Efficient Decentralized Multi-task Dataset Valuation via Model Merging

# Efficient Decentralized Multi-task Dataset Valuation via Model Merging
Source: [https://arxiv.org/html/2607.03346](https://arxiv.org/html/2607.03346)
Mohammadsajad Alipour Department of Computer Science Rensselaer Polytechnic Institute Troy, NY 12180, USA alipom@rpi\.edu&Mohammad Mohammadi Amiri Department of Computer Science Rensselaer Polytechnic Institute Troy, NY 12180, USA mamiri@rpi\.edu

###### Abstract

Accurate and efficient dataset valuation is essential for enabling fair and transparent data marketplaces, especially when multiple contributors provide data for training multi\-task models\. Most existing valuation methods, however, are limited to single\-task settings, overlooking scenarios where a buyer aims to optimize performance across multiple downstream tasks\. Moreover, traditional valuation approaches—such as Shapley\-based or retraining\-based methods—are computationally expensive and poorly suited for decentralized environments without a trusted central coordinator and with strict privacy constraints\. We propose DMVM \(Decentralized Multi\-task Valuation via Model Merging\), a novel framework that bypasses retraining and data sharing by leveraging task arithmetic to infer dataset contributions directly from model combinations\. Instead of retraining or sharing raw data, DMVM quantifies how models trained on different datasets combine in parameter space to infer each dataset’s marginal utility across multiple tasks\. This formulation yields a valuation process that is scalable, computationally efficient, and explicitly aligned with multi\-task generalization behavior\. To support decentralized deployment, we introduce a secure aggregation protocol that enables collaborative valuation without revealing individual model parameters or private data\. We also provide theoretical error bounds characterizing the approximation quality of DMVM and validate our framework through comprehensive experiments on computer vision and natural language processing tasks\.

## 1Introduction

Dataset valuation has emerged as a central challenge in the design of modern data marketplaces, where multiple sellers offer proprietary datasets and buyers seek to improve model performance on specific target tasks\. Accurately estimating the value of a seller’s data prior to purchase is vital for fair pricing, informed decision\-making, and sustaining trust in the marketplace\. This estimation aims to quantify the marginal utility a dataset provides to a learning algorithm for a specific task, guiding buyers toward high\-impact acquisitions and incentivizing high\-quality data contributions from sellers\.

Prior work has made substantial progress toward formalizing the problem of data valuation\. Game\-theoretic methods such as Data Shapley\[[21](https://arxiv.org/html/2607.03346#bib.bib24)\]offer theoretically principled metrics by averaging marginal contributions across all data subsets\. Influence\-function techniques estimate the effect of training samples on model predictions\[[31](https://arxiv.org/html/2607.03346#bib.bib54),[73](https://arxiv.org/html/2607.03346#bib.bib53)\], while gradient\-based methods \(e\.g\., reinforcement learning or utility\-based proxies\) seek to capture a dataset’s alignment with target tasks\[[69](https://arxiv.org/html/2607.03346#bib.bib55),[6](https://arxiv.org/html/2607.03346#bib.bib28)\]\. While theoretically compelling, these methods face critical obstacles when applied to real\-world data marketplaces, as discussed next\. A primary barrier to practical deployment is computational overhead: estimating marginal contributions, especially via Shapley\-based methods, typically requires retraining or fine\-tuning models across an exponential number of dataset combinations\. This becomes infeasible at scale, particularly for large models or a large number of sellers\. A second challenge is data privacy\. Most valuation techniques assume centralized access to raw data, an unrealistic expectation in many real\-world marketplaces involving sensitive or regulated datasets, e\.g\., medical imaging or personal data\. In such settings, sellers are unlikely to permit raw data sharing, even for valuation purposes\.

Beyond scalability and privacy, a third and often overlooked limitation is the narrow focus on single\-task valuation\. Most existing methods assess a dataset’s utility with respect to a single predefined task\. However, in realistic settings, buyers often pursue multi\-task learning objectives \(e\.g\., multi\-task vision models performing both detection and segmentation, or multi\-task language models performing sentiment analysis, topic classification, and summarization\)\. Valuation in a multi\-task learning context is inherently more complex because a dataset’s utility may differ across tasks, and its value cannot be captured by a single performance metric\. Despite the growing importance of multi\-task models in practice, there has been little work addressing dataset valuation in this setting, leaving a significant methodological gap\[[2](https://arxiv.org/html/2607.03346#bib.bib60),[75](https://arxiv.org/html/2607.03346#bib.bib62)\]\.

Meanwhile, model merging has recently emerged as a strategy for combining models fine\-tuned on different datasets without requiring access to, or retraining on, combined data\[[25](https://arxiv.org/html/2607.03346#bib.bib25),[63](https://arxiv.org/html/2607.03346#bib.bib68),[41](https://arxiv.org/html/2607.03346#bib.bib79),[62](https://arxiv.org/html/2607.03346#bib.bib40),[3](https://arxiv.org/html/2607.03346#bib.bib47)\]\. Starting from a common base, models are fine\-tuned independently; their parameter updates, often represented as task vectors\[[25](https://arxiv.org/html/2607.03346#bib.bib25)\]are then merged—via averaging, interpolation, or more advanced techniques such as task\-vector trimming or balancing—to approximate the effect of joint training\[[67](https://arxiv.org/html/2607.03346#bib.bib26),[70](https://arxiv.org/html/2607.03346#bib.bib81)\]\. Empirically,Taoet al\.\[[56](https://arxiv.org/html/2607.03346#bib.bib7)\]showed that merging models fine\-tuned on disjoint datasets can closely approximate the performance of a model trained on the combined data\. They further show a strong linear correlation between the accuracy of the merged model and that of the model trained on the full data mixture, indicating that model merging can serve as an efficient and reliable proxy for mixture fine\-tuning or multi\-task learning\. Moreover,Zhouet al\.\[[74](https://arxiv.org/html/2607.03346#bib.bib51)\]provide theoretical support for this intuition by deriving an error bound between the weights of a model trained via multi\-task learning and a model merged with task\-arithmetic\.

