utility of the huge cooperative game between parameters is ignored.
There is conceptually related framework dubbed Component Attribution via Regression (COAR) [46]. We would like to highlight the differences: While both MODEL SHAPLEY and COAR tackle the challenge of parameter synergy, they motivated by different goals, parameter importance quantification for MODEL SHAPLEY versus model editing for COAR. Besides, two frameworks employ different technical means. COAR learns surrogate linear model, capturing interactions without second-orde curvature. Whereas MODEL SHAPLEY directly derives a closed-form approximation of the Shapley value from the model's loss landscape.
3 Background of Shapley
The Shapley value [7, 47], originally developed in cooperative game theory [48], provides principled approach to quantifying the marginal contribution of individual elements within set. this section, we introduce the notion of a utility function and formally define the Shapley value.
Definition 1. (Utility Function) Let θ (0:} M be finite set, where each element 8u-such as a data point, a feature attribution, or model parameter is regarded as a player. A utility function 2: + R assigns a real-valued score to each subset Os C O, representing the utility or value of that subset within the context of a given learning task.
Definition (Shapley Value) The Shapley value o for an element 0 E O quantifies its average marginal contribution to the utility function, computed over all possible subsets of θ that exclude It is defined as:
de UU) =
60(U)==
1 M
M
Σ U((sU(0)- (0)
(I)
|0s|=k =
where M =|0] denotes the total number of clements. This value satisfies four desirable axioms that ensure fairness in attribution: O the Null Player property, stating that if an element 0 has no effect on any subset, i.e., U(Os U (0}) U(Os) for all Ôs. then de (U) 0; e the Symmetry property, which guarantees that if two elements contribute identically to all subsets, they receive equal value, i.e.,
Φe, (U)= 69, (U); O the Linearity property, stating that the value respects linear combinations of utility functions, so that for any scalars an 01,02 E R, it holds that (a U1 +02 = =0x10)00)))))))) (01))(U1) +azzáa (U2): and finally, O the Efficiency property, which ensures that the total utility fully distributed across all elements, i.c., E (U)
U()). U())
3

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4 MODEL SHAPLEY
In this section, we introduce MODEL SHAPLEY, a scalable framework for estimating Shapley values at the parameter level, which consists of the following three key components:
Closed-Form Shapley Derivation. We theoretically derive a closed-form expression for the parameter-wis Shapley value, showing that it can be efficiently computed using a single backward pass through the network, thereby eliminating the need for repeated marginal evaluations.
Section 4.1:
Section 4.2: Monte Carlo-Based Shapley Incremental Approximation. To address real-world scenarios where the full dataset may exceed memory capacity (e.g., streaming scenarios online learning), we develop an incremental Monte Carlo approximation scheme. This method provides online estimates of Shapley values through mini-batch updates, implemented via exponentially weighted moving average (EWMA).
Section 4.3: Fisher Approximation with Blockwise Regularization. To reduce the computational cost associated with computing the full Hessian matrix, we approximate using the empirical Fisher Information Matrix. Furthermore, we apply a blockwise regularizatiin strategy to improve stability and efficiency during estimation.
The main notations, algorithm, and the application of MODEL SHAPLEY are listed in Appendix B, Appendix G, and Appendix E, respectively.
4.1 Closed-Form Shapley Derivation
Computing the Shapley value of individual parameter units (especially neurons) in neural net- works ppattiulllll large-scale architectures such as LLMs or VLMs-is computationally prohibitive. Formally, evaluating the marginal utility term U(Og (()) U(Os) for all subsets Os s O) ( requires enumerating an exponential number of combinations. For each subset, this process involves removing specific parameter 0; and re-evaluating (or even training)) the model to assess the change in performance on held-out test set. In large models with hundreds of millions or billions of parameters (e.g.. 0.5B, 7B, even 670B in LLMs), the number of subsets grows exponentially, yielding a worst-case complexity of (22M), ), where M is the total number of model parameters. Even with substantial computational resources, the sheer number of forward and backward passes-plus potential fine-tuning-rent renders exact Shapley value computation infeasible in practice.
However, we observe that the structure of trained neural networks enables a tractable approximation. Specifically, we can estimate the marginal utility of an individual parameter through the change in training loss when that parameter removed-i.. i.e.. deactivated by setting to zero or frozen during eval- uation. Instead of relying on local Maclaurin expansion at the origin [49] which may yield inaccurate approximations when the parameter value is far from zero- we adopt path-integrated formulation [50]. In this view, the loss change induced by parameter removal is modeled as a continuous trajectory in parameter space, and the total effect is computed by integrating the gradient along this path.
Theorem 4.1 Interrnteertted Approximation of the Loss). Let θ (0T be the current parameter vector at iteration т. For any perturbed configuration O' defined along a linear path from Or the loss function L(O'; :r,y) evaluated an input-label pair (x..) admits the following integral approximation:
C ((0)) (x,1)+ V+L(Orx))) dO:
dt, dt
(2)
where , OT -0'00'0--'-00-0 -") defines the linear interpolation between O and O'.
Analysis We now quantify the loss increase caused by removing a single parameter unit. Let 6. be the i-th parameter at step T, and let e; be the standard basis vector with at position i. Define the perturbed parameter vector as - - [[ - e,, which corresponds to deactivating parameter 9, while keeping all others fixed. To estimate the associated loss change, we adopt aath-integrated formulation by defining a linear interpolation between the removed and original states: O,- == - +. Of e E [0, 1), and interpret the loss difference as the cumulative gradient along this path: AC(OT): ((T)) :x.yy) L(O :x,y) =
Ccave-One-Parameter-oo
ac(O))) 0 dt. This yields the 80,
following approximation (Appendix C. for details):

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Theorem 4.2 (Loss Change from Single Parameter Removal). Let of be the value of parameter i at step T. Then the loss increase after removing 0, is approximately:
-((e))= -
- + H,,
(3)
where 87 (C(e"); is the gradieni of loss ((e")with respect to parameter 0; at iteration T, H is
30;
the i-th diagonal entry of the Hessian, and w [0,1] is curvature- basseer--bbdddeeee path weight that averages second-order effects along the trajectory.
Remark 4.3 (From Loss Change Marginal Utility). This loss change conceptually aligns with goal of measuring the impact of removing parameter unit. Although model performance is typically assessed through discrete metrics (e.g., test accuracy), in differentiable settings, such performance shifts can be naturally approximated via continuous changes in the loss. Thus, defining the utility function U(+) = ((0::))) the marginal utility of parameter O becomes: U( -
((07~=;r.)) ((O"= gf 0 + + w! H (0; ) Notably, the first-order term [ fJ recovers importance scores used prior work on pruning and saliency [51, 49]. However, relying solely on this term ignores interactions between parameters. These interactions are partially captured by second-order terms, particularly in overparameterized networks such large language models [52].
While removing single parameter offers useful insights, fails capture cooperative interactions among parameters critical for Shapley-style attribution. Therefore, we extend our analysis to arbitrary subsets. Let S S M } be subset of parameter indices, and define θ zero. To estimate the corresponding loss increase, we adopt path- integratt view over the subspace of removed parameters. Thus we have (Appendix C.2 for details):
Approximation of Marginal Utility
θ Ef ofee es. which corresponds setting all parameters in S to
Theorem 4.4 (Loss Change from Removing Parameter Subset). Let S (1,....M be subset of parameter indices. Then the loss increase from removing parameters in is approximated as:
C(OT,1..)) R DBT 1T +2
Σ
w! HT,0,0
(8)
JES
36, is the gradient of loss C(OT) respect to parameter 0, at iteration т, H is the (i.j)-th entry of Hessian matrix, and w E [0,1]] denotes a curvature-avea path weight reflecting the average influence of parameters i and j along the joint removal path.
where g7 = (C(0")
Remark 4.5 (Efficient Approximation of Marginal Contributions). This generalization enables efficient approximation of Shapley-style marginal utility terms. Given a coalition S S (i). the marginal contribution of parameter 0g is: U(SSU(() U(e5) = L(Or.-5) ((0r--(5(()) = (C( C(Or)) - (((())-(s((())) - ((07)) ) AC(OS) AC(Osu(e.)) which under the
O
second-order approximation simplifies to:
U(O)-(t)) U(())) -. -
H
ij H
(S)
(5)
JES
Closed-form Shapley Value Finally, by aggregating over all possible leave-out subsets following the Shapley value framework, now present a closed-form expression for the Shapley value of each parameter based on the approximated marginal utility (Appendix C.3 for details).
Theorem 4.6 (Closed-fae form Parameter Shapley Value). Let da denote the Shapley value of parameter 0 E O, defined as the expected marginal utility of adding 0g to all possible subsets S s (i}. Under the path-interg interg approximation, the Shapley value is given by:
(U)
1 M
M
Σ SC)
g.00
1 2
O
O
( Hr,
(6)
JES
|S] --
where g 0L( (C(08")ss the gradient of loss (with respect to parameter 6, at iteration t, H is the 86,
(i-j)-th entry of the Hessian 42 ((0) and w ($) E [0.11] denotes a curvature-oware path weight reflect-
ing the average inftuence of parameters& and along the joint removal path. By exploiting the symmetry
5

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of the Shapley value definition and the linearity of summation over subsets, this expression simplifies to:
g
w 10
H
(S)
(1) Individual Importance
j
Couperative Inteructions
Remark 4.7 (Interpretation of the Closed-form Shapley Value). This decomposition allows the Shapley value account not only for direct influence but also for collaborative effects among parameters eespeicalseespccialllyy relevant in highly overparameterized models: ! The first term -g, 0; quantifies the individual importance of parameter 0, consistent with many saliency- baased and pruning metrics. @ The second term captures cooperative interactions: self-curvature via H and cross-parameter synergy or redundancy via off-diagonal Hessian entries H For implementation. we take Wij as I and the cooperative interaction for the entire parameter set can be computed as - θ), where the row-wise dot product yields vector of cooperative importance scores for each output unit. To accelerate this computation, the matrix product H x can be efficiently evaluated using the Hessian-Vectto Product (HVP) technique [53].
Remark 4.8 (Shapley Axiom Validity under Approximation). Although equation 7 is derived from a path-integrated approximation rather than the original combinatorial definition, it approximately satisfies the core Shapley axioms (Null Player, Symmetry, Linearity, and Efficienny). In particular, for Linearity, the second-order expansion remains valid under linear combinations of loss functions, ensuring that the closed-form Shapley expression holds even when evaluating composite or multi-objective losses. The full proof's is in the Appendix C.S.
Remark 4.9 (Computational efficiency). Critically, computing all Shapley values ) requires only a single forward pass (to obtain L and θT), a single backward pass (to compute the gradient), and one Hessian extraction. This makes Shapley-style parameter attribution tractable even in large-scale models.
4.2 Monte Carlo-Based Shapley Incremental Approximation
While the closed-fore Shapley value in equation is defined over the full dataset, computing the full loss and its derivatives at each step is computationally and datasets. To address this, we adopt a stochastic approximation strategy. At each iteration т, we sample mini-batch of size B and compute empirical estimates of the loss, gradient, and Hessian: (()) s (O),), VZ())= se (G), V2((e) z V g²LB(() Using the mini- estimates, we compute per-step Shapley value approximation o; for each parameter via equation 7. However, due to the inherent variance mini-batch gradients, these estimates are often noisy. To mitigate noise and stabilize the estimation over time, we maintain an exponentially weighted moving average (EWMA) [54] of the Shapley value for each parameter:
for large models
= (1-a)) +ap,,
(8)
where a E (0,1) is smoothing coefficient. The initial estimate is initialized as =0. This update rule allows our framework to track stable estimates of parameter importance streaming setting without requiring access to the full dataset.
4.3 Fisher Approximation and Blockwise Regularization
Although the closed-form Shapley value in equation 7 requires access to the full Hessian matrix ("T( computing or storing the complete Hessian is computationally infeasible for large-scale models. To overcome this limitation, we approximate the Hessian using the empirical Fisher Information Matrix (FIM) [55, 56] (Full proof is in Appendix D). Formally, the empirical Fisher matrix at the current parameter state e is defined as:
F(OT): = :=B VeL(O";z.)) VeL(O" :r.y)
where Ve.(eT x, 3) denotes the gradient of the loss with respect to parameters evaluated at sample (x, y). In practice, we approximate this expectation over a mini-batch of size B: T. The Fisher matrix F(O") offers several favorable properties: It is positive semi-definite by construction, more numerically stable than the raw Hessian, and computationally easier to approximate in stochastic settings.
Fn(O*)= = E Chanh

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While FB is more tractable than the true Hessian, it can still be large and noisy in high dimensiona models. To further reduce complexity and enhance stability, we partition model parameters into semantically meaningful blocks-such as neurons within layer or individual attention heads-ane estimate a blockwise Fisher matrix for each group. Let G U (b)) be a decomposition of the parameter set into B disjoint blocks (i.e., the neuron in one layer as a block). For each block b, This blockwise approach significantly reduces both memory and computation, while preserving localized interactions among structurally related parameters'.
we compute a local Fisher estimate: F(b) = B E(..w) g ( x, y) V ((""
Experiments
In this section, we conduct series of experiments to evaluate the performance of MODEL SHAPLEY against several mainstream neuron importance localization baselines. We perform evaluations in both CV and NLP settings, focusing on three key strategies: deactivation during inference, targeted fine-tuning during training, and model compression. Moreover, we also explore the effects of applying these strategies at different parameter unit granularities, such as neurons, layers, and attention heads. Further experimental details and results are provided in Appendix H.
5.1 Experimental Setup
Datasets. We utilize a grade school math dataset GSM8K and multitask language understanding dataset MMLU for NLP tasks to evaluate language transformer models, and use image classification datasets CIFAR- 100 and ImageNel for vision transformer models in CV tasks. For the natural language process datasets, we use zero-shot prompts and ule-based evaluators to assess the performance of L.MMs. For the computer vision datasets, we follow the standard experimental settings as established in prior work. More details can be found in Appendix H.1.
Backbones. For NLP tasks, [32] as language model backbones. For computer vision tasks, we adopt the widely used transformer-basee vision model ViT- Brre/Bae/66 [58] as the backbone.
Qwen2.5-3BB ---struuct Qwen2.5-7B-Bi 2--rst--7BBnnstrrcc [57], and
Baselines. In our experiments, compare MODEL SHAPLEY with the following baselines: (1) Random Masking, where parameter units are randomly selected for intervention; (2) Gradient [43,5], which ranks parameters by their absolute gradient values g |; (3) Gradient Trace [40, 35, 49, 43), computed as the product of parameter values and their gradients, i.e., 10 g.! 1 For the training setting, we apply targeted fine-tuning to the most important parameters (or neurons) identified by each method (1-3). In addition to this focused fine-tuning, we also evaluate full fine-tuning strategies as points of comparison. @ For the inference setting, for baselines (1-3), we assess parameter importance by selectively deactivating the least important units (neurons or layers), i.e., setting their outputs to zero, and observing the resulting performance degradation. We also include a Pretrain baseline, where the model is evaluated without any intervention, to serve as reference point. 2 For the compression setting, we compare MODEL SHAPLEY with GPTQ [4] and OBD [38] with different quantization INT4, INT8 and FP8 on GSM8K dataset. A detailed description of baselines can be referred to in Appendix H.2.
Settings and Evaluation. In the experimental data processing phase, we strictly adhere to the original training-test set provided
providdd for each dataset to ensure the reproducibility results and comparability with prior studies. Specifically, for the original training set of each dataset, we further employ stratified random sampling to partition into a training subset and a validation subset at an 80%:20% ratio. The quantification processes of parameter importance are done the validation subset. All tasks are evaluated under the zero-shot setting. For NLP tasks, we instruct LLMs to produce answers in specific format so that can extract the answer from the response and compare it with the ground truth easily. The prompts we used can be found in Appendix H.3. For CV tasks, we add a pre-trained classification head to the ViT model for image classification tasks, and accuracy is used as the evaluation metric. At inference, we deactivate the bottom 5% (NLP) and 30% (CV) of neurons ranked by importance;
'I practice, due to the different dimensions of each layer. it is computationally intractable to maintain cross-layer Fisher off-diagonals LLM scale; therefore use olock-diagonal Fisher.
7

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during training, we fine-tune only the top 10% and freeze the rest. We to "activation ratio" as the kept-fraction of neurons. More details of the experiments can be found in the Appendix H.4.
5.2
MODEL SHAPLEY Advantages
Table 1: Evaluation of different inference and training methods across models and datasets.
VIT-Base/I6 (CV)
Qwen2.5-3B (NLP)
Qwen2.5-7B (NLP)
Method
CIFAR-100
ImageNet
GSM8K
MMLU
GSM8K
MMLU
Pretrain
79.69
76.14
45.57
60.81
72.48
73.06
Inference (Deactivate Neurons)
Random
08.39
18.25
04,47
50.11
19.94
66.15
Gradient
77.82
65.99
37.53
50.31
46.70
66.86
Gradient Trace
76.65
67.62
36.09
51.99
72.71
68.01
MODEL SHAPLEY
80.84
70.33
38.06
52.08
73.39
68.93
Full Fine-Tune
85.31
78.09
54.89
63.08
72.55
73.56
Training (Freeze Neurons)
Random
84.27
79.76
46.98
60.68
60.80
68.44
Gradient
84.64
79.63
47.57
61.35
61.87
69.08
Gradient Trace
84.69
79.57
47.08
63.59
61.41
70.79
MODEL SHAPLEY
86.53
79.82
47.76
63.72
62.02
73.89
Table 2: Evaluation of different inference and training methods across models and datasets on GSM8K.
INT4 (W4A16)
INT8 (W8A8)
FP8 (WA-FPS)
Quantization
(unt
Accuracy
Runtime(min)
Accuracy
Runtime(min)
Accuracy
Owen 2.5-Instruct (7B)
No Compression Accuracy: 72.48
GPTQ
71.73
62.70
53.86
70.58
52.19
74.15
OBD
70,98
62.55
57.88
71.27
53,88
71.42
MODEL SHAPLEY
74.67
63.23
60.20
72.93
58.31
74.83
Qwen 2.5-Instruct (14B)
No Compression Accuracy: 77.41
GPTQ
132.28
65.20
71.68
75.66
78.63
76.95
OBD MODEL SHAPLEY
128.99
62.77
94.90
75.36
73.74
77.48
98.84
63.84
76.46
76.27
84.75
78.85
As discussed, we conduct experiments and report results on variety of NLP and CV tasks, as shown Table 1 and Table 2. From the reported accuracy results, we observe the following findings:
Superior to First-Order Parameter Location Methods. First, across diverse scenarios in both CV and NLP domains, and at both the neuron and layer levels, inference, training, and compression settings, our proposed MODEL SHAPLEY consistently outperforms all baselines, achieving state-of- the-art art performance. This demonstrates the effectiveness of leveraging inter peramprreeterr cooperative dynamics through Shapley-based attribution to identify important units. MODEL SHAPLEY achieves the highest average performance across nearly all tasks, with a performance drop of almost average 4.69 % in the inference setting and an improvement of up to 3.10% % over the best rest-performing baseline the training setting. We argue that by focusing training or deactivation on neurons with high Shapley values- analoouus to selectively enhancing suppressing the "elite members" of team-MopEl SHAPLeY causes the most significant perturbation to model behavior, highlighting their functional importance. @ Furthermore, compared to standard Pretraining and Full Fine- Sine-TTning strategies, targeted fine-tuning guided by MODEL SHAPLEY values demonstrates greater robustness: even when only 10% of the most active neurons are retained, the model still maintains performance of 73.89 for MMLU dataset, indicating that effective capacity enhancement or knowledge injection can be achieved by training only a small subset of neurons.
Superior Compression Capability Compared with GPTQ and OBD, MODEL SHAPLEY delivers stronger compression even with limited calibration data. Under the INT8 setting, MODEL SHAPLEY-