In multi\-task setting, the ideal way to assess a dataset’s value is to measure its marginal improvement to a jointly trained multi\-task model; that is, how much performance gain it contributes when all tasks are optimized jointly\. However, such joint training is computationally prohibitive in decentralized marketplaces, as it requires access to all datasets and repeated retraining across combinations of sellers\. To overcome this bottleneck, we draw inspiration from recent advances in model merging, which approximate the effects of joint multi\-task training without the cost of training on combined datasets\. Starting from a shared initialization, models are fine\-tuned independently on individual datasets, and their parameter updates, often represented as task vectors, are combined through arithmetic operations in parameter space\. Empirical and theoretical studies have shown that this process can closely replicate the outcomes of true multi\-task learning\[[35](https://arxiv.org/html/2607.03346#bib.bib50),[25](https://arxiv.org/html/2607.03346#bib.bib25),[74](https://arxiv.org/html/2607.03346#bib.bib51),[43](https://arxiv.org/html/2607.03346#bib.bib59)\], suggesting that model merging can serve as an efficient proxy for evaluating cross\-task contributions\.

Building on this insight, we introduce DMVM \(Decentralized Multi\-task Valuation via Model Merging\), a framework that formalizes multi\-task dataset valuation through the lens of model merging\. DMVM estimates each dataset’s marginal utility by analyzing how its corresponding model influences the aggregated performance across a buyer’s target tasks\. Our formulation enables fully decentralized valuation without raw data exchange, incorporates a privacy\-aware protocol to protect individual models during valuation, and provides a theoretical error bound that characterizes the approximation gap between DMVM\-based valuation and the ground\-truth multi\-task valuation\. We validate our framework through simulations of a decentralized data marketplace, showing consistent value estimates across vision and language benchmarks\. Our main contributions are:

- •Efficient and decentralized valuation framework\.We introduce DMVM, a novel approach for multi\-task dataset valuation through model merging\. It operates without retraining or centralized data access, enabling a fully decentralized deployment, without a trusted broker or third party\.
- •Privacy\-aware protocol\.DMVM incorporates a secure aggregation protocol enabling collaborative valuation without revealing raw data or individual models, ensuring compatibility with privacy\-sensitive applications\.
- •Theoretical guarantees\.We derive error bounds that characterize the approximation quality of DMVM relative to ground\-truth multi\-task marginal contributions\.
- •Empirical evaluation\.We demonstrate stable and reliable valuation performance of DMVM across representative vision and language benchmarks\.

By bridging the domains of dataset valuation and model merging, DMVM establishes a scalable, privacy\-aware, and theoretically grounded foundation for dataset valuation in multi\-task marketplaces\.

## 2Related Work

The proliferation of data\-driven machine learning has created a need to quantify the economic value of data, both to fairly compensate contributors and to efficiently acquire new, high\-quality data\. This research area can be broadly categorized into methods for valuing individual data points, techniques for valuing entire datasets, and the design and operation of data marketplaces\.

Data Valuation: Valuing Individual Points\.A dominant approach for data valuation is the Shapley value\[[51](https://arxiv.org/html/2607.03346#bib.bib82)\], a concept from cooperative game theory that fairly assigns value to each data point based on its average marginal contribution to a machine learning model’s performance across all possible data subsets\[[21](https://arxiv.org/html/2607.03346#bib.bib24)\]\. This method is axiomatically fair, satisfying properties such as symmetry and additivity\. The primary drawback of the standard Data Shapley is its exponential computational complexity,O​\(2N\)O\(2^\{N\}\), which is prohibitive for modern\-scale datasets of sizeNN\. To overcome this, several approximations have been developed\[[28](https://arxiv.org/html/2607.03346#bib.bib21)\]\. Sampling\-based methods estimate the Shapley value by sampling random permutations of the data, but they may still require numerous model retrainings\[[21](https://arxiv.org/html/2607.03346#bib.bib24),[40](https://arxiv.org/html/2607.03346#bib.bib22)\]\.Jiaet al\.\[[27](https://arxiv.org/html/2607.03346#bib.bib45)\]proposed an exact and efficient algorithm for computing Shapley values specifically for K\-nearest neighbour models, reducing the complexity toO​\(N​log⁡N\)O\(N\\log\{N\}\)\. Recent works further extend Shapley\-based valuation to graph\-structured data\[[11](https://arxiv.org/html/2607.03346#bib.bib30),[55](https://arxiv.org/html/2607.03346#bib.bib29)\]\.

Dataset Valuation\.While point\-level valuation supports curation and debugging, dataset valuation addresses how much an entire contributor’s dataset adds to utility within a cooperative game among data owners\. This line of work focuses on valuing entire datasets as a single unit, which is distinct from aggregating individual data point values\. The Shapley value can also be applied at the dataset level, measuring the marginal contribution of one seller’s entire dataset to a coalition of other sellers\. To address the computational cost, DU\-Shapley\[[19](https://arxiv.org/html/2607.03346#bib.bib8)\]was proposed as an efficient proxy specifically for dataset valuation, using a discrete uniform approximation that is more efficient than generic Monte Carlo sampling\. Another line of work is task\-agnostic dataset valuation methods that assess data without relying on a specific model or validation set, avoiding issues such as unavailable validation data\. Instead, they estimate the value based on the dataset’s intrinsic distributional properties\. The PriArTa framework evaluates sellers by computing the Wasserstein distance between augmentation\-invariant latent distributions of the buyer’s and seller’s datasets\[[26](https://arxiv.org/html/2607.03346#bib.bib9)\]\. Another related study introduces a fundamental valuation framework that characterizes datasets through two complementary metrics: relevance, capturing statistical similarity, and diversity, capturing statistical difference\[[4](https://arxiv.org/html/2607.03346#bib.bib10)\]\. These are estimated by comparing the dataset’s second\-moment properties, which can be done privately by sharing only principal components\. This methodology was later extended to graph data\[[18](https://arxiv.org/html/2607.03346#bib.bib23)\]\.

Data Marketplace\.A growing line of research explores the design and economic principles of data marketplaces\. Building such markets is challenging because data possess unique characteristics: they can be replicated at near\-zero cost, their value arises from complex interactions among datasets, and their usefulness for a given task is difficult to evaluate beforehand\[[37](https://arxiv.org/html/2607.03346#bib.bib44),[57](https://arxiv.org/html/2607.03346#bib.bib27),[10](https://arxiv.org/html/2607.03346#bib.bib33),[7](https://arxiv.org/html/2607.03346#bib.bib32),[1](https://arxiv.org/html/2607.03346#bib.bib31),[38](https://arxiv.org/html/2607.03346#bib.bib41)\]\. To address this, some frameworks propose a model\-based pricing approach, where the marketplace sells machine learning model instances rather than the raw data itself\[[7](https://arxiv.org/html/2607.03346#bib.bib32)\]\. The Dealer marketplace framework\[[37](https://arxiv.org/html/2607.03346#bib.bib44)\]formalizes an end\-to\-end system where data owners’ compensation is a function of both their Shapley value and their privacy sensitivity\. These mechanisms, however, often rely on a trusted broker to access all datasets and perform the valuation\. This introduces a fundamental dilemma: data owners must share their data to have it valued, but a buyer no longer needs to purchase data they have already observed\. Privatizing these computations via multi\-party computation \(MPC\)\-style protocols closes the loop for deployable and trustable data markets\[[57](https://arxiv.org/html/2607.03346#bib.bib27),[45](https://arxiv.org/html/2607.03346#bib.bib48)\]\.

Model Merging\.Model merging builds on the observation that neural network parameters often exhibit approximately linear structure across tasks\.Ilharcoet al\.\[[25](https://arxiv.org/html/2607.03346#bib.bib25)\]first introduced task arithmetic, showing that subtracting a pre\-trained model from its fine\-tuned counterpart yields a task vector whose addition or subtraction can edit model behavior, enabling multi\-task composition or unlearning without retraining\.Liet al\.\[[35](https://arxiv.org/html/2607.03346#bib.bib50)\]provided the first theoretical justification for task arithmetic, proving that the effectiveness of adding or subtracting task vectors depends on how related the underlying tasks are and showing that properly scaled combinations can generalize to unseen tasks\. In an empirical study,Taoet al\.\[[56](https://arxiv.org/html/2607.03346#bib.bib7)\]proposed Merge to Mix, which leverages merging as a surrogate for fine\-tuning on dataset mixtures\. By correlating the performance of merged models with that of mixture\-trained ones, they enabled efficient and data\-free selection of optimal dataset combinations\.Zhouet al\.\[[74](https://arxiv.org/html/2607.03346#bib.bib51)\]reframed task\-arithmetic as an efficient form of approximate multi\-task learning\. They showed that merging task vectors is equivalent to performing one step of gradient descent on a combined multi\-task objective, and derived a second\-order bound quantifying the difference between the merged model and the model obtained from full multi\-task training\. This analysis formally explains why task arithmetic can closely approximate joint optimization\. Despite these advances, existing model merging studies have not been explored in the context of dataset valuation, nor leveraged to quantify marginal data contributions in decentralized multi\-task settings\.

![Refer to caption](https://arxiv.org/html/2607.03346v1/neur.drawio.png)Figure 1:Overview of the DMVM framework for decentralized multi\-task dataset valuation\. Each seller trains a local model on its private dataset, and using a secure protocol, they produce a protected merged model for each coalition\. The buyer evaluates the merged models on its target validation tasks to estimate seller contributions via Shapley values, without accessing sellers’ private data or individual models\.
## 3Preliminaries

In a data marketplace, multiple sellers offer datasets of varying relevance and quality, and a buyer seeks to assess the contribution of each dataset to the performance of a learned model\. This process, known as*dataset valuation*, provides a quantitative basis for fair pricing and informed data acquisition\. A principled valuation metric should capture the*marginal utility*of each dataset; that is, how much additional benefit it brings when combined with others\. We consider a data marketplace consisting of a set ofnndata sellers, where each selleriiholds a private dataset𝒟i\\mathcal\{D\}\_\{i\}, and the collection of all datasets is denoted by𝒟=\{𝒟1,…,𝒟n\}\\mathcal\{D\}=\\\{\\mathcal\{D\}\_\{1\},\\dots,\\mathcal\{D\}\_\{n\}\\\}\. These datasets are not accessible to the buyer prior to purchase due to privacy or commercial constraints\. The buyer is interested in a set of downstream tasks𝒯\\mathcal\{T\}, where each taskt∈𝒯t\\in\\mathcal\{T\}has a validation set𝒟tval\\mathcal\{D\}^\{\\text\{val\}\}\_\{t\}\. The buyer’s objective is to train a model𝜽\\bm\{\\theta\}that generalizes well across all tasks in𝒯\\mathcal\{T\}\.