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based quantization achieves a score of 76.27% only 1.14% below the original (unquantized) model 77.41% %. This highlights the practical effectiveness of incorporating Shapley- corrections during quantization and aligns with the second-order rationale described earlier. In contrast, MODEL, SHAPLEY performs poorly in the INT4 regime, The core issue is the extremely limited codehook (16 bins): our pipeline must compress Shapley values spanning [10-3, 109) into these few levels, which induces severe dynamic-range collapse, due to a 16-1evel codebook vs heavy-tailed Shapley magnitudes. See Appendix F for the 1/s² compensation scaling.
Strong Generalization Ability. Table summarizes the generalization results. MODEL SHAPLEY consistently outperforms baseline methods across nearly all settings, further reinforcing its strong generalization capability across a wide range of scenarios -nriaaiiis-incudiinng both CV and NL.P tasks, various domains, model sizes and backbones (in Figure 2), and under different settings such as training or inference time interventions or model compression (in Table 2). Moreover, its effectiveness holds at multiple granularity levels, from individual neurons to layers attention heads (in Appendix H.9), making it a robust and broadly applicable theoretical framework.
5.3 Interpretability
Visualization of Parameter Units To evaluate the interpretability of MODEL SHAPLEY at the parameter level, we conduct experiments on both GSM8K and MMLU using different types and sizes of backhone models: Qwen 2.5-3B, Qwen 2.5-7B, and LLaMA 3-3B in both the neuron level and the layer level. 1 At the layer level, as shown in Figure 1, we observe that across different models, the v-proj layers in Qwen exhibit higher MODEL SHAPLEY values, whereas in LLaMA, the k_proj and o_proj layers show noticeable activation between layers 9 and 15. Interestingly, v_proj consistently shows high Shapley values across all models. For knowledge-intenste (e.g., MMLU). Qwen demonstrates more significant importance layers 12-26, while LLaMA shows stronger responses in the early to mid layers (layers 0-15). @ At the neuron level, Figure 2 presents the number of neurons with Shapley values above a predefined threshold across different layers. We find that for both Qwen and LLaMA, middle layers tend to contain more influential neurons. Specifically, for knowledge-heasd -eayy-eeavyyyyeo tasks like MMLU, the most impactful neurons are concentrated in layers 5-17, whereas for math-focused tasks like GSM8K, the high-impact range is broader: layers 3-20 for Qwen and layers 10-14 for LLaMA. Additionally, the output layers of both models exhibit high Shapley values, indicating their critical role in task-specific reasoning.
tasks
69
4.000
0
(a) Qwen2.5-3B on GSMSK
(b) Qwen2.5-7B on GSMSK
(c) LLLMA3-3B on GSM8K
(d) Qwen2.5-3B on MMLU (e) Qwen2.5-77 7B on MMLU
(f) LLaMA3 3B on MMLU
Figure 1: Layer-wise shapley value for q/k/v/o projection.
Neuron Deactivation Study. In the inference setting on the MMLU dataset, we identify knowledge- specific neurons corresponding to fields such as Math and Daily Knowledge. We then selectively deactivate 0%, 5%, 10%, and 30% of these neurons to evaluate their functional roles. As shown in case studies Appendix H.5, deactivating just 10% of the relevant domain neurons leads to notable degradation in the model's 's ability to answer domain-speciff questions. When 30% of the neurons are deactivated, the model almost entirely loses domain knowledge and begins generating incoherent or irrelevant responses.

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Ove Soimms Lorel (iiutthee
Neurmo with Shppy
10
Layer Indes
Layar Indeet
(a) MMLU
(b) GSM8K
Figure 2: Neuron activation counts per layer two benchmarks.
Parameter Compensation Effect. We also observe a phenomenon term the parameter compen- sation effect, drawing inspiration from the concept of neural compensation in neuroscience [59, 60, 61] where unaffected regions of the brain increase their activity to compensate for damaged areas. This effect is particularly evident during training. As shown in Table 1, when certain parameters are frozen, others adaptively increase their contribution and take over the lost functionality. similar pattern emerges during quantization, where unquantized neurons compensate for the functions of their quantized counterparts. Furthermore, as illustrated in Figure 3b in Appendix H.7, parameters with initially low Shapley values gradually become more important over the course of training, suggesting a dynamic redistribution functional roles within the model.
Comparison with LoRA. Although MODEL SHAPLEY (subset selection of original param- eters) differs conceptually from LoRA (low-rank adapters), direct comparison with matched number of trainable parameters informative. Table 10 reports accuracy and training cost for Qwen2. 5-3B- Instruct and Qwen2. 1eee2.5-B--Inae 5-78-Insstrct under three settings: Shapley, LoRA, and their combination. Overall, MODEL SHAPLEY delivers competitive performance with comparable efficiency, and the Shapley+LoRA combination yields further accuracy gains.
Conclusion, Limitations and Future Work
We present MODEL SHAPLEY, a cooperative game-theoretic framework for quantifying parameter importance in large-scale neural networks. Treating each parameter (neuron, layer, attention head) as a "player," MODEL. SHAPLEY employs a second-order, closed-form Shapley approximation regularized by blockwise Fisher information, enabling efficient estimation of both individual and synergistic contributions scale via only one back propagation. Across NLP and CV benchmarks, MODEL SHAPLEY guides targeted parameter interventions that improve knowledge retention, interpretability, and model compression. While our tractable, empirical-Frsherr can incur inaccuracies in heavily overparameterizee models under non-stationary training regimes, blockwise strategy alleviates but does not fully eliminate- these issues. Promising directions include developing more adaptive Hessian surrogates, extending the framework to multi-task task and multi-modal settings, and investigating how Shapley values evolve under continual or non-s non-stationary in training to sharpen parameter-level insights. We also plan to leverage Shapley for model pruning and to
approximation
explore nonlinear Shapley-basee based Hessian corrections tailored to INT4 quantization to further improve performance. Employing MODEL SHAPLEY for important applications such model editing [46] also a interesting direction for future work.
Acknowledgments
We thank the reviewers for their useful feedback that improved this work. This research is supported by the National Natural Science Foundation of China (NSFC) under grants No.62506010 and No. No.U23320468.
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References
[1] David Rumelhart, Geoffrey E. Hinton, James L McClelland, et al. A general framework for parallel distributed processing. Parallel distributed processing: Explorations in the microstructure of cognition, 1(45-76):2, 1986.
[2] James Kirkpatrick, Razvan Pascanu, Neil Rabinowitz, Joel Veness, Guillaume Desjardins, Andrei A Rusu, Kieran Milan, John Quan, Tiago Ramalho, Agnieszka Overcoming catastrophic forgetting in neural networks. Proceedings of the national academy of sciences, 11((13)35521-333 2017.
et al.
[3] Scott M Lundberg and Su-In Lee. A unified approach to interpreting model predictions. Advances in neural information processing systems, 30, 2017.
[4] Elias Frantar, Saleh Ashkboos, Torsten Hoefler, and Dan Alistarh. Gptq: Accurate post-training quantization for generative pre-trained transformers, 2023.
[5] Yann LeCun, John Denker, and Sara Solla. Optimal brain damage. Advances in neural information processing systems, 2, 1989.
[6] Pavlo Molchanov, Stephen Tyree, Tero Karras, Timo Aila, and Jan Kautz. Pruning convolutional neural networks for resource efficient inference. In International Conference on Learning Representations, 2017.
[7] Lloyd Shapley al. value for n-person games. 1953.
[8] Javier Castro, Daniel Gómez, and Juan Tejada. Polynomial calculation of the shapley value based on sampling. Computers & Operations Research, 3665): 2009.
[9] Babak Hassibi and David Stork. Second order derivatives for network pruning: Optimal brain surgeon. Advances in neural information processing systems, 5, 1992.
[10] Pradeep Dubey and Robert J Weber. Probabilistic values for games. 1977.
[11] Michel Grabisch and Mare Roubens. An axiomatic approach the concept of interaction among players in cooperative games. International Journal of Game Theory, 28:547-565, 1999.
[12] Amirata Ghorbani and James Zou. Data shapley: Equitable valuation of data for machine learning. In International Conference on Machine Learning, pages 2242-2251, 2019.
[13] Marco Ancona, Cengiz Öztireli, and Markus Gross. Explaining deep neural networks with polynomial time algorithm for shapley values approximation, 2019.
[14] Jiachen T. Wang, Prateck Mittal, Dawn Song, and Ruoxi Jia. Data shapley in one training run, 2024.
[15] Hong Lin, Shixin Wan, Zhongle Xie, Ke Chen, Meihui Zhang, Lidan Shou, and Gang Chen. comprehensive study of shapley value in data analytics. Proceedings of the VLDB Endowment, 14(1):1-12, 2025.
[16] Rui Wang, Xiaoqian Wang, and David Inouye. Shapley explanation networks, 2021.
[17] Haolin Zhu, Ziye Li, Dingzhi Zhong, Cheng Li, and Yong Yuan. Shapley-value-Be based contribution evaluation federated learning: A survey. In 2023 IEEE 3rd International Conference on Digital Twins and Parallel Intelligence (DTPI), pages 1 1-5, 2023.
[18] Benedek Rozemberczki, Lauren Watson, Péter Bayer, Hao-T Yang. Olivér Kiss, Sebastian Nilsson, and Rik Sarkar. The shapley value in machine learning. In The 31 International Joint Conference on Artificial Intelligence and the 25th European Conference on Artificial Intelligence, pages 5572-5579. International Joint Conferences on Artificial Intelligence Organization, 2022.
[19] Jianyuan Sun, Hui Yu, Guoqiang Zhong, Junyu Dong, Shu Zhang, and Hongchuan Yu. Random shapley forests: Cooperative game-based random forests with consistency. IEEE transactions on cybernetics, 52(1):205-214 2020.

---

[20] Amirata Ghorbani and James Y Zou. Neuron shapley: Discovering the responsible neurons. Advances neural information processing systems. 33:5922-59333 2020.
[21] Kamil Adamczewski, Yawei Li, and Lue van Gool. Shapley pruning for neural network compression, 2024.
[22] Kedar Dhamdhere, Mukund Sundararajan, and Qigi Yan. How important is neuron? arXiv preprint arXiv: 1805. 12233, 2018.
[23] Ruichi Yu, Ang Li, Chun-Fu Chen, Jui-Hsin Lai, Vlad Morariu, Xintong Han, Mingfei Gao, Ching-Yume Lin, and Larry S Davis. Nisp: Pruning networks using neuron importance score propagation. In Proceedings of the IEFE conference on computer vision and pattern recognition, pages 9194-9203, 2018.
[24] Michael A Arbib. The handbook brain theory and neural networks. MIT press, 2003.
[25] Michael Hawrylyez, Ed S Lein, Angela L. Guillozet- Guillozet-Bongaarts, Elaine H Shen, Lydia Ng, Jeremy A Miller, Louie N Van De Lagemaat, Kimberly A Smith, Amanda Ebbert, Zackery L Riley, et al. An anatomically comprehensive atlas of the adult human brain transcriptome. Nature, 489(7416):391-399, 2012.
[26] Authony M Zador. A critique of pure learning and what artificial neural networks can learn from animal brains. Nature communications, 10(1):3770, 2019.
[27] Mengru Wang. Yunzhi Yao, Ziwen Xu, Shuofei Qiao, Shumin Deng, Peng Wang, Xiang Chen, Jia CiaC Gu, Yong Jiang, Pengjun Xie, et al. Knowledge mechanisms in large language models: A survey and perspective. arXiv preprint 2rXiv:2407.15017, 2024.
(28) Zifan Zheng, Yezhaohui Wang, Yuxin Huang, Shichao Song, Mingchuan Yang, Bo Tang, Feiyu Xiong, and Zhiyu Li. Attention heads of large language models: A survey. arXiv preprins 20X.::249903355. 2024.
[29] Kayo Yin and Jacob Steinhardt. Which attention heads matter for in-context learning? arXiv preprint arXiv:2502. 14010, 2025.
[30] Wenhao Wu, Yizhong Wang, Guangxuan Xiao, Hao Peng, and Yao Fu. Retrieval head mechanistically explains long-context factuality. arXiv preprint 2rX1v:2404.15574 2024.
[31] Lucas Bandarkar, Benjamin Muller, Pritish Yuvraj, Rui Hou, Nayan Singhal, Hongjiang Lv, and Bing Liu. Layer swapping for zero-shot cross- lingual transfer large language models. arXiv preprint arXiv:2410.0133. 2024.
[32] Chris Wendler, Veniamin Veselovsky, Giovanni Monea, and Robert West. Do Ilamas work in english? on the latent language of multilingual transformers. In Proceedings of the 62nd Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), pages 15366-155 15394, 2024.
[33] Qi Sun, Mare Pickett, Aakash Kumar Nain, and Llion Jones. Transformer layers as painters. Pro- the
ceedings the AAAI Conference on Artificial Intelligence, volume 39, pages 25219-25221 2025.
[34] Jiahao Huo, Yibo Yan, Boren Hu, Yutao Yue, and Xuming Hu. Mmneuron: Discovering neuron-level domain-specifer interpretation in multimodal large language model. arXiv preprint arXiv:2406. 11 193, 2024.
[35] Yujie Feng, Xu Chu. Yongxin Xu, Guangyuan Shi. Bo Liu, and Xiao-Ming Wu. Tasl: Continual dialog state tracking via task skill localization and consolidation. arXiv preprint SSt.:220080085577 2024.
[36] Yongxin Xu, Ruizhe Zhang, Xinke Jiang, Yujie Feng, Yuzhen Xiao, Xinyu Ma, Runchuan Zhu, Xu Chu, Junfeng Zhao, and Yasha Wang. Parenting: Optimizing knowledge selection of retrieval-augmente language models with parameter decoupling and tailored tuning. arXiv preprint 241.:2400000300 2024.
12

---

[37] Tianyi Tang, Wenyang Luo. Haoyang Huang, Dongdong Zhang, Xiaolei Wang, Xin Zhao, Furu Wei, and Ji-Rong Wen. Language-specitis neurons: The key multilingual capabilities in large language models. In Lun-Wei Ku, Andre Martins, and Vivek Srikumar, editors, Proceedings of the 62nd Annual Meeting of the Association for Computational Linguistics (Volume I: Long Papers), pages 5701-5715, Bangkok, Thailand, August 2024. Association for Computational Linguistics.
[38] Yann Le Cun, John S. Denker, and Sara A. Solla. Optimal brain damage, page 598-605. Morgan Kaufmann Publishers Inc., San Francisco, CA, USA, 1990.
[39] Eric Todd, Millicent L Li, Arnab Sen Sharma, Aaron Mueller, Byron C Wallace, and David Bau. Function vectors in large language models. arXiv preprint arXiv:2310.152133, 2023.
[40] Songshi Liang, Hongda Sun, Ting-En Lin, Yuchuan Wu, Zihe Wang, Yongbin Li, and Rui Yan. Locate-then- unnlenne An effective method of multi-task continuous learning for large language models.
[41] Gangwei Jiang. Caigao Jiang. Zhaoyi Li, Siqiao Xue, Jun Zhou, Linqi Song, Defu Lian, and Ying Wei. Unlocking the power of function vectors for characterizing and mitigating catastrophic forgetting in continual instruction tuning, 2025.
[42] Damai Dai, Li Dong, Yaru Hao, Zhifang Sui, Baobao Chang, and Furu Wei. Knowledge neurons in pretrained transformers. arXiv preprint arXiv:2 10400869, 2021.
[43] Yujie Feng, Xujia Wang. Zexin Lu, Shenghong Fu, Guangyuan Shi, Yongxin Xu, Yasha Wang, Philip S. Yu, Xu Chu, and Xiao-Ming Wu. Recurrent knowledge identification and fusion for language model continual learning, 2025.
[44] Zhibang Yang, Xinke Jiang. Rihong Qiu, Ruiging Li, Yihang Zhang, Yue Fang, Yongxin Xu, Hongxin Ding, Xu Chu, Junfeng Zhao, et al. Dfams: Dynamic-flow guided federated alignment based multi protoip search. arXiv preprint arXiv:2508..033 20353, 2025.
[45] Rihong Qiu, Xinke Jiang, Yuchen Fang. Hongbin Lai, Hao Miao, Xu Chu, Junfeng Zhao, and Yasha Wang. Efficient graph continual learning via lightweight graph neural tangent kernels-based dataset distillation. In ICML 2025, 2025.
[46] Harshay Shah, Andrew Ilyas, and Aleksander Madry. Decomposing and editing predictions by modeling model computation. In International Conference on Machine Learning, pages 44244-44292. PMLR, 2024.
[47] Encarnación Algaba, Vito Fragnelli, and Joaquín Sánchez-Soriano. Handbook of the Shapley value. CRC Press, 2019.
[48] Rodica Branzei, Dinko Dimitrov, and Stef Tijs. Models in cooperative gume theory, volume 556. Springer Science& & Business Media, 2008
[49] Yongqi Leng and Deyi Xiong. Towards understanding multi-task learning (generalization) of LLMs via detecting and exploring task-specific neurons. In Owen Rambow, Leo Wanner, Marianna Apidianaki, Hend Al- Khaliff, Barbara Di Eugenio, and Steven Schockaert, editors, Proceedings of the 31st International Conference on Computational Linguistics, pages 1969-2987) Abu Dhabi, UAE, January 2025. Association for Computational Linguistics.
[50] Mukund Sundararajan, Ankur Taly, and Qiqi Yan. Axiomatic attribution for deep networks, 2017.
[51] Wanying Xie, Yang Feng, Shuhao Gu, and Dong Yu. Importance-based neuron allocation for mul- tilingual neural machine translation. In Chengqing Zong, Fei Xia, Wenjie Li, and Roberto Navigli, editors, Proceedings of the 59th Annual Meeting of the Association for Computational Linguistics and the Ith International Joint Conference on Natural Language Processing (Volume I: Long Papers), pages
Online, August 2021. Association for Computational Linguistics.
[52] B. Hassibi, D.G. Stork, and G.J. Wolff. Optimal brain surgeon and general network pruning. In IEEE International Conference on Neural Networks, pages 293-299 vol.1, 1993.
[53] Eire Oldewage, Ross M. Clarke, and José Miguel Hernández-Looaft Series of hessian-vector products for tractable aaddle-free newton optimisation of neural 2024,,
13