If the buyer had access to the datasets of a subset of sellers𝒮⊆\[n\]:=\{1,…,n\}\\mathcal\{S\}\\subseteq\[n\]:=\\\{1,\.\.\.,n\\\}, a model could be trained via multi\-task learning \(MTL\) as𝜽𝒮MTL=arg⁡min𝜽​∑i∈𝒮ℒ​\(𝜽,𝒟i\),\\bm\{\\theta\}^\{\\text\{MTL\}\}\_\{\\mathcal\{S\}\}=\\arg\\min\_\{\\bm\{\\theta\}\}\\sum\_\{i\\in\\mathcal\{S\}\}\\mathcal\{L\}\(\{\\bm\{\\theta\}\},\\mathcal\{D\}\_\{i\}\),whereℒ\\mathcal\{L\}denotes the loss function\. The corresponding utility of the subset𝒮\\mathcal\{S\}is defined as the average performance of the resultant model across all downstream tasks:

u​\(𝒮\)=1\|𝒯\|​∑t∈𝒯Perft​\(𝜽𝒮MTL,𝒟tval\),\\displaystyle u\(\\mathcal\{S\}\)=\\frac\{1\}\{\|\\mathcal\{T\}\|\}\\sum\_\{t\\in\\mathcal\{T\}\}\\mathrm\{Perf\}\_\{t\}\(\\bm\{\\theta\}^\{\\text\{MTL\}\}\_\{\\mathcal\{S\}\},\\mathcal\{D\}^\{\\text\{val\}\}\_\{t\}\),\(1\)wherePerft\\mathrm\{Perf\}\_\{t\}denotes a task\-specific evaluation metric \(e\.g\., accuracy or negative loss\) on tasktt\. Alternatively, utility can be expressed in terms of validation loss asu​\(𝒮\)=−∑t∈𝒯ℒ​\(𝜽𝒮MTL,𝒟tval\)\.u\(\\mathcal\{S\}\)=\-\\sum\_\{t\\in\\mathcal\{T\}\}\\mathcal\{L\}\(\\bm\{\\theta\}^\{\\text\{MTL\}\}\_\{\\mathcal\{S\}\},\\mathcal\{D\}^\{\\text\{val\}\}\_\{t\}\)\.The marginal contribution of selleriito a coalition𝒮\\mathcal\{S\}of other sellers is given byu​\(𝒮∪\{i\}\)−u​\(𝒮\),u\(\\mathcal\{S\}\\cup\\\{i\\\}\)\-u\(\\mathcal\{S\}\),which quantifies the added utility from including dataset𝒟i\\mathcal\{D\}\_\{i\}when training jointly with datasets of the sellers in𝒮\\mathcal\{S\}\. The Shapley value\[[51](https://arxiv.org/html/2607.03346#bib.bib82)\]provides a unique and principled method for aggregating marginal contributions across all possible coalitions\. For sellerii, it is defined as:

ϕi=1n​∑𝒮⊆\[n\]∖\{i\}u​\(𝒮∪\{i\}\)−u​\(𝒮\)\(n−1\|𝒮\|\),\\displaystyle\\phi\_\{i\}=\\frac\{1\}\{n\}\\sum\_\{\\mathcal\{S\}\\subseteq\[n\]\\setminus\\\{i\\\}\}\\frac\{u\(\\mathcal\{S\}\\cup\\\{i\\\}\)\-u\(\\mathcal\{S\}\)\}\{\\binom\{n\-1\}\{\|\\mathcal\{S\}\|\}\},\(2\)where the summation averages sellerii’s marginal contribution across all subsets𝒮\\mathcal\{S\}that excludesii, weighted by their cardinality\. This formulation satisfies key fairness axioms, including symmetry, linearity, and additivity, and is therefore widely regarded as the*ground\-truth reference*for data valuation\[[21](https://arxiv.org/html/2607.03346#bib.bib24)\]\. Despite its strong theoretical foundations, computing the Shapley valueϕi\\phi\_\{i\}directly is computationally intractable in practice\. It requires training models on all2n−12^\{n\-1\}possible subsets𝒮⊆\[n\]\\\{i\}\\mathcal\{S\}\\subseteq\[n\]\\backslash\\\{i\\\}to evaluateu​\(𝒮\)u\(\\mathcal\{S\}\), which becomes exponentially expensive asnngrows\. Moreover, in real marketplaces, buyers do not have access to the raw datasets of sellers prior to purchase\. Therefore, the utility functionu​\(𝒮\)u\(\\mathcal\{S\}\)cannot be computed without violating confidentiality\. These challenges motivate the development of scalable and privacy\-aware alternatives to the classical Shapley formulation\. In the following section, we propose DMVM, a framework designed to overcome both computational and privacy limitations while maintaining theoretical faithfulness to multi\-task marginal contributions\.

## 4Methodology

This section introduces our proposed framework for decentralized dataset valuation\. Section[4\.1](https://arxiv.org/html/2607.03346#S4.SS1)presents an efficient proxy for approximating the utility functionu​\(⋅\)u\(\\cdot\)without requiring model training\. Section[4\.2](https://arxiv.org/html/2607.03346#S4.SS2)then introduces a privacy\-aware protocol that enables secure valuation without exposing individual models\.

### 4\.1Model Merging as a Proxy for Utility

We assume that all sellers either adopt a common pretrained model𝜽pre\\bm\{\\theta\}^\{\\text\{pre\}\}provided by the buyer or agree on a shared architecture\. Each selleriilocally fine\-tunes this model on their private dataset𝒟i\\mathcal\{D\}\_\{i\}, yielding a personalized model𝜽i\\bm\{\\theta\}\_\{i\}\. To estimate the utility of any seller subset𝒮⊆\[n\]\\mathcal\{S\}\\subseteq\[n\], the buyer merges the corresponding fine\-tuned models\{𝜽i:i∈𝒮\}\\\{\\bm\{\\theta\}\_\{i\}:i\\in\\mathcal\{S\}\\\}into a single model𝜽𝒮merge\\bm\{\\theta\}^\{\\text\{merge\}\}\_\{\\mathcal\{S\}\}using a merging strategy, specifically,*task arithmetic*\. For brevity, we use𝜽\\bm\{\\theta\}to denote full model parameters and their associated weight matrices\. Building on recent theoretical and empirical findings\[[56](https://arxiv.org/html/2607.03346#bib.bib7),[74](https://arxiv.org/html/2607.03346#bib.bib51)\], we approximate the true utilityu​\(𝒮\)u\(\\mathcal\{S\}\)with the proxyu^​\(⋅\)\\hat\{u\}\(\\cdot\)defined asu^​\(𝒮\)=1\|𝒯\|​∑t∈𝒯Perft​\(𝜽𝒮merge,𝒟tval\),\\hat\{u\}\(\\mathcal\{S\}\)=\\frac\{1\}\{\|\\mathcal\{T\}\|\}\\sum\_\{t\\in\\mathcal\{T\}\}\\mathrm\{Perf\}\_\{t\}\\\!\\left\(\\bm\{\\theta\}^\{\\text\{merge\}\}\_\{\\mathcal\{S\}\},\\mathcal\{D\}^\{\\text\{val\}\}\_\{t\}\\right\),or equivalently, in terms of loss asu^​\(𝒮\)=−∑t∈𝒯ℒ​\(𝜽𝒮merge,𝒟tval\)\.\\hat\{u\}\(\\mathcal\{S\}\)=\-\\sum\_\{t\\in\\mathcal\{T\}\}\\mathcal\{L\}\(\\bm\{\\theta\}^\{\\text\{merge\}\}\_\{\\mathcal\{S\}\},\\mathcal\{D\}^\{\\text\{val\}\}\_\{t\}\)\.The merged model𝜽𝒮merge\\bm\{\\theta\}^\{\\text\{merge\}\}\_\{\\mathcal\{S\}\}is obtained as𝜽𝒮merge=𝜽pre\+α​∑i∈𝒮\(𝜽i−𝜽pre\),\\bm\{\\theta\}^\{\\text\{merge\}\}\_\{\\mathcal\{S\}\}=\\bm\{\\theta\}^\{\\text\{pre\}\}\+\\alpha\\sum\_\{i\\in\\mathcal\{S\}\}\(\\bm\{\\theta\}\_\{i\}\-\\bm\{\\theta\}^\{\\text\{pre\}\}\),whereα\\alphais a scaling hyperparameter\[[25](https://arxiv.org/html/2607.03346#bib.bib25)\]and\(𝜽i−𝜽pre\)\(\\bm\{\\theta\}\_\{i\}\-\\bm\{\\theta\}^\{\\text\{pre\}\}\)denotes the task vector associated with sellerii\. Using this utility proxy, we approximate the Shapley valueϕi\\phi\_\{i\}of each selleriias:

ϕ^i=1n​∑𝒮⊆\[n\]∖\{i\}u^​\(𝒮∪\{i\}\)−u^​\(𝒮\)\(n−1\|𝒮\|\)\.\\displaystyle\\hat\{\\phi\}\_\{i\}=\\frac\{1\}\{n\}\\sum\_\{\\mathcal\{S\}\\subseteq\[n\]\\setminus\\\{i\\\}\}\\frac\{\\hat\{u\}\(\\mathcal\{S\}\\cup\\\{i\\\}\)\-\\hat\{u\}\(\\mathcal\{S\}\)\}\{\\binom\{n\-1\}\{\|\\mathcal\{S\}\|\}\}\.
This formulation enables the buyer to estimate each dataset’s contribution without retraining a model for all2n−12^\{n\-1\}possible subsets\. By approximating utilityu​\(⋅\)u\(\\cdot\)withu^​\(⋅\)\\hat\{u\}\(\\cdot\), the computational burden is reduced from2n2^\{n\}model trainings and inferences to2n2^\{n\}model summations and evaluations, yielding substantial efficiency gains while preserving a close approximation to the ground\-truth Shapley value, as established in the following theorem:

###### Theorem 4\.1\.

Assume that MTL is performed via full\-batch gradient descent, and that the difference between the merged model𝛉𝒮merge\\bm\{\\theta\}^\{\\text\{merge\}\}\_\{\\mathcal\{S\}\}and the MTL\-trained model𝛉𝒮MTL\\bm\{\\theta\}^\{\\text\{MTL\}\}\_\{\\mathcal\{S\}\}is approximated up to second\-order terms \(i\.e\., higher\-order components are negligible, with remainder dominated by𝒪​\(η3\)\\mathcal\{O\}\(\\eta^\{3\}\), whereη\\etais the learning rate for MTL\)\. If the loss functionℒ\\mathcal\{L\}isLL\-Lipschitz continuous, then the approximation error between the true and proxy Shapley values satisfies:

\|ϕi−ϕ^i\|≤\(2​α​n2−2​α​n\+2​n\+α−1\)​L​C,\\displaystyle\|\\phi\_\{i\}\-\\hat\{\\phi\}\_\{i\}\|\\leq\(2\\alpha n^\{2\}\-2\\alpha n\+2n\+\\alpha\-1\)\\,LC,whereC=\(h\+22\)​Hmax​GmaxC=\\binom\{h\+2\}\{2\}H\_\{\\max\}G\_\{\\max\}\. Here,hhdenotes the number of optimization steps \(i\.e\., epochs\), andGmaxG\_\{\\max\}andHmaxH\_\{\\max\}are upper bounds on the gradient and Hessian norms, respectively\.