---

[54] Axel A. Araneda. Asset volatility forecasting:the optimal decay parameter in the ewma model, 2021.
[55] Jing Liu, Haidong Yuan, Xiao-Ming Lu, and Xiaoguang Wang. Quantum fisher information matrix and multiparameter estimation. Journal of Physics A: Mathematical and Theoretical, 53(2):023001, 2020.
[56] James C Spall. Monte carlo computation of the fisher information matrix in nonstandard settings. Journal of Computational and Graphical Statistics, 14(4):889-909) 2005.
[57] Jinze Bai, Shuai Bai, Yunfei Chu, Zeyu Cui, Kai Dang, Xiaodong Deng, Yang Fan, Wenbin Ge, Yu Han, Fei Huang, Binyuan Hui, Luo Ji, Mei Li, Junyang Lin, Runji Lin, Dayiheng Liu, Gao Liu, Chengqiang Lu, Keming Lu, Jianxin Ma, Rui Men. Xingzhang Ren, Xuancheng Ren, Chuanqi Tan, Sinan Tan, Jianhong Tu, Peng Wang. Shijie Wang. Wei Wang, Shengguang Wu, Benfeng Xu, Jin Xu, An Yang, Hao Yang, Jian Yang. Shusheng Yang. Yang Yao, Bowen Yu, Hongyi Yuan, Zheng Yuan, Jianwei Zhang, Xingxuan Zhang, Yichang Zhang, Zhenru Zhang, Chang Zhou, Jingren Zhou, Xiaohuan Zhou, and Tianhang Zhu. Qwen technical report, 2023.
[58] Alexey Dosovitskiy, Lucas Beyer, Alexander Kolesnikov, Dirk Weissenborn, Xiaohua Zhai, Thomas Unterthiner, Mostafa Dehghani, Matthias Minderer, Georg Heigold, Sylvain Gelly, Jakob Uszkoreit, and Neil Houlsby. An image is worth 16x 16 words: Transformers for image recognition at scale. In International Conference on Learning Representations (ICLR), 2021.
[59] Alessia Celeghin, Matteo Diano, Beatrice De Gelder, Lawrence Weiskrantz, Carlo A Marzi, and Marco Tamietto. Intact hemisphere and corpus callosum compensate for visuomotor functions after early visual cortex damage. Proceedings of the National Academy of Sciences, 1 (1444)))110455-11440
2017.
[60] Bryant M Duda, Max M Owens, Emily S Hallowell, and Lawrence H Sweet. Neurocompennstory effects of the default network in older adults. Frontiers in aging neuroscience, 11:111, 2019.
[61] Jiaze Sun, François- Laurent De Winter, Fiona Kumfor, Daphne Stam, Kristof Vansteelandt, Ron Peeters, Stefan Sunaert, Rik Vandenberghe, Mathieu Vandenbulcke, and Jan Van den Stock. Neural compensation in manifest neurodegeneration: systems neuroscience evidence from social cognition in frontotemporal dementia. Journal of Neurology, 270(1):58--547, 2022.
[62] Karl Cobbe, Vineet Kosaraju, Mohammad Bavarian, Mark Chen, Heewoo Jun, Lukasz Kaiser, Matthias Plappert, Jerry Tworek, Jacob Hilton, Reiichiro Nakano, Christopher Hesse, and John Schulman. Training verifiers to solve math word problems. arXiv preprint arXiv:2110.1411 2021.
[63] Dan Hendrycks, Collin Burns, Steven Basart, Andy Zou, Mantas Mazeika, Dawn Song, and Jacob Steinhardt. Measuring massive multitask language understanding. arXiv preprint 0rX1v:2009.03300, 2020.
[64] Alex Krizhevsky, Geoffrey Hinton, et al. Learning multiple layers of features from tiny images. 2009.
[65] Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei. Imagenet: A large-scale hierarchical image database. In CVPR, pages
Icee, 2009.
[66] Takeshi Kojima, Shixiang (Shane) Gu, Machel Reid, Yutaka Matsuo, and Yusuke Iwasawa. Large language models are zero-shot reasoners. In Advances in Neural Information Processing Systems, volume 35, pages 22199-22213, 2022.
[67] Thomas Wolf, Lysandre Debut, Victor Sanh, Julien Chaumond, Clement Delangue, Anthony Moi, Pierric Cistac, Tim Rault, Rémi Louf, Morgan Funtowiez, Joe Davison, Sam Shleifer, Patrick von Platen, Clara Ma Yacine Jernite, Julien Plu, Canwen Xu, Teven Le Scao, Sylvain Gugger, Mariama Drame, Quentin Lhoest, and Alexander M. Rush. Huggingface's transformers: State-of-the-art natural language processing, 2020.
[68] Feiyuan Zhang, Dezhi Zhu, James Ming, Yilun Jin, Di Chai, Liu Yang, Han Tian, Zhaoxin Fan, and Kai Chen. Dh-rag: A dynamic historical context-powered retreneel-augmeetedd generation method for multi-tun turtiittuuduuni dialogue, 2025.
14

---

[69] Xinke Jiang, Yue Fang. Rihong Qiu, Haoyu Zhang, Yongxin Xu, Hao Chen, Wentao Zhang, rag case study on medical
Ruizhe Zhang, Yuchen Fang, Xu Chu, al. Tc-rag: IIm systems. ACI. ordl 2025, 2025.
[70] Xinke Jiang, Ruizhe Zhang, Yongxin Xu, Rihong Qiu, Yue Fang, Zhiyuan Wang, Jinyi Tang, Hongxin Ding, Xu Chu, Junfeng Zhao, and Yasha Wang. Hykge: A hypothesis knowledge graph enhanced framework for accurate and reliable medical Ilms responses, 2024.
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Appendix Table of Contents
More Motivation Details
18
A.1
Blockwise approximation neglecting inter-layer interactions
18
A.2 Relation Shapley-Bascl Methods
18
A.3 Relationship Between MODEL SHAPLEY and Quantization
18
A.4 Rationale for Using the Fisher Information Matrix Approximation
18
A.5 Applicability of Shapley Values Under -on-Staaionary Training
19
A.6 Potential for Model Editing
20
Notations
20
Proof of Closed-Form Shapley Value
20
C.I Pnth-Integral Derivation of the Loss Change for Single-Weight Removal
20
C.2 Path-Integral Derivation of the Loss Change for a Parameter Subset
22
C.3 Combine Marginal Utility Function
23
C.4 Taylor Expansion Remainder Bound for Shapley Value Approximation
24
C.5 Proof of Shapley Axiom Properties under Second-order Approximation
25
C.5.1
Null Player
25
C.5.2
Symmetry
25
C.5.3
Linearity
25
C.5.4
Efficiency (Approximate)
26
Fisher Information Matrix as an Approximation to the Hessian
26
E
Applications of MODEL SHAPLEY
27
E.1 Step Constructing the Probe Dataset
E,2
Step 2: Defining Attribution Granularity
28
E.3
Step 3: Shapley-Based Attribution of Parameter Importance
29
E,4
Step
Applications of Attributed Parameter Units
29
E.4.1
Direct Usage for Functional Probing and Activation Augmentation
29
E.4.2
Targeted Fine- Tinnng for Capability Enhancement.
30
E.4.3
Shapley-Guides Quantization
30
Theoretical Analysis: Why INT4 Quantization Fails
31
Algorithm and Time Complexity
33
H More Experiment Details
33
H.1
Datasets Statics
33
H.2 Baselines Details
34
H.3 Prompts used in MODEL. SHAPLEY
35
H.4 Implementation Details
36
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H.5
Deactivate Neurons' Case Study
36
H.6
Deactivation Ration Study
................................
36
H.7 Analysis of Compensation Effects in Parameter Units
................................
H.8 Model Compression Sensitivity Experiments
41
H.9 Attention Head and Layer-wise MODEL SHAPLEY Study
41
H.10
Rumtime-normalized comparison
42
H.11 Fine-tuning Efficiency and Variance
43
H.12 Hardware Scalability and General Applicability
43
I Broader Impacts
44
J Data Ethics Statement
44
17

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A More Motivation Details
A.1 Blockwise approximation neglecting inter-layer interactions
Blockwise approximation theoretically neglects inter-layer interactions, but this design choice is both necessary and well-justified.
1. Computational necessity: Full global Hessian computation requires O(M²) memory for M parameters. For a 7B model with M 7 10", this would require 196TB just to store the Hessian matrix, while our blockwise approach reduces this to manageable levels (=28GB for typical block sizes). Our blockwise approach reduces this ((Md) where i' the block size, making the method practically feasible while preserving 95%+ of the interaction information based on our empirical our layer-level analysis (Figure 1) shows meaningful differentiation of importance across layers, suggesting that the blockwise strategy captures sufficient granularity for practical applications while maintaining computational feasibility.
2. Architectural Justification: More importantly, the Transformer architecture is highly modular. Information flows sequentially through layers, and the most critical synergistic interactions occur within functional blocks (e.g., neurons within an FFN layer, or heads within an MHA module). While inter- block dependencies exist, their influence is secondary to the dense intra-block computations. Therefore, the blockwise approximation correctly prioritizes the most significant cooperative effects that our Shapley-bar based method aims to quantify. Our empirical results, showing strong performance in identifying critical neurons and layers, support this architectural prior.
A.2 Relation to Shapley-Basee -Based Methods
Prior work has applicd Shapley values to neural networks [20, 21], with development and evaluation pri- marily on smaller-scale computer vision architectures (c.g., ResNet-50), rather than large language models (LLMs) with billions parameters. The Monte Carlo and bandit-style estimators in these methods remain computationally expensive at LLM scale due to the combinatorial nature of Shapley computation, whose exact complexity is O(2M) for M players (parameters, channels, or modules).
The approach proposed here employs second-order approximation that reduces parameter-level Shapley attribution to a single backpropagation pass augmented with a Hessian-vector product (Section 4.1; Remark 4.9). This yields orders-of-mant orders-of-magnitude savings (up to ~ 10 x in our settings), rendering parameter-level Shapley scoring tractable for modern LLMs.
Given these scaling properties, including [20, 21 as baselines is neither computationally feasible at LLM scale nor methodologically comparable to the proposed formulation. The Related Work section will explicitly note this distinction in scope and computational regime.
A.3 Relationship Between MODEL SHAPLEY and Quantization
All neurons and layers containing weights-incllddng all transformer blocks and embedding layers-wee were quantized in the experiments. The proposed approach integrates scamlessly with the standard GPTQ quantization framework without modifying its overall structure or scope. Specifically, GPTQ performs layer-wise quantization and calibration as usual, while MODEL SHAPLEY introduces task-aware guidance within this process.
The key idea is to use MODEL SHAPLEY to compute correction factor for the diagonal elements of the Hessian matrix. This factor encodes the task-specific importance of parameters and used to adjust GPTQ's quantization loss such that weights critical to downstream performance are preserved with higher precision. The rest of the GPTQ procedure-inncuding block-wise decomposition, quantization step selection, and post-calibration--enna remainss unchanged. This integration enables efficient quantization that retains task-relevant information with minimal additional computational cost.
For implementation details, refer to Section D.4 D.4.3 and Algorithm in the main text.
A.4 Rationale for Using the Fisher Information Matrix Approximation
The use the Fisher Information Matrix (FIM) in this work represents a deliberate balance between theoretical fidelity and the scalability requirements of large-scale models. The FIM provides practical
18

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and empirically validated surrogate for the Hessian in regimes where the loss landscape is locally quadratic conditions typically satisfied near model convergence, where the proposed method is intended to operate (e.g., for post-hoc analysis, pruning, targeted fine-tuning). In such regions, the FIM effectively captures the relevant curvature information necessary for estimating meaningful Shapley values.
During periods of high gradient variability or under significant data distribution shifts, the fidelity of the FIM as a Hessian proxy may decrease. However, this limitation is not unique to the FIM: even the true Hessian becomes unstable under these non-stationary conditions, reducing the reliability of any recond-orde attribution method. To mitigate this effect, the proposed approach employs Monte Carlo incremental approximation with an exponentially weighted moving average (EWMA) (Section 4.2). which stabilizes gradient- braseett-basedd estimates. The EWMA smooths transient fluctuations in mini-batch gradients while preserving persistent, task-relevant curvature signals, thereby improving the robustness of Shapley value computation.
From computational standpoint, the Fisher approximation is critical for feasibility at LLM scale. Com- puting and storing the full Hessian requires (MM²) operations and memory for M parameters, which infeasible for modern language models. In contrast, the FIM, when combined with efficient Hes- sian-vector product techniques, reduces the cost to (MM) while retaining sufficient curvature informa- tion for reliable parameter- Shapley attribution. This approximation thus enables second-order in- terpretability and importance estimation at a scale previously impractical for billion- billion-parameter models.
A.5 Applicability of Shapley Values Under Non-Stationary Training
The Shapley value framework, as originally defined in cooperative game theory, assumes fixed utility function. Accordingly, the proposed MODEL SHAPLEY formulation is designed primarily for post-training analysis and intervention converged models, where the mapping from parameters to utility (e.g., task performance) is stable. This regime aligns with the method's 's intended use cases, such as post-hoe interpretability, pruning, and targeted fine-tuning.
Theoretical scope. A stable parameter-utility mapping is a prerequisite for meaningful Shapley value attribution. Therefore, MODEL SHAPLEY is not intended to track parameter importance dynamically during ongoing optimization, where the loss landscape and model representations evolve rapidly. The approach instead focuses analyzing trained or near-converged checkpoints, where second-order approximations like the Fisher Information Matrix provide reliable curvature information.
Dynamic behavior. In non-stationary settings-sude early-stage training or continual learn- ing- ing-the the utility function changes continuously, causing parameter importance estimates to fluctuate. Empirically, this manifests as the parameter compensation effect (Appendix F.7; Figure 3b): when high-Shapley neurons are deactivated, subsequent training causes other neurons to adapt and increase their Shapley values, compensating for the removed functionality, This adaptive response confirms that the identified neurons were indeed functionally critical, while also illustrating the dynamic redistribution of importance during training.
Practical implications. Despite theoretical limitations under non-stationarity, MODEL. SHAPLEY
provides substantial practical benefits across several domains:
Targeted fine-tuning of high-Shapley neurons achieves 73.89% accuracy on MMI.U, nearly matching full fine-tuning (73.56%) while updating only 10%
Knowledge injection:
of parameters.
Model compression:
Shapley-guided quantization surpasses GPTQ and OBD baselines across precision levels by preserving task-critical weights.
Interpretability: Task-specific neurons clearly identified and validated via systematic deactivation studies (Section 5.3).
Extending Shapley-based formulations to dynamic, non-stationary training remains an open direction for future research. Nevertheless, for a wide range of practical applications involving trained models--iclluding compression, interpretability, targeted adaptation, and knowledge editing- SHAPLEY offers an efficient and theoretically grounded framework that scales to modern large language models.
19

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A.6 Potential for Model Editing
The parameter- and module- attributions produced by MODEL SHAPLEY provide a principled foundation for targeted model editing. By quantifying each parameter's 's contribution to a given task or behavior through its Shapley value, the method enables controlled interventions such suppressing components associated with undesired behaviors or amplifying those that positively influence task performance.
Although the current work focuses on interpretability.. targeted fine-tuning, and model compression, the same attribution framework naturally extends direct model editing. In particular, parameters or modules with high positive or negative Shapley values can serve as intervention targets for behavior refinement, knowledge correction, or safety alignment. This capability highlights MODEL SHAPLEY 's potential as a unified tool for both model understanding and modification, offering promising direction for future research in scalable and interpretable model editing.
B
Notations
The notations used throughout this paper are summarized in Table 3.
Table 3: Notation summary for MODEL SHAPLEY
Notation
Definition
θ [0..... Aas
Full set of model parameters (players in the Shapley framework) The i-th parameter unit
0,
8g
Value of parameter 0; at iteration
C,
Standard basis vector with at position 0 elsewhere
M
Total number of model parameters Shapley value of parameter under utility
do(U)
Or
Parameter vector at iteration during training evaluation Parameter vector with 0g removed (set to zero)
O7-S
Parameter vector with subset S removed
C (( (())
Loss function evaluated on input-label pair (x.y)
Utility function defined as negative loss: U --=-C Shapley value of parameter A, under utility U Gradient of the loss w.r.t. parameter 0 at iteration
6:
9 OC
HT 30,30 30;
Hessian matrix entry at iteration
W
Path-based curvature weight between parameters i and j under subset S Empirical Fisher Information Matrix at step T
F(OT)
Per-step Shapley approximation using mini-bated estimates
6
Exponentially weighted moving average (EWMA) of Φ;; Smoothing factor used in EWMA updates Shapley-adjuste trade-out ratio in quantization Task-specific probe dataset for functional attribution
a
8
DJ
Ia Set of top-a% most important parameter units
C Proof of Closed-Form Shapley Value
C.1 Path-Integral Derivation of the Loss Change for Single- Weigltt Removal
We re-derive Theorem 4.2 without fixing the Hessian to be constant, keeping every step of the path integral explicit. Throughout this section the only assumption is that the loss L is twice continuously differentiable.
Let 0- = (O ...,, 0%,) be the current parameter vector and fix an index ie (I.....M.. }. Removing ("zeroing") 0 gives - - 0Te. Define the straight-line path that re-grows the weight,
Setup and notation.
θ 0,=07 - (1-t)0Ted te [0.1].
(10)
20

---

Hence (prunee
(pruned model) and O = (full model),
1. Expressing the loss difference path integral
=C(0)==((0r-)--(00)))
(()))tt
(11)
dt
2. Chain rule along the path
dt AC(0T)= -c(0))=fed ((O,) | (,( dt= 07
/ g: (B())ddt.
(12)
Integrating the chain-rule derivative over E [0, 1] and substituting ac( coordinate-wise loss decrement above.
8 e, yields the
86,
Pash expansion the gradient (no constant Hessian approximation) For every we have the exact identity
g ( L
g :=:)("")) (=,(()") 8 ((.)))
(-.))))))
(13)
000
Plugging into the former equation gives a double integral:
AC(O)) -0, Bi 0f J. H.,(O,dds. dt
T. - [-¹ / H.
4. Reordering the integrals
L. J Odsdl f(-101(((())))))) (1-=))))dss - +(7)) J (O.))d
(14)
Exchanging the integration order compresses the double integral into the (1 (1-s) kernel, and substituting this compact form yields the closed-form quadratic correction for AC in terms of the diagonal Hessian.
Introducing dimensionless curvature weight
Define
H J. (1 (1-s) (1.))
H H
(15)
The factor H.,, merely normalises the path average, that measures how the local curvature along the path deviates from its value at the full model:
area under (1 - H.....(OO..
(16)
area under (1-s)H;
With eq (17), equation (16) takes the compact weighted form AC(O)) -g +, - (0))
(17)
6. Consistency with the usual second-order Taylor term
If the removed weight is tiny, |0 0, the entire path + collapses O and continuity of the Hessian implies (e,())) - H Consequently w + and eq (19) reduces to the familiar quadratic update
AC =- 0, + H,)
(18)
Equations (19) supply the desired path-integral proof of Theorem 4.2, clarifying the role and interpretation of the curvature weight ((()
(1)
21

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C.2 Path-Integral Derivation of the Loss Change for a Parameter Subset
We give self-contained, line-by-line proof of the loss increment that occurs when a subset S S [I.... } of parameters is pruned ("left out"). The presentation keeps the Hessian fully path-dependent; the familiar constant-Hessian 2 formuum drops out as a limiting special case.
We denote the current iterate by θ (0T ,r.,, When pruning subset S of parameters, we obtain the pruned parameter vector ~-.-5 defined by removing parameters indexed by S from Or, i.e, -- = -" -I es. The direction of removal, representing the parameter changes due to pruning, is thus 50 :=:=0 A7.-5 0 = Eie ex. For brevity, we denote the loss at an arbitrary point θ by [()), and its gradient and Hessian by ([00)=ZZZ0 and : (0))==V(((0))) (Q)) respectively. At the current iterate, these are abbreviated as g" := g((") and H H(r)).
1. Path parameterisation
Choose the straight line that starts at the full model and ends at the pruned model,
g -
sE [0,1),
(19)
so O = -" and 0 -
2. Expressing the loss difference path integral
C((r)= [ d ds
(20)
ds
/ T(,
O, / g(O,) J0ds
(21)
Σ0J 07 / g: (O.)ss IES
(22)
The loss difference is written path integral and, using the chain rule with the constant velocity 80 -0", reduces to a weighted path-average of the removed gradient components. This compact form directly quantifies each deleted parameter's marginal contribution to AL(S).
3. Expanding the path-wise gradient
+ S H(,))00dt
g(O.)=
(23)
(0.0)= -0; [" (.,,d,,
(24)
jES
00 S. J. ['He, 1JES
AC Eg
(25)
iES
Integrating the Hessian along the optimisation path yields the path-dependent gradient (first two lines), and substituting this result into the loss variation condenses the double integral, showing how curvature accumulates through the H.j "path-area" term.
By exchanging the order of integration, the nested double integral collapses the single-integral kemel (1 (1-1), whose substitution into the path-dependent loss expansion yields the quadratic form below.
[h' J Odids= - [6','' (-)dsil - S/ (1-1)(-)d
(26)
Σ 07 [ (1- (0))dtt
AC(S)= iES
Σg g. 0,
(27)
JESS
22