This theorem establishes a formal link between MTL and task\-arithmetic as the model merging method, in the context of Shapley\-based valuation \(the proof is provided in Appendix[A\.4](https://arxiv.org/html/2607.03346#A1.SS4)\)\. It shows that, under mild assumptions, the error between DMVM and ground\-truth valuation, grows quadratically in the number of sellers and linearly inα\\alpha\. Moreover, settingα=1n\\alpha=\\frac\{1\}\{n\}makes the error grow linearly withnn\. However, this comes at the cost of restricting the performance of the merging algorithm, sinceα\\alphais typically treated as a tunable hyperparameter, e\.g\., selected via binary search\. Furthermore, even the2n2^\{n\}summations and evaluations can be replaced by an arbitrary finite number of Monte Carlo \(MC\) samples while retaining a controlled approximation error, as established in the following theorem:

###### Theorem 4\.2\.

Letϕ¯i\(m\)\\bar\{\\phi\}\_\{i\}^\{\(m\)\}denote the MC estimator ofϕ^i\\hat\{\\phi\}\_\{i\}obtained frommmindependent samples of the marginal contribution terms in the Shapley expansion\. Assume each sampled marginal contribution lies in an interval of width at mostRR\. Under the assumptions of Theorem[4\.1](https://arxiv.org/html/2607.03346#S4.Thmtheorem1), for anyδ∈\(0,1\)\\delta\\in\(0,1\), with probability at least1−δ1\-\\delta, we have:

\|ϕi−ϕ¯i\(m\)\|≤\(2​α​n2−2​α​n\+2​n\+α−1\)​L​C\+R22​m​log⁡\(2δ\)\.\\displaystyle\|\\phi\_\{i\}\-\\bar\{\\phi\}\_\{i\}^\{\(m\)\}\|\\leq\(2\\alpha n^\{2\}\-2\\alpha n\+2n\+\\alpha\-1\)\\,LC\+\\sqrt\{\\frac\{R^\{2\}\}\{2m\}\\log\\\!\\left\(\\frac\{2\}\{\\delta\}\\right\)\}\.

Theorem[4\.2](https://arxiv.org/html/2607.03346#S4.Thmtheorem2)shows that DMVM can admit a further sampling\-based acceleration beyond the deterministic approximation in Theorem[4\.1](https://arxiv.org/html/2607.03346#S4.Thmtheorem1)\. In particular, the total error decomposes into a deterministic merging error and a statistical sampling error that decays as𝒪​\(1/m\)\\mathcal\{O\}\(1/\\sqrt\{m\}\)\. Proof for this theorem is provided in Appendix[A\.6](https://arxiv.org/html/2607.03346#A1.SS6)\. Finally, although the utility proxyu^​\(⋅\)\\hat\{u\}\(\\cdot\)avoids raw data sharing, computing the merged model𝜽𝒮merge\\bm\{\\theta\}^\{\\text\{merge\}\}\_\{\\mathcal\{S\}\}still requires access to the individual fine\-tuned models\{𝜽i\}i∈\[n\]\\\{\\bm\{\\theta\}\_\{i\}\\\}\_\{i\\in\[n\]\}\. To prevent model leakage during this process, we introduce a privacy\-aware protocol in the next section\.

### 4\.2Privacy\-Aware Merging via Secure Aggregation

While the utility proxyu^​\(⋅\)\\hat\{u\}\(\\cdot\)enables efficient valuation, it still requires access to the individual fine\-tuned models\{𝜽i\}i∈\[n\]\\\{\\bm\{\\theta\}\_\{i\}\\\}\_\{i\\in\[n\]\}for computing the merged model𝜽𝒮merge\\bm\{\\theta\}^\{\\text\{merge\}\}\_\{\\mathcal\{S\}\}\. Directly sharing these models would violate seller privacy and undermine the decentralized design of the marketplace\. To address this, we design a*two\-fold privacy\-aware protocol*that integrates secure MPC with an*unmergeable transformation*mechanism inspired by recent advances in model obfuscation\[[30](https://arxiv.org/html/2607.03346#bib.bib11),[61](https://arxiv.org/html/2607.03346#bib.bib12),[8](https://arxiv.org/html/2607.03346#bib.bib65)\]\.

Overview\.Our protocol incorporates two components: \(1\) a*secure aggregation*mechanism that allows the buyer to obtain only the aggregated merged model𝜽𝒮merge\\bm\{\\theta\}^\{\\text\{merge\}\}\_\{\\mathcal\{S\}\}without revealing any individual model𝜽i\\bm\{\\theta\}\_\{i\}; and \(2\) an*unmergeable transformation*step that prevents an adversarial buyer from reconstructing any seller’s model via algebraic manipulation of overlapping subsets of the merged models\. We first describe the secure aggregation process, then extend it to incorporate the unmergeable mechanism\.