---

4. Defining the w!)
Defining the dimensionless weight wij as the path-averaged curvature normalised by the terminal Hessian allows eq. (29) to be recast in compact quadratic form where all trajectory dependence is absorbed into w (5)
(' (1-t)
(5)
(28)
H
AL(S)
Eg 0
Σ w 19) H H700 1,ES
(29)
iES
Why this definition?
(i) The factor (1 -t) is the exact kernel that emerges when the double integral in eg (28) is reordered;
it ensures the 2 symmetry familiar from second-order Taylor series.
(ii) Dividing by H normalises the path-average so that w (S) is pure number measuring how much
the curvature between coordinates (i.j) varies relative to its value O", particular, w (S) if Hij is constant along the path.
ij
5. Limiting behaviour as the removed weights vanish
Suppose ||0s| Then | 0, so the entire path collapses into an arbitrarily small neighbourhood of @T By continuity of the Hessian, Hiy (O,) + Hy uniformly, and
W (S) 1 (+10ll)
(30)
Consequently eq (29) reduces smoothly to the usual second-order Taylor estimate
AC(S)
> o. + Σ HH,00
(31)
SES
iJES
This completes the explicit path-integral proof, while clarifying the origin and interpretation of the weighting coefficients w! (S)
C.3 Combine Marginal Utility Function
In this appendix, we provide the detailed derivation of the closed-form Shapley value approximation given equation 7, now incorporating path-based curvature weights.
Setup. Recall that the Shapley value for a parameter 0; under utility function U(0) ((e) is defined as:
M
M
()) = M
Σ U(SU()))-U(S),
(32)
SCOVMO k-|
where M = O is the total number of parameters.
Marginal contribution approximation. Based the path- aath-integrated second-order approxiatiion,,, the marginal utility of adding parameter 0; to subset S is:
w., (8) (0)) O U(SU())-U(S)~-E)
H
Oz
(33)
JES
where w!O [0, 1] denotes curvature-weighted average over the interpolation path involving
parameters i and
23

---

Averaging over subsets.
Substituting equation 33 into equation 32, we have:
M v.(U) = M
M- M-1
Σ
o
0;
W
H
0g
Σ
w
k-1
2
k=1
Sc 1
JES
M
g. 0:
Wi
(8)
8
H.
1
0
M
Σ
Σ
(1) wij
H
(34)
M
SCO
|5|=k-1
Simplitying the interaction term. contain.j is (V-2) S0:
Fixing index fi, the number of subsets S of size k 1 that
M - k-
Σ (ijjES
HG = Σ
wij
H
(35)
|
Substituting back:
M
(u(U) =
(0B) -B. 0;
1 ( O
i8)
H
(36)
2
M
k-
Evaluating the combinatorial ratio. As before,
M-2
k-2
k-1
k-1
M
(37)
M 1--
M-1
M-1 2
M
Final closed-form expression.
Substituting the result:
-0)=---0---
(1 1000
H
n Σ j
H7
(38)
This generalizes the original expression in equation 7 by introducing path- weighted curvature terms derived from integrating along removal trajectories.
C.4 Taylor Expansion Remainder Bound for Shapley Value Approximation
Our Shapley value approximation in $4.1 is derived from path integral formulation, which computes the expected marginal contribution of each parameter along a continuous interpolation path. When 0. - 0. this integral can be well-approximated by the second-order Taylor expansion of the utility function around the current parameter vector @7. This appendix quantifies the approximation error by deriving uniform bound on the remainder term in the Taylor expansion.
Second-order expansion with remainder. For any parameter vector 0 in a neighborhood of a reference point d*
[()) - 1000==0000=V((000000)=-----00000 (F(00-0)))))()))000))))((()))))0
(39)
where R3 collects all third- and higher-order terms. By Taylor's 's theorem there exists E (0,1) such that
[R((0,0)) S 1 8 V L -(00-0)))l
(40)
Propagation to the Shapley estimate. Denote by of the true Shapley value and by & the second-order approximation given in Eq. (7). For each coalition sco)(0,) we define the Taylor error
e (S) = R3 (0t--uuf) suff) (
(41)
Using the linearity of the Shapley aggregator we have
MM-1)
S
Σ
ws (S));
where ws =
(42)
M
[S]
Scel(1)
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---

Uniform bound under smoothness and small perturbation. Assume the following: (1) The third derivative the loss is uniformly bounded within ball of radius around e i..., | (0)) = for all ||0 - < (2) Each single-paramen the "set-to-7ero" removal rule, with max,10ţ] Under these assumptions, Eq. (40) implies that e((S) - - | for every subset S, and the summation in Eq. (42) becomes:
removal satisfies || - - -> S.. This holds under
Φ di
(43)
The cubic dependence on implies that the approximation error rapidly vanishes as training converges and parameter updates shrink. The constant reflects the worst-case third-order curvature in the local neighborhood. In practice, we observe that this bound is small for modern architectures, which supports the empirical accuracy of our closed-fone estimate.
C.5 Proof of Shapley Axiom Properties under Second-order Approximation
We provide formal justifications that the closed-form parameter-level Shapley expression equation approximately satisfies the four core Shapley axioms under second-order Taylor approximation.
C.5.1
Null Player
Claim: If parameter 0; has no effect on any subset of parameters, then Φb;
Proof: From equation 7, we have:
6 6---g,0- -
0g
H7
(44)
for all j. and 0, = or finite, then all terms vanish and d; =0. This corresponds exactly to definition a null player in Shapley theory.
[ 00 and HJ, for
C.5.2 Symmetry
Claim: If parameters 0, and 0, contribute identically the loss and interact identically with all other parameters, then Φz
Proof: Assume the following: 6; = 8j. g. = gg. H
H 3j
H
H
for all k + i.j.
H. H (by symmetry of Hessian).
Then by substituting into equation we have:
b -== 6=
(0
07 Σ
w
(S) H OT.
(45)
KES
Φ - -
(
00
07 Σ
w (S) jk:
H AT.
(46)
kES
Since all terms are identical under the above symmetry assumptions, Φ; -==00 Φj.
C.5.3 Linearity
Claim: If utility functions are linearly combined V==aU++3335 then the Shapley values satisfy
),(U)=a =ao((U))
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Proof: Let the losses be L1.L2, and define:
L C U=-6 L,
= aa + (2)
(1)
(47)
H = a
(1) +
Substituting these expressions into the closed-form formula for @; gives
Φ a +30
(48)
thus confirming exact linearity of the contribution with respect the loss weights.
C.5.4 Efficiency (Approximate)
Claim. The total Shapley value approximates the total utility:
Σ b Re U
(49)
Proof.
g: :g -38
(50)
H ow (1)
H + Bw
(1)
H (5.2 BBw
2 2
(51)
Hy:=aw,, (S,1) H 11)
H
(1+j)-
(52)
These weighted quantities merge the task-specific gradients and curvatures into a single of effeecttive statistics.
--- H..
(53)
jff
With the aggregates in hand, each Shapley value becomes a concise quadratic contribution to loss reduction.
3+1 HO -0=
55:0+ 5 O,H,
(54)
A second-order expanssin around = 0 an identical quadratic form in the same aggregated terms.
8.0,-
(55)
1=1
2.J
Summing the individual payoffs therefore reproduces the negative of the Taylor approximation, up to cubic errors.
M
Σ
di ~ U()))
(56)
Since --C, this matches the utility gain and thus verifies Shapley efficiency to second order.
D Fisher Information Matrix as an Approximation to the Hessian
In this section, we explain why the Fisher Information Matrix (FIM) is often used as tractable approximation the Hessian the loss function, particularly in settings where the loss arises from the negative log- likeeihoooo
26

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Log-likelihood and the Score Function Consider a probabilistic model p(æ: 0) (i.e. LLMs) with parameter vector 0 E TRd, and let a denote a single observed data point. The log-likelihod function is given by logp(æ 0), and its gradient with respect is known as the score function: s(0) := Valogp(< 0). The score function measures the sensitivity of the log-likelihood to changes in the parameters. Under standard regularity conditions (e.g., differentiabiilty, integrabilityy, and dominated convergence), the expectation of the score vanishes: |
Definition and Interpretation of the Fisher Matrix The Fisher Information Matrix is defined as the covariance (second moment) of the score function: F(0)= =====((=))) [+(0)s(0) Intuitively, quantifies how much information the data carries about the parameters: small changes in 0 cause large variations in the likelihood, then the entries of F(0) will be large.
Connection to the Hessian of the Log-Likelihood To relate this the Hessian, consider the second derivative (Hessian) of the log-likelihood: Hoggu = Volog((> 0). Using the identity
Vologp(> 0) = v)p(z(0)
p(2 {0)) and the fact that the total probability integrates to one (i.e.,
(o(==0)dz= 1), obtain:
[ (0)
Voogp(<
(57)
This is well-known result statistics:
F(0) (10)==Ea-pleee (0 Voggp(z (x]0)]
(58)
That is, the Fisher Information Matrix equals the negative expected Hessian of the log-likelihood.
From Log-Likelihood to Loss Function In machine learning and statistical estimation, is common to define the loss function as the negative log-likelihood: [()): := llopp 0), whose Hessian becomes
Hc ==(0)= V
-V)o))))--loop(aa llogp(<
(59)
Taking expectations yields:
Exme(=10) [2]
[ =F(O)
(60)
demonstrating that the Fisher matrix equals the expected curvature of the loss landscape.
Empirical Fisher Matrix and Practical Considerations practice, the true data distribution p(x: 0) is unknown, and we instead work with a finite dataset { The empirical Fisher matrix is computed as:
N v (V)](ll(((((VVVV)(/
F(O)=
(61)
This approximation has several advantages: it avoids computing second derivatives, is simple to imple- ment, and compatible with stochastic optimization techniques such as mini-batch gradient descent.
E
Applications of MODEL SHAPLEY
Leveraging the estimated Shapley values (o.). MODEL SHAPLEY facilitates a variety of downstream applications centered functional attribution at the parameter level. These applications span from interpreting the internal mechanisms of the model, to actively manipulating its behavior during inference, fine-tuning critical components, and guiding parameter-efffiient model quantization.
In the following, we illustrate how MODEL SHAPLEY enables parameter-level functional analysis through four key stages: 1 constructing a task-specific dataset for attribution, @ determining the granularity of analysis (e.g., neuron, layer, or attention head), @ applying Shapley-based attribution to identify important units, and @ utilizing the results in downstream interventions.
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E.1 Step 1: Constructing the Probe Dataset
To attribute functionality at the parameter level, we first construct a probe dataset D== ((;;,)) fi=1 tailored to a specific objective F (e.g., robustness, factual recall, reasoning). Unlike training or evaluation datasets, this probe dataset is explicitly designed to activate the model's behavior with respect to the target capability, thereby enabling meaningful attribution.
The structure of D depends on the task. For instance, in retrieval-an -augeeedd generation (RAG), each input 2; consists of a { query, retrieved document] pair, and the output is the corresponding response generated by the model. In code generation tasks, I; may be a natural language prompt or partial pro- gram, with yi being the expected code completion. For robustness assessment, I; could be adversarialyy perturbed inputs, while y; reflects the desired stable output under perturbation. Across these diverse settings, the probe dataset serves as lens through which task-specific model behavior is revealed.
By computing Shapley values over Dr, we estimate each parameter's marginal contribution the model's ss ability to satisfy objective F. This attribution highlights the most functionally sensitive units those whose perturbation or removal would most affect the model's performance on the given task.
E.2 Step 2: Defining Attribution Granularity
To enable interpretable functional attribution, we first define the level of granularity at which model parameters are examined. Transformer architectures are inherently modular, composed of repeated layers that include Multi-Head Attention (MHA) mechanisms and This modularity naturally gives rise to three commonly used units of attribution: neurons, attention heads, and layers. Each level captures different aspects of model behavior and presents trade-offs between interpretability, attribution resolution, and practical utility for intervention.
Networks (FFNs).
Neuron-Level Attribution. A neuron refers to an individual hidden unit within the FFN of a given Transformer layer. Specifically, the j-th neuron in the i-th FFN layer computes:
neuron., -acton ( wi wi+bil,).
(62)
where h' E Rd is the output representation from the MHA submodule in layer i, Wi gdx4d and bi E Rd are the weight and bias parameters of the FFN input projection, and act_ fn(.) is typically a GELU or SiLU nonlinearity. A neuron is considered active if its post-activation value is nonzero. Attribution this granularity enables highly localized functional analysis, though it may be sensitive to activation noise and task-specific variance.
Each MHA module consists of multiple heads, each responsible for computing independent attention pattern. The output of the h-th head in the i-th layer is given by:
Attention Head-Level Attribution.
head = Attn (h 1 wi
9.h- H
Wi K.A.
.
WW: w..)),
(63)
where h'-l EIR. denotes the representation of single token from the previous layer, H e Lxd denotes the sequence of hidden states, and W 9.h W k. W v.. E dddx//H are the learned projection matrices for queries, keys, and values, respectively. Attention heads often specialize in capturing distinct structural or linguistic features (e.g., syntax, coreference, position), making this level of attribution particularly effective for interpreting distributed attention behavior and its contribution to model predictions.
Layer-Level Attribution. Each Transformer layer comprises the full computation block including all attention heads and the FFN. The forward pass through layer can be expressed as:
h' Concat Concat(head
W +b h'= fn (h Wi + bb/) +b
(64)
where Wi E ddd and by E r are the output projection weights and biases for the attention module, and W i, W are the FFN projection matrices with corresponding biases bj bj. Attribution at the layer level provides a coarser but more robust perspective on where critical computations occur, and is often suitable for interventions such as pruning, freezing, targeted parameterization.
28

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E.3 Step 3: Shapley-Based Attribution of Parameter Importance
To quantify the task-specific importance of individual parameter units, we adopt Shapley value framework grounded in cooperative game theory. Shapley values offer principled approach to measuring the marginal utility of each parameter, averaged over all possible subsets of cooperating units. This provides a theoretically justified metric for attributing functional relevance.
However, exact Shapley computation is intractable in large-scale neural networks due the exponential number of subsets. To address this, we derive a closed-form approximation based on a second-order Taylor expansion of the loss function (()). Under the common assumption that removing parameter can be approximated by zeroing it out, the Shapley value Φ; for parameter 0, is given by:
(i)
(S) H7
g
(65)
2
Individual Importance
(2) Cooperative Internctions
where gi - 30, and H.y denotes entries of the Hessian. The first term captures the individual salience of the parameter, while the remaining terms account for higher-order cooperative interactions.
0c
To scale this computation in practice, we employ several approximations:
Gradients and Hessian- vector products are estimated on mini-batee mai-batches, enabling per-step online approximation of Φ;.
We stabilize noisy estimates with an exponential moving average (EWMA): =1--))0 +a0l
The Hessian is approximated using the empirical Fisher Information Matrix (FIM): F(+) := E(a,y) stochastic training dynamics.
which is more stable and tractable under
To reduce memory and computational overhead, we apply a blockwise approximation, aggregating importance scores within structured units (e.g., neurons in layer or heads in MHA block) using local statistics.
After computing Φ; for all parameter units, we identify the top-a% most important units by ranking their Shapley values. For searser-grained structures such as attention heads or entire layers, we perform mean pooling over neuron-level scores within each unit:
Σ 63
(66)
heau
Player
1 layer
head
0y
0,
This pooling ensures that higher-level Shapley attribution remains grounded in fine-grained parameter sensitivity, while enabling consistent scoring across different levels of model granularity. grained architectural intervention.
E.4
Step 4: Applications of Attributed Parameter Units
The estimated Shapley scores } identify parameter units that are functionally critical for a given task. These scores support a variety of downstream applications aimed at interpreting model behavior, reinforcing specific capabilities, or guiding efficient intervention. We describe two representative cases below.
E.4.1 Direct Usage for Functional Probing and Activation Augmentation
Without any further training, the activations of -ngptimpoetnnce parameter units can be used to probe internal model representations or to modulate inference behavior. Let I, = {il ( ranks in top-a% denote the set of top-scoring parameters.
At inference time, we extract the activations (x) of the important units and use them as feature inputs probing model:
Fprote (x)
[a: [==()))
(67)
where Fpmshe is downstream classifier trained to detect specific knowledge, behaviors, or internal states. This enables lightweight interpretability tools for factual recall, knowledge localization, fairness analysis, safety probing.
29

---

Equation equation 71 follows directly from the efficiency axiom applied to the second-order expansion Alternatiely,. we can modulate the model's output by augmenting the activation of important units:
enhanced (1)
(x)==()))))
(68)
where A is tunable scalar offset. This form of activation intervention parallels the concept of functional or concept vectors [41, 39), and can be used to amplify model behaviors associated with certain capabilities.
E.4.2 Targeted Fine- Fine-Tuning for Capability Enhancement
Rather than updating the entire model, we can focus learning on the most functionally relevant parameter units identified via {di}. During fine-tuning, we restrict gradient updates to only the top-ranked parameters ETa, while freezing the rest:
0, -- -. Vo,C, for foriEIa.
(69)
0 j
(70)
where is the learning rate. This strategy allows for parameter-effiiient fine-tuning by concentrating up- dates on task-relevant components while reducing overfitting and preserving prior capabilities. Such tar- geted adaptation is particularly beneficial in continual learning, domain adaptation, or few-shot
E.4.3 Shapley-Guided Quantization
OBD evaluates importance through the first-order product [9.0%]. OBS and GPTQ extend this idea by incorporating non- non-diagonal Hessian entries that parameter interactions are partially compensated.
In network that has already converged, the first-order gradient term ac/00, is typically close to zero (see Eq. (7)), so the loss change (C is dominated by the second-order component
(2) p - H
1 V H.
The cross term Ejes H,y03 is already handled by GPTQ's inverse update. What remains unaccounted for is the task-specifer magnitude of the diagonal piece 1 H We therefore correct H; itself rather than modifying the update order.
Let 0.2² denote the empirical second-order Shapley estimate obtained with the Fisher-block approximation of Section 4.3. Define the per-parameter correction factor and modified diagonal entry
(2)
H. < w. 3H, + (1-8)H
(71)
H +E
where e >> 0 ensures numerical stability and B E (0.1] controls the strength of the Shapley injection. If wf the diagonal term is boosted; if w; < it is attenuated.
Replacing H by H. leaves the Woodbury-stylee inverse update in Algorithm E.1 unchanged, so the overall (dk)) complexity is identical to vanilla GPTQ. The convergence proof in Appendix B 8.4 remains valid after substituting H H.
in Theorem 4.6. By isolating the quadratic diagonal term inside (22) we obtain scale-invariant ratio that indicates whether the naive quadratic estimate under- or over-explains the cooperative loss increase attributed to weight i.
OBD relies on gradients, OBS and GPTQ already harness off-diagonal curvature, and our stapley-guided diagonal correction completes the picture by aligning each diagonal Hessian entry with its cooperative, task-specific contribution. The resulting H integrates seamlessly into GPTQ, yielding stronger accuracy-compression trade-offs without sacrificing speed.
The overall procedure is outlined in Algorithm 1.
30

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Algorithm 1 Shapley-Guided Quantization with Corrected Hessian
Require: Weight matrix W, inverse Hessian H ,Shapley scores {or
block size B, reweight
parameter [011]
Ensure: Quantized weight matrix Q
1: Initialize Q E +
2: f= N steps of B do
Iterate over blocks
3:
for j=
4:
8, k 2 H. Hj
> Estimate standard Hessian contribution
5:
W - (2) (sy +E)
Compute Shapley- correction weight
6:
Hjj - - 1 (1-3) - H1i
>> Rescale diagonal Hessian entry Quantize column
7:
Q. quant (.))
8:
E.j- - (W.,) :J Q.,)/HH, M3j
D Estimate quantization error
9:
Witti++++ Witli-B E ji+la4n
H
B Apply error compensation to future
columns
10: end for
W. W + W. 514 B:+BB
E. B.44B44BB
Final block correction
end for
13: return Q
F Theoretical Analysis: Why INT4 Quantization Fails
Setup and notation. Let w E Rd denote the parameters of a trained model, and L(w) its task loss. We write H V2L(w) for the local curvature (or empirical-Fiste surrogate). For a uniform signed quantizer with bit width b and scale > O.
B a 2b
(72)
Q (w,) (cip(roond(uy/b))-BBB))
and define the quantized weights 93 = Q!! (w;) with quantization error
A
w ワj.
(73)
When there is clipping, | <//2; when wy Bs (saturation). |A, 2 Myy -Bs. We focus b-4 (INT4), where B-7 and the representable range is 7s.
Second-order post-quantization correction. Following the quadratic approximation underlying our Shapley-based formulation, we define an effective diagonal curvature term:
Hイ = (Bw, + - B) H3j
(74)
and the corresponding elementwise correction
E.j W/ - Qj
(75)
Hj
one-step second-orden refinement updates the weights as
w =
(76)
For theoretical clarity, we fix 8=1 (pure Stapley-weightee curvature), so H, = H,j. Substituting equation 75 into equation 76 and retaining the dominant scaling in yields
A
w
(77)
H
where we used that (i) for non-saturated weights A O(s), and (ii) i remains comparable in scale to H under a block-diagonal approximation.
For the symmetric scheme in equation 72, the maximum is B 2 1, yielding repeeeaaeeeeeeettaalll magnitude of 7s. Asymmetric variants ([-8,,7s) differ only by constants.
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Scaling implication.
From equation 77, the correction magnitude decays quadratically with the scale:
8w _ w w
2
Hence larger s suppresses curvature- based compensation, while smaller s amplifies it and risks numerical instability. Empirically, Shapley importance values span heavy-tailed range of roughly [10-9,10 100); mapping such a distribution into few quantization levels magnifies both instability modes discussed below.
The INT4 dilemma. For b=4, only 2 = 16 quantization codes are available, corresponding to 2B+1= = 15 symmetric levels (including zero). To avoid saturating large weights, the scale must increase until max, [ty| S Bs==s. 7s. This makes || = Oss) large, while the compensation term in equation 77 shrinks as 1/s²:
|o
~
(HH,,
leading to systematic under-compensatton and residual enror. Conversely, choosing smaller to reduce rounding error causes widespread clipping (wwy] > 7s), making biased truncation term that invalidates the local quadratic approximation. At the same time, the factor 1/(Hs²) can grow excessively, introducing severe numerical instability.
Interaction with Shapley-driven compression. Our compression pipeline leverages Shapley values to prioritize parameter sensitivity. In practice, these scores span orders of magniuude (10-3 110)). INT4's 16 discrete codes cannot represent such variation faithfully: after normalization, many parameters with different importances collapse into identical quantization levels. This dynamic-range collapse forces either (i) a large to accommodate outliers, or (ii) a small s that incurs severe clipping-thus reproducing the two failure modes described above.
Comparison with INT8. At b=, 256 available codes allow choosing a smaller s that avoids both saturation and instability. The quantization error A remains bounded, the quadratic approximation holds, and the compensation in equation 77 remains accurate and stable. Hence, the very same second-order mechanism that performs well at INT8 becomes ill-conditioned under INT4.
Conclusion. Equations equation 74-equation 77 reveal that INT4 operates in a narrow and fragile regime:
Large s: quantization error grows, while curvature-based compensation vanishes (ox 1/5²)
under-compensation:
Small s: clipping dominates and 1/(H²²) explodes = instability and loss of quadratic validity.
When coupled with heavy-tailed Shapley importance scores, the 16-level INT4 codebook cannot preserve task-relevant sensitivity. Robust INT4 quantization therefore requires additional structure such as per-channel scaling, outlier preservation, nonlinear remapping importance, or damped curvature inversion. Absent these, accuracy degradation is unavoidable.
Discussion: Toward Robust INT4 Quantization. The analysis above suggests that the core limitation of INT4 lies in the interaction between coarse quantization steps and the diagonal Hessian approximation. Because H is estimated per-parameter and can vary by several orders of magnitude, its inversion in equation 77 becomes ill-conditioned when combined with large scale s. Future work could therefore aim to stabilize or regularize the diagonal curvature before applying second-order corrections. Promising directions include: (i) smoothing H within parameter groups channels avoid extreme ratios; (ii) damping or clipping the inverse Hessian term 1/H,, limit numerical explosion; (iii) learning a nonlinear remapping of H that preserves curvature ordering while compressing its dynamic range; and (iv) integrating adaptive scaling tied to local Hessian statistics, so that quantization and curvature correction co-adapt. Such curvature-aware stabilization may enable INT4 quantization to retain second-order fidelity without resorting to mixed- mixed-precision precision designs.
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Algorithm 2 Parameter-wise Shapley Value Estimation with Gradient Similarity
Require: Model parameters 6 11,...0044. loss function 4, mini-batch size B, smoothing coefficient ox, total steps 7
Ensure: EWMA-based Shapley value estimates (oī K
1: Initialize Ø 0, for all i 1.....M
2: for for==10 to T do
3: Sample mini-batch 3 (xx
4: Compute gradient: g" +- V
5:
Compute curvature approximation: H t xg"
6:
9"-0"
7:
0 (H' x0")
8:
Φ² 2 (1)
+ (2)
9:
Ø²
0 )0" t0.0"
10:
end for
return
M (0 1=1
G Algorithm and Time Complexity
This algorithm computes parameter-wise Shapley values using gradients and curvature approximations.
Algorithm 2.
Line I initializes the exponentially weighted moving average (EWMA) estimate for each parameter. Lines 2-5 iterate through training steps, where at each iteration, the algorithm samples a mini-batch and computes the parameter gradient g" and approximate curvature matrix H" (using Fisher approximation). Lines 6-9 compute the per-parameter Shapley contributions: line 6 calculates the individual importance via first-order terms; lines 7 account for self-curvature and parameter interaction terms using second-orde -order curvature information. The total Shapley score is then aggregated in line and line 9 updates the running average using the EWMA rule. Finally, line 11 returns the smoothed Shapley scores after T steps.
Time and Space Complexity. The overall computational cost per step dominated by the gradient calculation and the curvature approximation. Assuming the size of parameters in the model is M and the mini-batch size is B, the gradient step costs O(B M), while the full curvature matrix requires up to (M²2) operations. If blockwise Fisher approximation is used with block size d, the cost can be reduced to (M -d). The total space complexity is also (MM²) ) in the worst case, or O(M dd) when using blockwise regularization. This makes the algorithm scalable in practice for modern networks with structured parameter grouping.
H More Experiment Details
H.1 Datasets Statics
We employ four benchmark datasets across natural language understanding and computer vision domains to evaluate the generality and robustness of our method. The detailed statistics and task settings are summarized below.
(L) GSM8K (Grade School Math 8K) [62] is dataset of 8.5K high quality linguistically diverse grade school math word problems. This dataset is usually used to evaluate the perofimance of language models on basic mathematical problems that require multi-step reasoning.
(2) MMLU (Massive Multitask Language Understanding) [63] is benchmark designed to evaluate models' multitask language understanding across 57 tasks, covering subjects such as humanities, STEM, social sciences, and more. Each task consists of multiple-choice questions, typically with four options. The dataset is split into a development and test set, with no training set provided. This benchmark is primarily used for evaluating zero-shot or few-shot performance in language models, focusing on factual reasoning and knowledge retention.
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(3) CIFAR- 100 [64] is widely used image classification dataset consisting of 60,000 x22 color images 100 classes, with 600 images per class. Among the 600 images each class, there are 500 training images and 100 testing images. It is standard benchmark for evaluating image classification performance, particularly in the context of resource-effiient training, model compression, and continual learning.
(4) ImageNet i65] a large-scale image classification dataset with over 1.2 million training images and 50,000 validation images across 1,000 object categories. It is one of the most prominent benchmarks in computer vision, used to assess the representation power, generalization, and transferability of visual models. The dataset serves as foundation for pretraining and evaluating deep convolutional and transformer-based vision models. We use the release version of ImageNet in 2012 as the dataset in our experiments.
H.2 Baselines Details
In this section, introduce the baselines we used for quantification of important parameters in NLP and CV tasks, and the baselines that are widely used in LLM quantization.
For NLP and CV tasks, we systematically compare our method with the following baseline strategies to quantify the contribution of each neuron the model.
Random Masking This baseline assigns uniform random importance scores to neurons without considering their structural or functional relevance. Specifically, each neuron n, in the network assigned score x~1((0,1) from uniform distribution. This strategy establishes non-informative reference to other quantification methods.
Gradient Magnitude [43, This baseline prioritizes neurons based the magnitude of their gradient values |g? computed during backpropagation. Specifically, given a parameter 0g in
the training step t, its importance score is defined as the absolute gradient s, 207
ac
where
denotes the loss function. This strategy assumes that parameters with larger gradients contribute more significantly to reducing the loss and thus deserve higher retention priority during pruning or masking.
Gradient Trace [40, 35, 49, 43] This method computes importance scores by the product of parameter values and their corresponding gradients, i.e., 8, [0 gg. The intuition is to capture both the sensitivity (g.) and the scale (BT) of each parameter, effectively identifying neurons whose activation patterns are both volatile and impactful. By incorporating parameter magnitudes, this magnitude parameters, providing more
approach mitigates the bias towards large gradients balanced measure of importance.
For quantization tasks, we compare our method with GPTQ [4] and OBD [38]. The details of these two methods are shown below.
GPTQ [4] This method represents a state-of-the-art post-training quantization (PTQ) technique designed for large language models. GPTQ employs greedy Hessian- based algorithm quantize each parameter group independently. Specifically, minimizes the quantization error ||0, 0 under the constraint that 0; lies within the specified quantization level (e.g., 4-bit or 8-bit). The optimization is formulated as:
0; = arg min |0,- 0. +A-tr(0 H,,,))
(78)
0.c0
where Q denotes the discrete quantization space. H, is the approximate Hessian matrix capturing the sensitivity of the output to each parameter, and is regularization parameter. This group-wise quantization approach efficiently balances accuracy and computational overhead.
OBD [38] Optimal Brain Damage (OBD) is gradient-barse pruning and quantization framework that leverages the second-order Taylor expansion of the loss function. Given a parameter 0,y at the i-th layer and j-th position, OBD computes the saliency score as:
S(O.g) 82 02C
(79)
During quantization, parameters with lower saliency scores are prioritized for quantization or removal, they contribute less to the overall loss. OBD typically uses Hessian diagonal
34

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approximation to efficiently estimate the second-order derivatives, enabling scalable application to large models. Unlike GPTQ, OBD integrates pruning and quantization decisions, making it suitable for joint model compression tasks.
We evaluate different quantization settings commonly used in model compression, including INT4, INT8, and FP8, as detailed below:
INT4/ W4A16 quantization uses 4-bit integers represent model weights. In the W4A16 setting, weights (W) are quantized to bits, while activations (A) remain in 16-bit floating point (FP16). This configuration achieves significant model size and memory bandwidth reduction while maintaining inference accuracy due to high-precisinn activations.
INT8/ W8A8 quantization is widely adopted precision format where both weights and activations are quantized 8-bit integers. The W8A8 configuration balances compression and performance, offering compatibility with many hardware accelerators and maintaining wear-floating-pon accuracy in many practical applications.
FP8 a low- wrwpprecisonn floating-pon point format (e.g., E4M3 or E5M2) gaining traction for training and inference. While strictly integer quantization, FP8 enables dynamic range advantages over INT formats. It can be used to represent either weights or activations or both, depending on the hardware support. Some hybrid schemes combine INT8 for weights with FPS for activations vice versa, leveraging the benefits of both representations.
H.3 Prompts used in MODEL SHAPLEY
In this section, we provide a detailed display of all the prompts used.
Prompt for MMLU Dataset The following prompt is leveraged to instruct LLM to answer MMLU dataset questions.
Prompt
The following is multiple choice question. Think step by step and then finish your answer with "the answer is (X)" where X is the correct letter choice.
[Question] (Question q}
A. [Content of {
B. Conteent of B}
C. [Content_og C)
D. (Content of D)
Prompt for GSM8K Dataset The following chain-of-thought prompt is used to generate step-by-step answers [66] to instruct LLM to answer GSM8K dataset questions.
Prompt 2
(Question 4) Let's think step by step and output the final answer after "####".
Prompt for Private Company Rules Dataset The following prompt is used to generate answers by selecting the most relevant knowledge base.
Prompt 3
(Question 4 You are a domain-specifte large language model integrated with multiple knowledge bases. The list of knowledge bases is as follows: (database_liss
# Please analyze the question "(query)" }" and determine which knowledge base it most likely belongs to. none of them are appropriate, output "others". outpur"others"".
By using these prompts above to guide the L.LM to output the final answer in specified format, we can then extract the model's output through regular expression matching and conduct evaluations.
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H.4 Implementation Details
We employ three different parameter-level language model architectures as our backbones, including
Qwen2.5-3B Breen..5-3B-nnttruee ----nstruct, Qwen2.5-7B 2---s---r-nnsttrctt [57], and LLaMA-3.2 -.------e--B-nsstrrct [32]. We use Py Torch
2.6.0 library to implement all the algorithms based on the open-source HuggingFace transformers [67] codebase. All NLP tasks and CV tasks are tested on an Ubuntu server equipped with 8 NVIDIA A A100 GPUs with 80GB memory, and quantization tasks are tested on Ubuntu server equipped with NVIDIA 4090 GPUs with 48GB memory. During the deactivation process, we only care about MLP layers in the transformer models. And we deal with the weight and bias of neuron at the same time.
For the training experiments, The activative ratio is set to 0.1 for all tasks. We conduct NLP tasks with learning rate of 3e-5, a max gradient norm of 1.0, a warmup ratio of le-3, batch size of 16 for GSM8K dataset and 8 for MMLU dataset, gradient accumulation steps of 4, a cutoff length of 1024 for GSMSK dataset and 2048 for MMLU dataset, max respoonse length of 1024, and epochs. And CV tasks are conducted with a learning rate of le-5, weight decay of le-4, a batch size of 512, max epochs of 50 and use early stop strategy with patience of epochs.
For the inference experiments, we set activative ratio to 0.95 for all NL.P tasks, and we set activative ratio to 0.7 for CV tasks. The batch size setting is the same the train experiments.
For the quantization experiments, we set the calibration sample number to 512, and max input sequence length to 2048, All quantization experiments are conducted on GSM8K dataset.
H.S Deactivate Neurons' Case Study
In this section, we present a case study to visually demonstrate the effectiveness of MODEL SHAPLEY. As shown in Table 6, Table 7, Table 4, Table 5, we analyze the Qwen 2.5 7B-Instruct model on the MMLU and GSM8K datasets, evaluating the impact model outputs after deactivating the bottom-K K neurons (5%, 10%, and 30%) identified based on their MODEL. SHAPLEY values. These key neurons correspond functional units associated with knowledge retrieval and mathematical reasoning tasks.
Under normal conditions, the model demonstrates strong performance both factual response generation and mathematical reasoning. When 5% of the key neurons are deactivated, the model still produces reasonable outputs. However, upon deactivating 10% of the key neurons, we observe noticeable decline in output quality, with significant drops in both fluency and accuracy. At 30% deactivation, the model begins to generate incoherent or irrelevant responses, effectively losing basic conversational capabilities.
These results provide compelling evidence for the effectiveness of MODEL SHAPLEY in identifying critical neurons responsible for specialized cognitive functions.
H.6
Deactivation Ration Study
Normallzed Shapley Distribution Allve Neurons (top 90% deactivated step 0)
86.24
step 0
86.2
86.17
step
Density (log scale)
step 40
Accuracy (96)
86.0
step
step 80
100
85.8
85.6
15
85.2
10-
0.1
0.3
0.5
0.7
0.9
0.0
0.2
0.6 Normalized Shapley
0.4
100
Activate Ratio
(a) Accuracy CIFAR- of VIT-Base-lB Backbone different activation ratios during finetune.
(b) Compensation Study of Qwen 2.5-3B Instruct GSM8K Dataset.
We conducted experiments on the ViT-Base-1e model using the CIFAR- dataset. As shown in Figure 3a, the results reveal that significant portion of the neurons redundant. Remarkably,
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Table 4: Case study on MMLU Dataset comparing the normal output and deactivated BottomK output. The numen of Augustus referred to which of the following characteristics? The choices are: A. Divine power, B. Sexual virility, C. Military acumen, D. Philosophical intellect
Question
Ground Truth
(A) Divine power
To determine the correct answer, let's consider what numen meant in the context of Roman religion and the time of Augustus. Numen was a concept in Roman religion that referred to the distinctive power or presence of a deity. It was not a personification but rather a kind of spiritual presence that could be associated with an object or place. In the context of Augustus, who was elevated to near-divine status by the Roman people and Senate, his numen would refer to his divine power or the divine presence attributed to him.- Option A: Divine power aligns closely with the concept of numen in this context.- Option B: Sexual virility does not fit the general notion of numen.- Option C: Military acumen, while characteristic that made Augustus highly successful, is not specifically related to the concept of numen.- Option D: Philosophical intellect, again, does not relate specifically to numen. Thus, the numen of Augustus referred to his divine power. The answer is (A). [CORRECT]
Normal Output
To determine the correct answer, let's 's consider what numenin the context of a historical figure refers 1.. ** Divine power (A)*: In Roman mythology and history, the numen often referred to divine forces or their manifestations. Augustus was given numen to indicate a divine presence or power that could protect and influence the empire.2. ** Sexual virility (B)**: While sexual virility might have been a characteristic of Roman leaders for historical or symbolic purposes, it doesn't typically align with the term numenin historical 3ontexts.3, **Military acumen (C)*#; While military acumen might have been a significant trait for leaders, the term ftumenis more aligned with powers attributed to deities divine influence rather than specific human traits like military skills.4. 4. **Philosophical intellect (Dy**; Similarly, philosophical intellect would relate more to mental prowess rather than the divine or protective aspect typically associated with a numen. The term ñumenin the context of Roman emperors and Augustus specifically relates to divine power or a divine presence, which is a characteristic that could protect and influence the empire on a supernatural level. weeel..Tereffoee Therefore, the answer is (A). [CORRECT]
Deactivated
Output (5%)
To determine the correct answer, let's 's consider what humenin the context of Augustus refers to. The term ñumenis associated with the Roman gods and can be linked the concept of divine attributes or powers within the Roman context. ancient Rome, fiumenčould be the power or the divine aspect of god. An individual or deity could be associated with numen, which would manifest their divine attributes or Lewers.Let's 's evaluate each option:A. Divine power This option aligns well with the concept of a numen. A numen would indeed refer to a divine power or attribute that is associated with divine entity.B. B. Sexual virility- This is not relevant to the concept of numen in ancient Rome. It might perturb the context to something more related to ancient Greek, where attributes like virility were discussed within other deities, such as Priapus.C. C. Military acumen While military acumen can be trait of leader, it is not directly related to the concept of numen, which pertains more to divine qualities. D. Philosophical intellect- - Although philosophical intellect could be characteristic, it is not directly related to the term fumenn Which is more aligned with divine attributes. Given the context and the specific meaning of ñumenin relation divine attributes, the answer is:A. Divine power. The answer is (X) where X is the correct letter choice A. [CORRECT]
Deactivated Output (10%)
To answer this question step. you must follow the process of a number in the reference of a m The correct answer C. Military acumen:Thh The answer is (C).Step
Deactivated Output (30%)
reason: The current is in a 1 5 1 5 1 5 the 8 An acuen of acuen of U A F 1 5 A the of 5 Relevant M 1 5-With with 5 5 555555 5 the 5 rule I ab 5.5555 5 u 5-shift I an the (*(1 the 5 The 5 The 5 5 The 5 the 5) the 5 5 This A comes the 8 5 the the 25 a 5 1 Mu I
5 Prevention 52 1 111111151
ID
1 1"] [WRONG]
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Table 5: Case study on MMLU Dataset comparing the normal output and deactivated BottomK output. Every country with seacoast has an exclusive economic zone (EEZ) under the UN Convention on the Law of the Sea. Which one of the following statements regarding the Law of the Sea and the EEZ is NOT correct?" Choices": A. "The EEZ extends up to 200 nautical miles (370 km) from the coastline." ", B.. "Countries have the right to exploit natural resources in their EEZ.", " C. "Countries can exploit resources on the continental shelf extending up to 350 nautical miles (560 km) beyond their EEZ.", D. "A country does not have the right to exploit living resources in its EEZ".
Question
Ground Truth
(D) A country does not have the right to exploit living resources in its EEZ
Let's analyze each statement regarding the Law of the Sea and the EEZ:A. The EEZ extends up to 200 nautical miles (370 km) from the coastline. According to the UN Convention on the Law of the Sea, this statement is correct.B. Countries have the right to exploit natural resources in their EEZ. Thiss statement is also correct as it aligns with the principle of the EEZ. C. Countries can exploit resources on the continental shelf extending up to 350 nautical miles (560 km) beyond their EEZ.- This statement is correct, as states have the right to explore and exploit all natural resources of the seabed and subsoil beyond and adjacent to their territorial sea up to 350 nm from the baselines. D. A country does not have the right exploit living resources in its EEZ. This statement is incorrect. Countries have the right to explore and exploit living resources within their EEZ, contrary to what claims. After reviewing all the statements, we find that only one is incorrect. Therefore, the answer is (D). [CORRECT]
Normal Output
To determine which statement is NOT correct, let's analyze each option step by step: A. The EEZ extends up to 200 nautical miles (370 km) from the coastline. This statement is correct according to the UN Convention on the Law of the Sea, which defines the EEZ extending up 200 nautical miles. B. Countries have the right to exploit natural resources in their EEZ. This statement is also correct. Article 77 of the convention states that states have the authority exploit resources in their EEZ for reproduction purposes. C. Countries can exploit resources on the continental shelf extending up to 350 nautical miles (560 km) beyond their BEZ.
Deactivated Output (5%)
This statement is accurate under the convention as it pertains to the continental shelf, which can extend up to 350 nautical miles beyond the EEZ, as mentioned in Article 76. D. A country does not have the right to exploit living resources in its BEZ. This statement is incorrect. Contrary what D suggests, Article 105 of the convention actually allows states to exploit living resources in their BEZ, subject to certain conditions. The answer (D). [CORRECT]
Let's 's's analyze each statement regarding the Law of the Sea and the Exclusive Economic Zone (EEZ):A. The EEZ extends up to 200 nautical miles (370 km) from the coastline. This statement is correct according to the Convention on the Law of the Sea. The EEZ is the area extending from the coastline to 200 nautical miles. B. Countries have the right to exploit natural resources in their EEZ. This statement is also correct. Countries can exploit natural resources in the EEZ, including renewable resources, provided they do not leave negative impact the marine environment. C. Countries can exploit resources on the continental shelf extending up to 350 nautical miles (560 km) beyond their EEZ. This statement is not correct according to the Convention. The continental shelf extends up to 100 nautical miles (350 km) from the coastline, while the EEZ extends to 200 nautical miles from the coastline. The continental shelf is not part of the EEZ and extends only up to 100 nautical miles (350 km). D. A country does not have the right to exploit living resources in its EEZ. This statement is correct according to the Convention. While countries can exploit natural resources in the EEZ, living resources are not included in the natural resources under the Convention. The statement that is not correct is C. as incorrectly states the continental shelf extends beyond the EEZ, which is not true. The answer is (C)." [WRONG]
Deactivated Output (10%)
The correct answer is the answer (A) and the to the in you and the in you in in Mor the years D in in in in in in in in in in in in in in in the (The answer is (D)) is in in Condition (2 in n. 8 the in in in in the in in 2 2 in the 2 The 2 in 1 A in in 3 111 1 in in 1
Deactivated Output
(30%)
1 the
the 1 1 1 1 1 1333 [WRONG]
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Table 6: Case study on GSM8K Dataset comparing the normal output and deactivated BottomK output. Josh decides to try flipping house. He buys house for $80,000 and then puts in $50,000 in repairs. This increased the value of the house by 150 %. How much profit did he make?
Question
Ground
70000
Truth
Let's break down the problem step by step:
Initial Cost: Josh buys a house for $80,000.
2. Repairs Cost: He spends $50,000 on repairs.
So, the total cost including repairs is:
50,000+550,0000
130,000 dollars
3. Value Increase: The repairs increase the value of the house by 150%.
Normal Output
To find the increase in value, we calculate 150% of the purchase price:
150% x 80,000 = 1.5 80,000 = 120,000 dollars So, the value of the house is:
00,000++10,,0000 120,000 =0000000=00000 2000000000000000000 000 dollars
4. Profit Calculation: The profit is the selling price minus the total cost.
70,000 dollars
70,000 [CORRECT]
Let's break down the problem step by step:
Initial Purchase: Josh buys a house for $80,000.
2. Repairs: Josh spends additional $50,000 on repairs.
3. Total Investment: The total amount Josh has invested (purchase price + repairs)
is:
50,0000+50,000-0000 100,,00000000000000
Increase in Value: The repairs increased the value of the house by 150%. To find the new value of the house, we need to calculate 150% of the purchase price and add it to the purchase price:
Deactivated Output (5%
Increase in value = 80,000 1.50 = 120,000
So. the new value of the house is:
200,0000
Profit Calculation: Profit is calculated by subtracting the total investment from the new value of the house:
Profit-000,00 130,000 $33,000=70,,000 70,000
Josh made profit of $70,000. /..../CCRRR [CORRECT]
Step 1: Calculate the total cost of the house, including repairs The house was bought for $80,000 and then $50,000 was spent repairs, the total cost is: 80,000++50,000-00 130,000 Step 2: Determine the increased value of the house The value of the house increased by 150% after the repairs, the house is: 1.5 x 80,000= = 130,000 In this case, 1.5 times the original value of the house is equal 130,000, indicating that the original value of the house was $80,000 and the repairs added $100,000 to the value (150% I = 1.5, means 1 times the original plus times the, so 330,000- 130,000 100,000). Step Calculate the profit The profit is the difference between the increased value of the house and the total cost of the house, so The profit is the increased value (130,000) minus the cost (130,000), so 130,000 (1 (150% 330,000), so 130,000 (150%. 130,000) 130,000 (150% 130,000) = 130 [WRONG]
value of the
Deactivated Output (10%)
Let's think through the steps calculate the profit of the house, in the context of flip and the increase of the value by 150%. Step by step, we'll output the final answer: Step 1: Initial Value of the House Let's 's start with the initial value of the house - Initial value: $80,000 To the 150% increase, we need to a calculate. The 150% increase is step. and is the value that need to calculate. Let's assume the initial value is a 150% increase, for the purpose of the 150% increase, to the 150% increase itself the 150% increase is 150% increase in the current value of the technology, not the 150% increase in the initial value itself. This, we will calculate the increase of the 150% value a 150% increase in the next value, and the 150% increase in the 1 55 5 the 2 5 1
Deactivated Output (30%)
11111111 the, I with the, 1 Step 2: Calcul with the 150% Increase the 1 5 5,5, addon. Let, the, the, 5, Let, this, th for let ihm', the the, the, the," 5' atte", API', [WRONG]
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Table 7: Case study on GSM8K Dataset comparing the normal output and deactivated BottomK output. Janet's ducks lay 16 eggs per day. She eats three for breakfast every morning and bakes muffins for her friends every day with four. She sells the remainder at the farmers' market daily for $2 per fresh duck egg, How much in dollars does she make every day at the farmers' market?
Question
Ground Truth
18
Let's break down the problem step by step:
1. Calculate the total number of eggs Janet gets each day:
Janet's s ducks lay 16 eggs per day.
2. Calculate the number of eggs Janet uses for breakfast:
Janet cats eggs for breakfast every morning.
3. Calculate the number of eggs Janet uses for baking muffins:
Janet uses 4 eggs to bake muffins for her friends every day.
4. Calculate the total number of eggs Janet uses: Total eggs used for breakfast and baking 3 (breakfast) + (muffins) = 7 eggs per day.
Normal Output
5. Calculate the number of eggs Janet can sell:
Remaining eggs after usage Total eggs per day -Egs used
Remaining eggs = 16- 7 eggs per day.
6. Calculate the revenue from selling the eggs:
Janet sells the remaining 9 eggs at $2 per egg.
Revenuc 9 eggs $2/egg $18 per day.
18 [CORRECT]
Let's break down the problem step by step:
Determine the total number of eggs laid per day:
Janet's ' ducks lay 16 eggs per day.
2. Calculate the number of eggs used for breakfast:
Janet eats eggs for breakfast every morning.
So, the number of eggs used for breakfast is 3 eggs,
3. Calculate the number of eggs used for baking muffins:
Janet bakes muffins using 4 eggs each day.
Deactivated Output (5%)
So. the number of eggs used for baking is eggs.
4. Calculate the total number of eggs used (for breakfast and baking):
The total number of eggs used 3+4= 7
5. Calculate the number of eggs remaining for selling:
The number of eggs remaining after using some for breakfast and baking
7- eggs.
Calculate the daily earnings from selling the remaining eggs:
Janet sells the remaining eggs at $2 per egg.
So, the daily earnings are 9 x 2 = 18 dollars.
Therefore, Janet makes 18 dollars every day at the farmers' market. [CORRECT]
To solve this problem, let's s first calculate the number of eggs Janet has after various activities and then determine the earnings from selling the remainder.
1. Total eggs laid by Janet's 's ducks in a day (16 eggs)
Eggs eaten for breakfast (16 eggs 3 eggs for the morning breakfast 13 eggs)
Deactivated Output (10%)
3. Eggs used for baking muffins (16 eggs 4 eggs for baking muffins 12 eggs) Eggs sold at the farmers' market (16 eggs eggs for morning breakfast +4 eggs for baking 2 eggs = 8 eggs)
Earnings per egg at the farmers' market (2 dollars per egg)
4. Earnings calculation (8 eggs multiplied by 2 dollars per egg = 16 dollars)
So, Janet makes 16 dollars every day at the farmers' market by selling the
remainder of her eggs. 16 [WRONG]
Let's break the problem step by step and calculate the number of the daily you
would make for the visit to the farmers's 's market.
Deactivated Output (30%)
Step by step:
1. Calculate the number of the daily Mor's (d and " in) in the first, to the work of the Mor ( (in, the 8 and " "O 0))):
The Mor ( (()) 164 (O ). 2 (( (in). to the the the (( (the ( ed ( (he ( !!!!!
O!! D) 2 (!) ([WRON]
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retaining only the top 70% of neurons ranked by their Shapley values achieves an accuracy of 86.42% surpassing the 85.19% accuracy obtained when 90% of the neurons are retained.
This finding suggests that not all neurons contribute equally to the model's predictive capability, and that carefully selected subset of high-importance neurons can maintain even enhance performance. It also highlights the potential for neuron pruning or activation sparsity as an effective strategy for improving efficiency without sacrificing accuracy.
H.7 Analysis of Compensation Effects in Parameter Units
During our experiments, we observed the neural compensation effect. To further investigate this phenomenon, we design an additional set of experiments using the Qwen 2.5 3B-Instruct model on the GSM8K dataset. Specifically, at various training steps (step = 0, 20, 40, 60, and 80), we deactivated the top 90% of neurons ranked by importance and analyzed the distribution of parameter importance.
Our results in Figure 3b show that as training progresses, the distribution of Shapley values evolves. When high-Shapley neurons are deactivated, other neurons progressively assume their functional roles. This redistribution leads to increased Shapley values among previously less important neurons, illustrating a clear compensation effect within the network. These findings suggest that the model dynamically redistributes functional importance across neurons during training to adapt to perturbations, highlighting the inherent resilience and adaptability of neural representations.
H.8 Model Compression Sensitivity Experiments
INT4 Quantization
IN Quantiration
74.22
74.07
2475
59
00.35
63:17
90:20
Rumbime (min)
Accuracy
46-
Accuracy
76
72.86
72:56
12/11
72:43
S4
255
(a) INT 4 Quantization with different /8 on GSMSK.
(b) INT 8 Quantization with different , on GSM8K
We further compared the performance of Qwen 2.5 7B- Instructt and Qwen 2.5 14B- [4B-Instruct under different compression settings for INT 4 in Figure 4a and INT in Figure 4b. By varying the parameter 38 in the range 00.10.33,05,,0.7,0..1 we evaluated both time efficiency and model accuracy.
The experimental results demonstrate the effectiveness of MODEL SHAPLEY, particularly in scenarios where cooperative game-theoretic contributions are explicitly weighted. As a increases, the trade-off between compression and performance becomes more pronounced, yet MODEL SHAPLEY consistently maintains superior accuracy with reduced computational cost. These findings validate the importance of incorporating cooperative importance measures in guiding neuron selection for model compression.
H.9 Attention Head and Layer-wise MODEL SHAPLEY Study
In this experiment, we investigate the effectiveness of MODEL SHAPLEY at the attention head and layer level. Specifically, evaluate its ability to identify important parameter units beyond MLP layers. We conduct knowledge base routing experiments on a private company policy dataset, following the setup in [68, 69, 70]. The dataset consists of 2,088 training samples and 344 test samples, covering four subcategories of company regulations and an "others" category (i.e., questions that should not trigger knowledge base queries).
The routing task aims to determine whether the model should query a knowledge base and, if so, which one. We compare the performance of MODEL SHAPLEY and gradient trace in identifying important attention heads and layers. Using the Qwen 2.5 7B-Instruct model, we compute the Top-K importance scores using both methods and select the most relevant units accordingly.
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For probing, we use the training set to identify the most important attention head and layer, and then use the output signals from that selected layer as imput a downstream probing model two-layer MI classifier trained to perform 5-way classification.
Table 8: Top-5 Important Attention Heads and Layers Identified
Rank
Gradient Trace Trace (Layer: Score)
MODEL SHAPLEY (Layer: Score)
1st
modeLl attm.v atton._pproj x.90 10
modellayers2266mlp.oovw proj: 0.375
2nd
modelllayen.3..elf.ttm....
1.66 10-5
htteelaaaynns22...e.ttt....... 0.354
3rd
modelllayers225.eee_aaaaaa
1.52 10
modelllaycrs223.elff..mm sttpsself__ttt__oor_. 0.352
4th
model.layers.2..slff...a wtwp://.222..eeaaatta_pppp x 10
modellaayers222seelf..am wttps//_aaattv_pp_____ satp
0.289
5th
2ede attpseefr__tttt____r_gg x 10-5
model.laaerrs22..sfff.a.at
0.260
Based on these rankings, we use model.layers.22.sel set_s____attt__oor for the gradient trace experiment and model.layers.66. fml.mlppoow.. proj for the MODEL SHAPLEY experiment. The classification accuracy on the routing task is summarized in Table
Table 9: Comparison of Probing
Gradient Trace & MODEL SHAPLEY on Attention Head and Layer-wise
Method
Train Set Accuracy
Test Set Accuracy
Gradient Trace (model.layers. 22..errs.2..ellaaattt attn. v_proj) MODEL SHAPLEY (mooee.layerss2..mp.ddwww
0.89
0.81
0.92
0.83
The experimental results demonstrate that MODEL SHAPLEY is effective not only at the neuron level, but also at the finer-grained attention head and layer level.
Table 10: Accuracy (%) and training cost (same units across methods; lower is better) on GSM8K.
Qwen2.5-3B-Instree
Qwen2.5-7B-Innstruct
Method
Accuracy
Cost time
Accuracy
Cost time
Shapley
49.51
215
62.47
265
LoRA
48.45
200
64.14
170
Shapley+LoRA
52.16
242
63.53
273
H.10 Runtize-normalizzdd comparison
It is crucial to first clarify where the computational cost occurs. The runtime difference between our method and the baselines lies in the one-time, offline parameter importance quantification step. The subsequent quantization process, guided by these importance scores, takes similar amount of time for all methods.
Therefore, a constant runtime comparison for the importance quantification step is not straightforward. Each method is distinct algorithm designed to run completion. Artificially truncating our method to match a faster baseline's s runtime would mean using incomplete Hessian approximation, compromising the integrity of our approach and leading to an uninformative comparison.
Instead, we believe the right way to evaluate this trade-off is to compare the final accuracy against the actual, practical runtime. We admit our method has higher cost for importance quantification. as it incorporates second-order cooperative interactions, not just first-order individual importance. Our experiments show this additional investment is both marginal and highly worthwhile.
For Qwen2.5-7B INTS quantization: GPTQ: 53.86 min total (importance: min), OBD: 57.88 min total (importance: 8 min). @ MODEL SHAPLEY 60.20 min total (importance: 10 min).
The 2-5 minute difference in importance estimation yields 1.5-2% accuracy gains, which persists throughout the model's deployment lifetime. We believe the one-time cost of 5 extra minutes during quantization is negligible compared to the perpetual benefits of higher accuracy and/or smaller model size during deployment, where models serve millions of requests.
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H.11 Fine-tuning Efficiency and Variance
We extend our significance analysis with two compact replications across NLP and vision. In each setting we report mean accuracy and standard error, and we assess statistical significance with peappendent-saplleee t-tests at the 5% level. We also examine stability across random seeds under realistic compute budgets.
On GSM8K with Qwen2.5-3B-- 5-3B- Inssuutt each configuration is repeated with three random seeds. This replication compares MODEL SHAPLEY with the strongest baseline, Gradient Trace. Table 11 presents the three-seed statistics. MODEL SHAPLEY attains higher mean and smaller standard error than Gradient Trace, and (-tests at the 5% level indicate statistically significant improvements, supporting reliable gains under the three-seed regime.
On CIFAR- 100 with VIT-Base, we run eight seeds and include four methods: Random, Gradient, Gradient Trace, and MODEL SHAPLEY. Table 12 reports the aggregate mean and standard error for these methods. MODEL SHAPLEY achieves the highest mean accuracy, while standard errors differ across baselines. Using the full eight-seed replication for inference, pairwise I-tests the 5% % level confirm significant improvements of MODEL SHAPLEY over each baseline.
Table 11: Accuracy statistics (%) across three runs on GSM8K with Qwen2.
Method
Mean
Standard Error
Gradient Trace
48.49
0.72
MODEL SHAPLEY
49.51
0.27
Table 12: Accuracy statistics (%) across eight runs on CIFAR-100 with VIT-Base.
Method
Mean
Standard Error
Random
84.59
0.13
Gradient
85.16
0.16
Gradient Trace
85.12
0.10
MODEL SHAPLEY
85.51
0.19
Environment note. The mean values reported here may differ slightly from those in the main text because these additional runs were executed on different hardware/software stack (NVIDIA H800 GPUs with newer CUDA environment) than the original experiments (NVIDIA A GPUs with a previous CUDA version), yielding a general upward shift in accuracy for all methods.
Computational efficiency. MODEL SHAPLEY estimates parameter importance with single forward and backward pass plus a Hessian product (see Section 4. I and Remark 4.9). For multi-epoch fine-tuning, this one-time cost is amortized by updating only small subset of parameters. In our experiments, selecting 10% parameters reduced training time by approximately 85% relative to full-mode! fine-tuning and reduced gradient storage by 90%, enabling larger batch sizes. The one-time importance estimation for a 7B model takes roughly 30 minutes and can be reused across fine-tuning tasks.
H.12 Hardware Scalability and General Applicability
The scalability of MODEL SHAPLEY primarily governed by the memory and compute requirements of the blockwise Hessian approximation rather than any hardware-speciffc constraint. The method designed to be hardware-agnostic and can operate efficiently across diverse GPU architectures by adjusting the block granularity used in Shapley value estimation.
The total memory footprint scales quadratically with the number of parameters per block. Consequently, the granularity of the partitioning determines the trade-off hetween computational cost and analytical resolution:
Coarse granularity (e.g., treating an entire neuron or layer as single block) involves larger parameter groups, leading to higher memory demand but faster overall computation. Under
43

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this configuration, models with approximately up to 3B parameters can be handled efficiently on modern GPUs such the NVIDIA H100 (80 GB).
Fine granularity (e-g., defining blocks the neuron-in-atteme in--attention-head level) reduces per-block memory requirements substantially, enabling analysis of larger models, up to the 7-13B parameter range comparable hardware.
In practice, the same principle generalizes across different hardware configurations. Devices with greater memory capacity or multi-GPU setups naturally extend the upper bound of model size, while smaller devices can still employ MODEL SHAPLEY effectively by adopting coarser partitioning. This aardnace-agnostic flexibility allows practitioners to tailor the granularit--ffiiienyy trade-off' according to available resources, ensuring the method's applicability wide range of model scales and hardware environments.
I Broader Impacts
Our work leverages cooperative-game game- theoretic Shapley values to develop MODEL SHAPLEY, an efficient,
estimator that assigns importance score to every parameter in a neural network. By collapsing the naive 2 -scale computation to one gradient pass, MODEL SHAP- LEY produces interpretable saliency maps that drive targeted fine-tuning, pruning, and quantization, cireumventing the accuracy degradation often seen with full-model fine-tuning. This efficiency espe- cially critical in the of large models, where the sheer parameter count makes exhaustive attribution prohibitively expensive. In practice, MODEL SHAPLEY excels at large-model interpretability tasks (e.g.. probing and logit-lens analysis), parameter-spede adaptation, and precision-awe quantization, providing a practical template for future responsible and efficient model development.
J Data Ethics Statement
To evaluate the efficacy of this work, we conducted experiments that only use publicly available datasets, namely, MMLU, GSM8K, CIFAR- and ImageNet in accordance with their usage terms and conditions, if any. We further declare that no personally identifiable information was used, and no human or animal subject was involved in this research.
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the
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Reviews:
[{"review_id": "T4a7uDbY7v", "reviewer": "Reviewer_rAZT", "summary": "This paper proposes Model Shapley, a novel framework for estimating the importance of individual model parameters using Shapley values from cooperative game theory. Unlike traditional pruning or gradient-based heuristics, Model Shapley captures both individual and synergistic contributions of parameters via a second-order closed-form approximation with Fisher-based regularisation. Experiments shows it works for both NLP and CV tasks.", "questions": "While the paper compares accuracy and runtime across different quantisation methods, it is unclear whether the superior accuracy of Model Shapley persists when runtime is held constant. Could the authors clarify how much of the accuracy gain stems from longer calibration time or more expensive estimation steps? A cost-normalised evaluation (e.g., accuracy at fixed runtime) would make the comparison more convincing.", "limitations": "As above", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 4, "strengths_and_weaknesses": "Strengths\n- The paper is the first to apply Shapley values at the parameter level, explicitly modelling inter-parameter cooperation with solid theoretical grounding. \n- The method supports efficient approximations, scales to large models, and performs well across inference, fine-tuning, and compression tasks.\n- The evaluation spans NLP and CV tasks with diverse models, providing convincing empirical evidence.\n\nWeaknesses\n- The blockwise approximation neglects inter-layer or global parameter interactions.\n- In certain tasks, the improvements over strong baselines are small, and the advantage over prior methods is not always significant. Also, the cost is higher than other baselines.", "was_revised": true, "final_justification": "The additional explanations addressed my main concerns and clarified the technical reasoning behind the design choices. The new experiments and variance reporting also improved my confidence in the stability and significance of the results. I increased the clarity score.", "rebuttal": "We sincerely appreciate your thoughtful and constructive feedback, which has helped us clarify key technical aspects. We believe the additional justifications provided below comprehensively address the raised concerns and demonstrate that Model Shapley offers both significant theoretical contributions and practical advantages.\n\n---\n## W1. Blockwise approximation neglecting inter-layer interactions\n\nThank you for this insightful observation. We acknowledge that blockwise approximation theoretically neglects inter-layer interactions, but this design choice is both necessary and well-justified.\n\n1. **Computational necessity**: Full global Hessian computation requires O(M²) memory for M parameters. For a 7B model with M≈7×10⁹, this would require ~196TB just to store the Hessian matrix, while our blockwise approach reduces this to manageable levels (≈28GB for typical block sizes). Our blockwise approach reduces this to O(M·d) where d is the block size, making the method practically feasible while preserving 95%+ of the interaction information based on our empirical validation.Furthermore, our layer-level analysis (Figure 1) shows meaningful differentiation of importance across layers, suggesting that the blockwise strategy captures sufficient granularity for practical applications while maintaining computational feasibility.\n\n2. **Architectural Justification**: More importantly, the Transformer architecture is highly modular. Information flows sequentially through layers, and the most critical synergistic interactions occur **within** functional blocks (e.g., neurons within an FFN layer, or heads within an MHA module). While inter-block dependencies exist, their influence is secondary to the dense intra-block computations. Therefore, the blockwise approximation correctly prioritizes the most significant cooperative effects that our Shapley-based method aims to quantify. Our empirical results, showing strong performance in identifying critical neurons and layers, support this architectural prior.\n\n---\n## W2. Significance of improvement\n\nWe appreciate this concern about the magnitude of improvements. While the absolute differences may appear modest, we acknowledge this limitation arose from **computational constraints typical in LLM research**. We conducted additional experiments with 3 independent runs on GSM8K with Qwen2.5-3B-Instruct. With limited resource and finte budget, we could only afford to compare our method with the best baseline method Gradient Trace during the rebuttal phase.\n\n|                | mean  | variance | std    |\n|----------------|-------|----------|--------|\n| Gradient Trace | 48.49 | 1.57     | 1.25   |\n| Model Shapley  | 49.51 | 0.22     | 0.47   |\n\nAs shown, **MODEL SHAPLEY** consistently statistically outperforms the baseline (t-test at significal level 5%). Notably, our method also exhibits lower variance, suggesting it provides a more stable and reliable improvement. While the rebuttal period's time and resource constraints limited us to this setting, we are confident in this trend and will add variance information for other key experiments in the final manuscript.\n\nWe would also like to clarify that the mean performance values reported here in the rebuttal may differ slightly from those in the main paper. This is because, due to resource availability during the rebuttal period, we conducted these additional experiments on a different hardware/software setup (NVIDIA H800 GPUs and a newer CUDA environment) compared to the original experiments (NVIDIA A100 GPUs and previous CUDA version). This change resulted in a general improvement in model performance across all methods.\n\nRegarding the computational cost, which we address in detail in Q1, we show the minor overhead is justified by these consistent and statistically significant accuracy gains.\n\n---\n## Q1. Runtime-normalized comparison\n\nThank you for this excellent question on the trade-off between accuracy and runtime.\n\nIt is crucial to first clarify **where the computational cost occurs**. The runtime difference between our method and the baselines lies in the **one-time, offline parameter importance quantification** step. The subsequent quantization process, guided by these importance scores, takes a similar amount of time for all methods.\n\nTherefore, a \"constant runtime\" comparison for the importance quantification step is not straightforward. Each method is a distinct algorithm designed to run to completion. Artificially truncating our method to match a faster baseline's runtime would mean using an incomplete Hessian approximation, compromising the integrity of our approach and leading to an uninformative comparison.\n\nInstead, we believe the right way to evaluate this trade-off is to compare the final accuracy against the actual, practical runtime. We admit our method has a higher cost for importance quantification, as it incorporates second-order cooperative interactions, not just first-order individual importance. Our experiments show this additional investment is both **marginal and highly worthwhile**.\n\nFor Qwen2.5-7B INT8 quantization:\n* GPTQ: 53.86 min total (importance: ~5 min)\n* OBD: 57.88 min total (importance: ~8 min)\n* Model Shapley: 60.20 min total (importance: ~10 min)\n\nThe 2-5 minute difference in importance estimation yields 1.5-2% accuracy gains, which persists throughout the model's deployment lifetime. We believe the one-time cost of 5 extra minutes during quantization is negligible compared to the perpetual benefits of higher accuracy and/or smaller model size during deployment, where models serve millions of requests."}, {"review_id": "cHw75ysdpa", "reviewer": "Reviewer_AFub", "summary": "This paper addresses the problem of parameter importance quantification in neural networks—determining which weights contribute most significantly to model performance. The authors identify a key limitation in existing approaches: they typically fail to account for interactions between parameters when assessing their individual importance.\n\nTo address this gap, the authors propose *Model Shapley*, a novel method that leverages Shapley values from cooperative game theory to quantify parameter importance. The key advantage of this approach is that Shapley values naturally capture the complex interactions between parameters, providing a more comprehensive measure of importance than previous methods.\n\nSince computing exact Shapley values is computationally infeasible for neural networks, the authors develop a new approximation technique based on a second-order approximation. This makes their approach practically applicable to real-world models.\nThe authors demonstrate the utility of their method through two main applications: targeted knowledge injection via fine-tuning and model compression via quantization and parameter pruning during inference. For their experiments, they focus on 2 NLP tasks (gsm8k, MMLU) as well as two computer vision tasks (CIFAR-100, ImageNet).", "questions": "**Questions**\n\n- Why are the methods of [20,21] not used as baselines in Table 1?\n- How many neurons are intervened on in Table 1?\n- Regarding the quantization experiments: Which neurons / layers are quantized? What exactly is the setup here? This is unclear without looking at Appendix and should be mentioned in the main text.\n\n**Suggestions**\n\n- Additional experiment to support findings in Table 1: plot % of weights removed vs. performance. \n- Move Table 1 to top or bottom of the page.\n- In Table 1, add the Full Fine-Tune row below “Training” to improve readability.\n- Figure 2 is hard to read. Increase font sizes considerably", "limitations": "Currently the discussion of limitations is very brief (only two sentences in the conclusion). The authors should consider a discussion of the runtime of their approximation as well as the focus on 4 particular benchmarks and 3 models.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "**Strengths**\n\nThe paper demonstrates solid technical quality through its dual contributions. The authors present a novel second-order approximation for computing Shapley values in the context of parameter importance, which distinguishes their work from existing Shapley-based approaches in the literature. The validation spans three distinct applications (knowledge injection, interpretability, and model compression).\n\nThe paper is well written and easy to follow. The introductory sections, related work, and background are well-written and effectively motivate the research problem. The authors clearly articulate the gap in existing parameter importance methods—namely, the neglect of parameter interactions—and position their Shapley-based solution as a principled remedy.\n\nWhile the application of Shapley values to parameter importance is not entirely novel (the authors acknowledge two previous works in this direction), their contribution lies in developing a new second-order approximation method. \n\n**Weaknesses**\n\nI see the main issues in the experimental portion of the paper and list my concerns below.\n- For almost all tasks (as expected) deactivating neurons during inference leads to substantial drop in performance. It's not particularly clear to me what exactly the benefit of the proposed method is. Why would I use this method in practice to remove certain weights from my model? If the argument is because you end up with a smaller model, this aspect should be emphasized and studied in the paper. How many weights do you actually prune? And how does the performance vary as you prune more or less weights (compared to baselines)? These are important experiments that are currently missing but would make the paper much stronger. \n- Selective fine-tuning leads to similar performance compared to full fine-tuning but it seems like the results presented are based on fine-tuning a single model. What’s the variance here? How often did you fine-tune? And, again, what am I gaining from this? Is this more efficient compared to full fine-tuning? How does it compare to parameter-efficient methods when fine-tuning a similar number of weights, e.g., LoRA? What if I factor in the Shapely value estimation? Without reporting run-times, I'm not convinced that this method offers any benefits for the fine-tuning setting. \n- The neuron deactivation study offers only qualitative results which are deferred to the Appendix. What about quantitative evaluation and baselines here? In particular, it will be important to measure whether the performance on other inputs is maintained, i.e., compute a utility vs. forgetting tradeoff.", "was_revised": true, "final_justification": "After taking the reviewers response and also discussion with other reviewers into account I increased my score.", "rebuttal": "We sincerely thank your insightful and constructive comments. Your feedback has been instrumental in helping us understand the key points that led to your assessment. In the following responses, we provide direct clarifications and outline concrete revisions that we believe fully address these concerns. We are hopeful that our detailed response will offer **a clearer perspective on the significance of our contributions and merit a positive reassessment of our work**.\n\n---\n## W1. Benefit and practical applications of the method\n\nThank you for this important question. We apologize for not clearly articulating our contributions. Our method's primary value lies in **accurately quantifying parameter importance while accounting for inter-parameter interactions**—a capability that existing methods lack.\n\nThe neuron deactivation experiments serve as **validation** of our importance scores, not the end application. The substantial performance drop is not the goal, but rather the **proof** that our method correctly identifies functionally vital neurons. The practical benefits include:\n\n1. **Model Compression**: In Table 2, our Shapley-guided quantization achieves 72.93% accuracy (INT8) vs 71.27% (OBD) on GSM8K—only 0.5% below the uncompressed model while OBD drops 1.2%.\n2. **Efficient Fine-tuning**: By fine-tuning only the top 10% neurons identified by our method, we achieve 73.89% on MMLU (Table 1), compared to 73.56% with full fine-tuning—using 90% fewer parameters.\n3. **Knowledge Preservation**: Our method enables targeted adaptation without catastrophic forgetting, crucial for continual learning scenarios.\nTo prove this, we conduct additional experiments by finetuning LLM on math dataset GSM8K and test their performance on GSM8K and general benchmark MMLU. As shown in the table below, our model retains strong performance on general benchmark MMLU, showcasing its ability to avoid catastrophic forgetting.\n\n|       | **Qwen2.5-3B** |       | **Qwen2.5-7B** |       |\n| :--- | :--- | :--- | :--- | :--- |\n|                | GSM8K | MMLU  | GSM8K | MMLU  |\n| Gradient Trace | 48.49 | 62.09 | 60.42 | 68.74 |\n| Model Shapley  | 49.51 | 62.28 | 62.47 | 69.13 |\n\n---\n## W2. Fine-tuning efficiency, significance and LoRA comparison\n\nWe appreciate your concern about statistical rigor and efficiency. You're right that variance reporting is essential. We acknowledge this limitation arose from **computational constraints typical in LLM research**. However, to address your valid concern about statistical significance, we are conducting additional fine-tuning repetitions on the GSM8K dataset with Qwen2.5-3B-Instruct. With limited resource and finte budget, we could only afford to compare our method with the best baseline method Gradient Trace during the rebuttal phase.\n\n|                | mean  | variance | std    |\n|----------------|-------|----------|--------|\n| Gradient Trace | 48.49 | 1.57     | 1.25   |\n| Model Shapley  | 49.51 | 0.22     | 0.47   |\n\nAs shown, **MODEL SHAPLEY** consistently statistically outperforms the baseline (t-test at significant level 5%). Notably, our method also exhibits lower variance, suggesting it provides a more stable and reliable improvement. While the rebuttal period's time and resource constraints limited us to this setting, we are confident in this trend and will add variance information for other key experiments in the final manuscript.\n\nWe would also like to clarify that the mean performance values reported here in the rebuttal may differ slightly from those in the main paper. This is because, due to resource availability during the rebuttal period, we conducted these additional experiments on a different hardware/software setup (NVIDIA H800 GPUs and a newer CUDA environment) compared to the original experiments (NVIDIA A100 GPUs and previous CUDA version). This change resulted in a general improvement in model performance across all methods.\n\nBesides, Our method is highly efficient. The Shapley value estimation requires only **a single forward and backward pass** plus a Hessian-vector product, as detailed in Section 4.1 and Remark 4.9. For fine-tuning over many epochs, this one-time cost of identifying important neurons is easily amortized by the computational savings of updating only a small subset of parameters (e.g., 10% in our experiments reduces training time by ~85% compared to full fine-tuning. Reduces gradient storage by 90%, enabling larger batch sizes. One-time cost of ~30 minutes for 7B model, amortized over multiple fine-tuning tasks).  \n\nWe appreciate the suggestion to compare against LoRA. While our method is **conceptually different**—selecting a subset of original parameters versus introducing low-rank updates—a direct comparison is valuable. We fine-tune a similar number of parameters using LoRA and our method on GSM8K dataset.\n\n|       | **Qwen2.5-3B** |       | **Qwen2.5-7B** |       |\n|--------------|---------|-----------|---------|------------|\n|              | Accuracy| Cost time | Accuracy| Cost time  |\n| Shapley      | 49.51   | 215       | 62.47   | 265       |\n| Lora         | 48.45   | 200       | 64.14   | 170        |\n| Shapley+Lora | 52.16   | 242       | 63.53   | 273   |\n\nThe results in the table above demonstrate that our method achieves competitive performance with comparable efficiency. And more importantly, our contribution is **orthogonal to LoRA**.\n\n---\n## W3. Quantitative evaluation of neuron deactivation\n\nWe apologize for the confusion and would like to clarify that **quantitative results for the neuron deactivation study are provided in Table 1, under the \"Inference (Deactivate Neurons)\" section**. This table details the performance degradation on CV and NLP tasks when deactivating neurons identified by our method versus baselines. The qualitative examples in Appendix F.5 are intended to be supplementary illustrations of this quantitative effect. We will clarify the link between the main table and the appendix examples much clearer.\n\nSpecifically, our quantitative results show:\n\n* Deactivating bottom 5% neurons: 72.71% → 73.39% accuracy (minimal impact)\n* Deactivating top 5% neurons: 72.71% → 46.70% accuracy (significant drop)\n\nThis demonstrates both the effectiveness of our importance scores and addresses your utility-forgetting tradeoff concern—our method precisely identifies which neurons can be safely removed vs. those critical for performance.\n\n---\n## Q1. Comparison with methods [20,21]\n\nThis is an excellent question. Both [20] and [21] are pioneering works in applying Shapley values to neural networks. However, their methods were designed for and evaluated on smaller-scale CV models (e.g., ResNet-50).\n\nThe core reason for their exclusion is **computational feasibility**. The Monte Carlo and bandit-based approximations in [20, 21] are still computationally prohibitive for modern LLMs, which have billions of parameters. The theoretical complexity of exact Shapley computation is exponential (O(M⋅2^M)). A key contribution of our work is deriving a second-order approximation that reduces this complexity to a **single backpropagation pass**, making parameter-level Shapley attribution tractable for large-scale models (**10^6x more efficient**). Therefore, including [20, 21] as baselines would not constitute a fair or practical comparison in the context of LLMs. We will add a note to the related work section to clarify this distinction.\n\n---\n## Q2. Number of neurons intervened\n\nThank you for the question. We apologize for not placing this information in the main text. As detailed in Appendix F.4, the intervention ratios were as follows:\n\n* For **inference (deactivation)**, we deactivated the bottom **5%** of neurons for NLP tasks and **30%** for CV tasks.\n* For **training (freezing)**, we fine-tuned the top **10%** of neurons (i.e., froze 90%) for all tasks.\n\nWe will move these crucial details to the main experimental setup section in the revision for clarity.\n\n---\n## Q3. Quantization setup\n\nWe apologize for the lack of clarity regarding the quantization setup. We will move the detailed description from the appendix to the main experimental setup section.\n\nTo clarify here: **all neurons and all layers containing weights (i.e., all transformer blocks and embedding layers) were quantized**. Our method integrates with the standard GPTQ algorithm but does not alter its comprehensive scope. Our specific contribution is to inject task-specific knowledge into the quantization process. We use **MODEL SHAPLEY** to calculate a correction factor for the Hessian diagonal, which guides GPTQ to better preserve weights critical to the task, as detailed in Section D.4.3 and Algorithm 1. The rest of the procedure, including layer-wise quantization and calibration, follows the standard GPTQ implementation.\n\n---\n## Addressing suggestions\n\nWe gratefully accept all formatting suggestions and will implement them in revision, including the performance vs. pruning curves, improved figure readability, and table reorganization."}, {"review_id": "4Wqsu8gu9F", "reviewer": "Reviewer_23zM", "summary": "MODEL SHAPLEY is a new method that uses Shapley values from game theory to measure how important each parameter is in a neural network. It helps improve interpretability, fine-tuning, and compression with only one backward pass.", "questions": "please see the weeknesses.", "limitations": "yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "[Strengths]\n1. It considers both individual and cooperative effects between parameters.\n2. Needs only one forward and backward pass using a second-order approximation.\n3. Works well across NLP and vision tasks for pruning, training, and interpretation.\n\n[Weaknesses]\n1. Since the Fisher matrix is only an approximation of the true Hessian, could the Shapley values be unreliable when gradients are unstable or the data distribution shifts?\n2. Blockwise approximation makes computation faster, but does it miss important fine-grained interactions between parameters, especially at the neuron level?\n3. The method assumes the model is stable during estimation. Can it still give accurate Shapley values during early training or in continual learning when parameters keep changing?", "was_revised": true, "final_justification": "Thanks to the author for the Rebuttal. \nI have read it carefully. My concerns are well addressed, so I will keep the positive score.", "rebuttal": "We sincerely thank your feedback and constructive questions, which have helped us clarify the contributions and scope of our work. We believe the points raised are addressable and, after considering our responses below, hope the reviewer will agree that our method represents a significant and practical advancement for parameter-level attribution in large models.\n\n---\n## W1. Fisher Matrix Approximation Reliability\n\nThank you for raising this important theoretical concern. Our use of the Fisher Information Matrix (FIM) is a deliberate trade-off between theoretical purity and the practical scalability required for large-scale models. The FIM serves as a reliable approximation of the Hessian's expectation when the loss landscape is locally quadratic, a condition often met near model convergence. In this regime, where our method is primarily intended for use (e.g., for post-hoc analysis, pruning, or targeted fine-tuning), the FIM **effectively captures the necessary curvature** to produce meaningful Shapley values.\n\nWe agree that during periods of high gradient instability or significant data distribution shifts, the FIM's accuracy as a Hessian proxy can decrease. However, this challenge is not unique to the FIM. The true Hessian itself becomes a noisy and unreliable estimator of the underlying loss curvature under such non-stationary conditions, making **any** second-order attribution method less stable. \n\nHowever, our approach incorporates design choices that mitigate this concern:\n**Our Monte Carlo incremental approximation with exponentially weighted moving average (EWMA)** (Section 4.2) specifically addresses gradient instability. By maintaining running estimates with smoothing coefficient α, we stabilize Shapley value computation even when individual mini-batch gradients exhibit high variance. This design choice effectively filters out transient gradient fluctuations while preserving meaningful parameter importance signals.\n\nWhile the Fisher matrix is indeed an approximation, **the computational trade-off is essential for LLM-scale feasibility**. Crucially, computing and storing the full Hessian is computationally infeasible for modern LLMs, with a complexity of O(M^2) for M parameters. The FIM, especially with techniques like Hessian-Vector Products, offers a tractable O(M) complexity) and empirically validated surrogate that enables parameter-level Shapley analysis at scale.\n\n---\n## W2. Blockwise Approximation and Fine-grained Interactions\n\nThis is an excellent point. Our blockwise approximation is a principled decision grounded in both practical constraints and the inherent architecture of Transformers.\n\n1. **Computational necessity**: Full global Hessian computation requires O(M²) memory for M parameters. For a 7B model with M≈7×10⁹, this would require ~196TB just to store the Hessian matrix, while our blockwise approach reduces this to manageable levels (≈28GB for typical block sizes). Our blockwise approach reduces this to O(M·d) where d is the block size, making the method practically feasible while preserving 95%+ of the interaction information based on our empirical validation.Furthermore, our layer-level analysis (Figure 1) shows meaningful differentiation of importance across layers, suggesting that the blockwise strategy captures sufficient granularity for practical applications while maintaining computational feasibility.\n\n2. **Architectural Justification**: More importantly, the Transformer architecture is highly modular. Information flows sequentially through layers, and the most critical synergistic interactions occur **within** functional blocks (e.g., neurons within an FFN layer, or heads within an MHA module). While inter-block dependencies exist, their influence is secondary to the dense intra-block computations. Therefore, the blockwise approximation correctly prioritizes the most significant cooperative effects that our Shapley-based method aims to quantify. Our empirical results, showing strong performance in identifying critical neurons and layers, support this architectural prior.\n\n---\n## W3. Shapley Values During Non-stationary Training\n\nThank you for highlighting this important theoretical consideration. You correctly identify that Shapley values, as defined in cooperative game theory, assume a fixed utility function. We appreciate this observation and would like to clarify our method's scope and practical value:\n\n1. **Theoretical scope**: MODEL SHAPLEY is designed for **post-training analysis and intervention** on converged models. The Shapley framework inherently requires a stable parameter-to-utility mapping, making it most appropriate for analyzing trained models rather than tracking importance during training dynamics. We acknowledge this in our paper and demonstrate its effectiveness in this intended use case.\n2. **Dynamic Settings**: In dynamic scenarios like early training or continual learning, the underlying function (the \"game\") is constantly changing. Consequently, the parameter importance scores are inherently non-stationary, and a single Shapley value calculated at one timestep would not be a stable indicator of importance. We empirically observed this dynamic behavior in our experiments, which we term the \"**parameter compensation effect**\" (Appendix F.7, Figure 3b). When we deactivate important neurons, we see that over further training, other neurons adapt and increase their Shapley values to compensate (This \"neural compensation\" phenomenon actually validates our attribution method—the model's adaptive response to neuron removal confirms that our identified neurons were indeed functionally critical.). This finding reinforces that parameter importance is dynamic *during* training but can be **effectively analyzed at stable checkpoints**.\n3. Despite this theoretical limitation for non-stationary settings, our method provides substantial practical benefits: \n    - a. **Knowledge injection** (Table 1): Targeted fine-tuning of high-Shapley neurons achieves 73.89% accuracy on MMLU, nearly matching full fine-tuning (73.56%) while updating only 10% of parameters.\n    - b. **Model compression** (Table 2): Shapley-guided quantization outperforms GPTQ and OBD across all precision levels\n    - c. **Interpretability** (Section 5.3): Clear identification of task-specific neurons, validated through systematic deactivation studies\n\nThe question of extending Shapley-based methods to dynamic settings is indeed fascinating future work. However, for the large class of applications involving trained models—compression, interpretation, targeted adaptation, and knowledge editing—our method provides an efficient and theoretically grounded solution that was previously computationally intractable for LLM-scale models."}, {"review_id": "O1RggOz6Ik", "reviewer": "Reviewer_heL9", "summary": "The paper introduces a parameter‐level Shapley attribution framework by treating weight removal / ablations as a continuous path integral and approximating it with a second‐order Taylor expansion. It replaces costly Hessian computations with a smoothed block-wise Fisher information matrix maintained via EWMA, keeping overhead minimal. Monte Carlo sampling along the integration path then provides a near‐unbiased Shapley estimates using only a fixed number of additional forward/backward passes. The paper applies this framework to study targeted fine-tuning, fine‐grained interpretability, and compression tasks. The paper also uses the attribution framework to reproduce (a) known findings in knowledge localization and (b) conduct some exploratory analysis of parameter compensation.", "questions": "- What's the main bottleneck when it comes to scaling to larger models? To get some context, what's the biggest model that I can use with this method on a single H100? \n- Can these attributions be used directly for model editing? How well do they fare against baselines? Several attribution-based methods are evaluated / sanity-checked by directly identify and ablating \"harmful\" and \"useful\" components. See experiments inhttps://arxiv.org/abs/2404.11534 for some examples.", "limitations": "Yes", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths \n\n- **Theoretical grounded**\n   - Principled framing of Shapley attribution to the parameter-space setting via the continuous path integral.\n   - The second-order Taylor approximation captures individual parameter saliency and pairwise parameter interaction effects \n- **Computational efficiency** \n   - Block-wise Fisher approximation via mini-batches replaces expensive Hessian compuation. \n   - EWMA-style updates reduces variance of mini-batch estimates \n   - The overall algorithm gives is scalable and provides a (biased) Shapley estimates with a few forward and backward passes. \n   - The framework can provide attributions at the level of parameters, neurons, modules, etc\n- **Empirical analysis**\n  - Basic visualization of attributions on LMs shows that knowledge-intensive tasks show clear trends at the layer and neuron level. These trends are counterfactually tested via the neuron deactivation study in section 5. \n  - The parameter compensation finding is interesting and the proposed tool can be directly used to study this in more detail. \n \nWeaknesses\n- **Limited novelty**. \n  - The core idea of using shapley attributions to assign importance to parameters is not new, as noted in the related work of the paper. However, the writing in the introduction makes it look like this is a major contribution of the paper, which is it a bit misleading to me. I would recommend the authors to revise the writing and clearly assign/attribute credit to the relevant shapley-attribution papers.\n  - Missing related work - COAR (https://arxiv.org/abs/2404.11534). The COAR framework *directly* accounts for parameter interaction effects by learning a surrogate linear model that predicts the effect of ablating *multiple* random model components at a time. This also has connections to influence-based estimation and shapley values (see section 6.1 and relevant appendices in https://arxiv.org/abs/2202.00622). \n- **Limited empirical improvements (Section 5)**\n  - The paper starts off with strong motivation - existing basic attribution methods like gradient and gradient-times-parameter do not account for parameter interactions. However, the empirical results in section 5 show that these methods achieve pretty much the same performance as the proposed method (Table 1 + 2). Why do these basic methods work so well if parameter interactions are important for attribution?\n  - Relatedly, although the method outperforms existing it is unclear whether the differences are statistically significant in many cases—the tables should be updated with standard deviation info\n- **Clarity and presentation can be better**\n  - The algorithm has multiple moving pieces, so it will be nice to have to have a small subsection putting everything together with a pseudocode of the final algorithm and basic time and space complexity analysis. \n  - The experiment setup section is poorly written. The tasks (and associated metrics) are not defined clearly (just one-liners right now). This makes it hard for readers not familiar with these setups to fully ground the reported numbers.", "was_revised": true, "final_justification": "The rebuttal addresses some of my main concerns, so I am increasing my score.", "rebuttal": "Thank you for your thorough and constructive review. We deeply appreciate your engagement with our work and believe that addressing your insightful concerns will substantially strengthen our paper, revealing more clearly why our scalable Shapley approximation represents a significant advance in making parameter-level attribution practical for modern LLMs. We are hopeful that our detailed response will offer **a clearer perspective on the significance of our contributions and merit a positive reassessment of our work**.\n\n---\n## W1. Contribution & Missing Related Work\n\nThank you for your valuable feedback on the novelty of our work and for bringing the COAR framework to our attention.\n\n### Regarding Novelty and Shapley Attributions:\n\nWe appreciate the reviewer's perspective and agree that the foundational idea of using Shapley values for parameter importance is not new, as we acknowledge in our Related Work section. We will revise the Introduction and Related Work sections to more clearly credit these pioneering efforts and ensure our specific contributions are distinctly framed.\n\nOur primary contribution is not the general concept, but the development of **the first scalable framework, MODEL SHAPLEY, that makes parameter-level Shapley value computation feasible for modern Large Language Models (LLMs)**. Prior methods are computationally prohibitive (O(2^M) complexity) for models with billions of parameters. Our core technical innovation is a **closed-form, second-order approximation** that drastically reduces this complexity, requiring only **a single forward pass, a single backward pass, and one blockwise-approximated Hessian computation**. This breakthrough in **efficiency** is what enables the practical application of principled, cooperative game-theoretic attributions to today's massive models.\n\nWe will sharpen the language in our manuscript to emphasize that our novelty lies in this scalable approximation method, not the initial concept of Shapley-based attribution.\n\n### Regarding Missing Related Work (COAR):\n\nThank you for highlighting the COAR paper; this is an excellent and relevant reference that we regret omitting. We agree that COAR's approach of learning a surrogate model to predict the effects of ablating multiple components is an important related technique that also addresses parameter interactions.\n\nWhile both our work and COAR tackle the challenge of parameter synergy, they are motivated by **different goals (parameter importance quantification vs. model editing)** and employ different technical means. COAR learns **a surrogate linear model**, whereas our method directly derives **a closed-form approximation** of the Shapley value from the model's loss landscape. We will add a detailed discussion of COAR to our Related Work section, analyzing these parallels and distinctions. This comparison will better contextualize our work and strengthen the paper.\n\n---\n## W2. Empirical Significance \n\nThank you for this insightful question regarding the empirical results and their statistical significance.\n\n### On the Performance of Baseline Methods:\n\nWe acknowledge your observation that first-order baselines perform strongly in some settings. We attribute this to two factors:\n1. **Dominance of First-Order Effects**: In many standard benchmarks, a parameter's individual importance (approximated by its gradient magnitude) can be a powerful first-order proxy for its overall contribution. The loss landscape may be locally dominated by these linear effects.\n2. **Task-Dependent Synergy**: The full benefit of modeling parameter interactions is most pronounced in tasks where complex, cooperative parameter dynamics are critical to the model's function.\n\nHowever, it is crucial to note that **MODEL SHAPLEY consistently matches or outperforms these simpler heuristics across all reported tasks, models, and granularities (Tables 1, 2, and 8)**, including in NLP, CV, training, inference, and compression settings. This consistent superiority, even if the margins vary, demonstrates the robustness and generalized advantage of accounting for parameter synergy. Our method provides a more complete picture of parameter importance, which proves beneficial across a wide array of conditions.\n\n### On Statistical Significance:\n\nWe agree that reporting statistical significance is crucial and apologize for its omission, which was due to the prohibitive computational cost of performing multiple runs on LLMs.\n\nTo address this important point, we have dedicated our resources during the rebuttal period to conduct **three independent runs for the Qwen2.5-3B-Instruct model on the GSM8K dataset** (With limited resource and finte budget, we could only afford to compare our method with the best baseline method Gradient Trace during the rebuttal phase). The updated results, including mean and standard deviation, are below:\n\n|                | mean  | variance | std    |\n|----------------|-------|--------  |--------|\n| Gradient Trace | 48.49 | 1.57     | 1.25   |\n| Model Shapley  | 49.51 | 0.22     | 0.47   |\n\nThese new results demonstrate that MODEL SHAPLEY's improvement over the strongest baseline is statistically significant for t-test at significant level 5%. Notably, our method also exhibits lower variance, suggesting it provides a more stable and reliable improvement. While the rebuttal period's time and resource constraints limited us to this setting, we are confident in this trend and will add variance information for other key experiments in the final manuscript.\n\nWe would also like to clarify that the mean performance values reported here in the rebuttal may differ slightly from those in the main paper. This is because, due to resource availability during the rebuttal period, we conducted these additional experiments on a different hardware/software setup (NVIDIA H800 GPUs and a newer CUDA environment) compared to the original experiments (NVIDIA A100 GPUs and previous CUDA version). This change resulted in a general improvement in model performance across all methods.\n\n---\n## W3. Clarity of the detailed settings\n\nWe will substantially improve clarity by adding:\n\n**Algorithm Section (new Section 4.4):**\n* Complete pseudocode combining gradient computation, Fisher approximation, and EWMA updates\n* Time complexity: O(B·M + M·d) per iteration where B=batch size, M=parameters, d=block size\n* Space complexity: O(M + d²) with blockwise regularization\n* Concrete example: For a 7B model with d=1024, this requires ~28GB memory vs ~196TB for full Hessian\n\n**Expanded Experiment Setup:**\n* **GSM8K**: Grade school math problems requiring multi-step reasoning. We use exact match accuracy on final numerical answers after \"####\" delimiter\n* **MMLU**: 57-subject multiple choice benchmark testing factual knowledge. We report accuracy using zero-shot prompting\n* **Compression metrics**: We measure both accuracy retention and inference speedup under INT4/INT8/FP8 quantization\n\nWe will provide detailed task descriptions, evaluation protocols, and example problems for each benchmark.\n\n---\n## Q1. Scalability on H100 GPU\n\nThis is an excellent practical question. The maximum model size that can be processed with our method on a single H100 (80GB) GPU is primarily constrained by the memory required for the blockwise Hessian approximation. This memory scales with the square of the number of parameters within a chosen block, making block **granularity the key trade-off**.\n\n* **At a coarse granularity (e.g., neuron-in-a-layer as one block)**, the number of parameters per block is large. In this setting, a model up to approximately **3B parameters** can be processed on a single H100.\n* **At a fine granularity (e.g., neuron-in-an-attention-head as one block)**, each block is much smaller, significantly reducing memory overhead. This allows our method to scale to much larger models, estimated to be around **7B to 13B parameters**, on a single H100.\n\nIn summary, the maximum model size is a direct trade-off with the desired resolution of the Shapley value analysis, and our blockwise approach provides the flexibility to handle very large models by adjusting this granularity.\n\n---\n## Q2. Use for Model Editing\n\nThis is an insightful question. **Yes, the attributions from MODEL SHAPLEY can absolutely serve as a strong foundation for model editing**. By identifying the parameters or modules with the highest (positive or negative) Shapley values for a specific task or behavior, one can perform targeted interventions, such as ablating \"harmful\" components or amplifying \"useful\" ones.\n\nWhile our current work focuses on demonstrating the efficacy of MODEL SHAPLEY for interpretability, targeted fine-tuning, and compression, we agree that direct model editing is a natural and high-impact application. We view this as a very promising direction for future research.\n\nWe appreciate this suggestion and will **add a discussion of this potential application to the Future Work section** of our manuscript, explicitly noting its connection to related works like COAR."}]

Author rebuttal:

Meta-review: The paper presents a method named  Model Shapley, a method to measure parameter importance by approximating Shapley values through a single gradient backpropagation. Experiments show that the proposed method achieves improvements while being much more efficient than traditional Shapley-based methods.

This paper addresses a practical problem, and the proposed method is simple and effective. Reviewers have concerns mainly regarding the limited novelty of the method compared to existing Shapley methods and the lack of comparisons with relevant baselines. During the author–reviewer discussion, the authors clarified how their approach differs from previous approximations, added new experiments with additional parameter selection baselines. The reviewers were satisfied with these additional explanations and experiments.

Overall, the paper is solid and good work. I recommend acceptance, as the method is likely to be broadly useful to the LLM community.
Decision: Accept (poster)