Secure Aggregation\.For brevity and simplicity, we setα=1\|𝒮\|\\alpha=\\frac\{1\}\{\|\\mathcal\{S\}\|\}; we have provided the algorithm for any arbitraryα\\alphavalue in Appendix[A\.1](https://arxiv.org/html/2607.03346#A1.SS1)\. For any subset𝒮⊆\[n\]\\mathcal\{S\}\\subseteq\[n\]with\|𝒮\|≥2\|\\mathcal\{S\}\|\\geq 2, each selleri∈𝒮i\\in\\mathcal\{S\}samples a random mask matrix𝒎i\\bm\{m\}\_\{i\}of the same dimension as their model parameters\. Each seller shares𝒎i\{\\bm\{m\}\}\_\{i\}with all other sellers in𝒮\\mathcal\{S\}and computes a masked version of their contribution as𝒆i=𝜽i\|𝒮\|−𝒎i,\\bm\{e\}\_\{i\}=\\frac\{\\bm\{\\theta\}\_\{i\}\}\{\|\\mathcal\{S\}\|\}\-\{\\bm\{m\}\}\_\{i\},which is sent to the buyer\. Meanwhile, all sellers compute the aggregate mask𝒎=∑i∈𝒮𝒎i\{\\bm\{m\}\}=\\sum\_\{i\\in\\mathcal\{S\}\}\{\\bm\{m\}\}\_\{i\}and send it to the buyer, who verifies the consistency across submissions\. The buyer then reconstructs the merged model as𝜽𝒮merge=𝒎\+∑i∈𝒮𝒆i=∑i∈𝒮𝒎i\+∑i∈𝒮\(𝜽i\|𝒮\|−𝒎i\)=1\|𝒮\|​∑i∈𝒮𝜽i\.\\bm\{\\theta\}^\{\\text\{merge\}\}\_\{\\mathcal\{S\}\}=\{\\bm\{m\}\}\+\\sum\_\{i\\in\\mathcal\{S\}\}\\bm\{e\}\_\{i\}=\\sum\_\{i\\in\\mathcal\{S\}\}\{\\bm\{m\}\}\_\{i\}\+\\sum\_\{i\\in\\mathcal\{S\}\}\\\!\\left\(\\frac\{\\bm\{\\theta\}\_\{i\}\}\{\|\\mathcal\{S\}\|\}\-\{\\bm\{m\}\}\_\{i\}\\right\)=\\frac\{1\}\{\|\\mathcal\{S\}\|\}\\\!\\sum\_\{i\\in\\mathcal\{S\}\}\\bm\{\\theta\}\_\{i\}\.This guarantees that the buyer receives only the aggregated model𝜽𝒮merge\\bm\{\\theta\}^\{\\text\{merge\}\}\_\{\\mathcal\{S\}\}, not individual𝜽i\\bm\{\\theta\}\_\{i\}’s\. Similarly, since only random masks are shared among sellers, no seller gains access to others’ models\.

Unmergeable transformation\.While the above protocol protects individual models, a malicious buyer could still recover a seller’s model by subtracting two merged models from overlapping subsets \(e\.g\.,\{θ1,θ2,θ3\}\\\{\\theta\_\{1\},\\theta\_\{2\},\\theta\_\{3\}\\\}and\{θ1,θ2\}\\\{\\theta\_\{1\},\\theta\_\{2\}\\\}\)\. To prevent this, we introduce a transformation that preserves functional behavior for inference but disrupts linear relationships among models\. We instantiate this for two common modules: multi\-layer perceptron \(MLP\) blocks and consecutive linear projections\.

Consider two consecutive layers of an MLP:MLP​\(x\)=𝑾\(2\)​σ​\(𝑾\(1\)​𝒙\+𝒃\(1\)\)\+𝒃\(2\)\\text\{MLP\}\(x\)=\\bm\{W\}^\{\(2\)\}\\sigma\(\\bm\{W\}^\{\(1\)\}\\bm\{x\}\+\\bm\{b\}^\{\(1\)\}\)\+\\bm\{b\}^\{\(2\)\}, whereσ​\(⋅\)\\sigma\(\\cdot\)is an element\-wise nonlinearity\. For weight matrices𝑾i\(1\)\\bm\{W\}^\{\(1\)\}\_\{i\}and𝑾i\(2\)\\bm\{W\}^\{\(2\)\}\_\{i\}held by sellerii, the goal is for the buyer to recover∑i∈𝒮𝑾i\(1\)\\sum\_\{i\\in\\mathcal\{S\}\}\\bm\{W\}^\{\(1\)\}\_\{i\}and∑i∈𝒮𝑾i\(2\)\\sum\_\{i\\in\\mathcal\{S\}\}\\bm\{W\}^\{\(2\)\}\_\{i\}without accessing individual weights\. To achieve this, all sellers in𝒮\\mathcal\{S\}jointly agree on a random permutation matrix𝑷𝒮\\bm\{P\}\_\{\\mathcal\{S\}\}\(orthogonal with one “1” per row/column and “0” elsewhere, satisfying𝑷𝒮−1=𝑷𝒮⊤\\bm\{P\}\_\{\\mathcal\{S\}\}^\{\-1\}=\\bm\{P\}\_\{\\mathcal\{S\}\}^\{\\top\}\)\. For mask matrices𝒎i\(1\)\{\\bm\{m\}\}^\{\(1\)\}\_\{i\}and𝒎i\(2\)\{\\bm\{m\}\}^\{\(2\)\}\_\{i\}, instead of sending𝑾i\(1\)\|𝒮\|−𝒎i\(1\)\\frac\{\\bm\{W\}^\{\(1\)\}\_\{i\}\}\{\|\\mathcal\{S\}\|\}\-\{\\bm\{m\}\}^\{\(1\)\}\_\{i\}and𝑾i\(2\)\|𝒮\|−𝒎i\(2\)\\frac\{\\bm\{W\}^\{\(2\)\}\_\{i\}\}\{\|\\mathcal\{S\}\|\}\-\{\\bm\{m\}\}^\{\(2\)\}\_\{i\}, each selleriitransmits:

𝑷𝒮​𝑾i\(1\)\|𝒮\|−𝒎i\(1\),𝑾i\(2\)​𝑷𝒮⊤\|𝒮\|−𝒎i\(2\)\.\\displaystyle\\frac\{\\bm\{P\}\_\{\\mathcal\{S\}\}\\bm\{W\}^\{\(1\)\}\_\{i\}\}\{\{\|\\mathcal\{S\}\|\}\}\-\{\\bm\{m\}\}^\{\(1\)\}\_\{i\},\\quad\\frac\{\\bm\{W\}^\{\(2\)\}\_\{i\}\\bm\{P\}\_\{\\mathcal\{S\}\}^\{\\top\}\}\{\{\|\\mathcal\{S\}\|\}\}\-\{\\bm\{m\}\}^\{\(2\)\}\_\{i\}\.
Having received∑i∈𝒮𝒎i\(1\)\\sum\_\{i\\in\\mathcal\{S\}\}\\bm\{m\}^\{\(1\)\}\_\{i\}and∑i∈𝒮𝒎i\(2\)\\sum\_\{i\\in\\mathcal\{S\}\}\\bm\{m\}^\{\(2\)\}\_\{i\}from the sellers, the buyer can only reconstruct∑i∈𝒮𝑷𝒮​𝑾i\(1\)\|𝒮\|\\frac\{\\sum\_\{i\\in\\mathcal\{S\}\}\\bm\{P\}\_\{\\mathcal\{S\}\}\\bm\{W\}^\{\(1\)\}\_\{i\}\}\{\|\{\\mathcal\{S\}\}\|\}and∑i∈𝒮𝑾i\(2\)​𝑷𝒮⊤\|𝒮\|\\frac\{\\sum\_\{i\\in\\mathcal\{S\}\}\\bm\{W\}^\{\(2\)\}\_\{i\}\\bm\{P\}\_\{\\mathcal\{S\}\}^\{\\top\}\}\{\|\{\\mathcal\{S\}\}\|\}instead of∑i∈𝒮𝑾i\(1\)\|𝒮\|\\frac\{\\sum\_\{i\\in\\mathcal\{S\}\}\\bm\{W\}^\{\(1\)\}\_\{i\}\}\{\|\{\\mathcal\{S\}\}\|\}and∑i∈𝒮𝑾i\(2\)\|𝒮\|\\frac\{\\sum\_\{i\\in\\mathcal\{S\}\}\\bm\{W\}^\{\(2\)\}\_\{i\}\}\{\|\{\\mathcal\{S\}\}\|\}\. Since𝑷𝒮⊤​𝑷𝒮=I\\bm\{P\}\_\{\\mathcal\{S\}\}^\{\\top\}\\bm\{P\}\_\{\\mathcal\{S\}\}=Iandσ​\(⋅\)\\sigma\(\\cdot\)operates elementwise, this transformation leaves the model’s inference behavior unchanged, but destroys the additive structure that would otherwise enable model reconstruction via subtraction\. Thus, any negation or subtraction of merged models yields uninformative parameter combinations\. For consecutive linear layers such as𝑸​𝑲⊤\\bm\{QK\}^\{\\top\}projection in transformer blocks\[[58](https://arxiv.org/html/2607.03346#bib.bib13)\]\(for every matrix𝑸\\bm\{Q\}and key matrix𝑲\\bm\{K\}\), a similar mechanism applies\. Sellers agree on a random scalarρ∈ℝ\\\{0\}\\rho\\in\\mathbb\{R\}\\backslash\\\{0\\\}, and instead of sending𝑸i\|𝒮\|−𝒎i\(1\)\\frac\{\\bm\{Q\}\_\{i\}\}\{\|\\mathcal\{S\}\|\}\-\\bm\{m\}^\{\(1\)\}\_\{i\}and𝑲i\|𝒮\|−𝒎i\(2\)\\frac\{\\bm\{K\}\_\{i\}\}\{\|\\mathcal\{S\}\|\}\-\\bm\{m\}^\{\(2\)\}\_\{i\}, selleriisendsρ​𝑸i\|𝒮\|−𝒎i\(1\)\\frac\{\\rho\\bm\{Q\}\_\{i\}\}\{\|\\mathcal\{S\}\|\}\-\\bm\{m\}^\{\(1\)\}\_\{i\}and𝑲iρ​\|𝒮\|−𝒎i\(2\)\\frac\{\\bm\{K\}\_\{i\}\}\{\\rho\|\\mathcal\{S\}\|\}\-\\bm\{m\}^\{\(2\)\}\_\{i\}to the buyer\. This also preserves the functionality of𝑸​𝑲⊤\\bm\{QK\}^\{\\top\}while making the individual components unmergeable, and therefore negations would result in useless and meaningless weights\.

Table 1:Comparison of our methods and baselines \(mmis the number of subsets that are sampled\)\.Secure Shapley Estimation\.With these mechanisms in place, the buyer never accesses any raw data or individual model weights\. We define the privacy\-aware Shapley estimate of selleriias:

ϕ~i=1n​∑𝒮⊆\[n\]\\\{i\},\|𝒮\|≥2u^​\(𝒮∪\{i\}\)−u^​\(𝒮\)\(n−1\|𝒮\|\)\.\\displaystyle\\tilde\{\\phi\}\_\{i\}=\\frac\{1\}\{n\}\\sum\_\{\\mathcal\{S\}\\subseteq\[n\]\\backslash\\\{i\\\},\|\\mathcal\{S\}\|\\geq 2\}\\frac\{\\hat\{u\}\(\\mathcal\{S\}\\cup\\\{i\\\}\)\-\\hat\{u\}\(\\mathcal\{S\}\)\}\{\\binom\{n\-1\}\{\|\\mathcal\{S\}\|\}\}\.\(3\)whereu^​\(⋅\)\\hat\{u\}\(\\cdot\)is computed via the above secure aggregation mechanism with unmergeable transformation\. In the above approximation, we restrict the cardinality to\|𝒮\|≥2\|\\mathcal\{S\}\|\\geq 2, thereby excluding the case\|𝒮\|=1\|\\mathcal\{S\}\|=1to guarantee that no individual model is shared from a seller to the buyer, which is required for privacy\. The following results bound the deviation betweenϕ~i\\tilde\{\\phi\}\_\{i\}and the true Shapley valueϕi\\phi\_\{i\}with the proof provided in Appendix[A\.5](https://arxiv.org/html/2607.03346#A1.SS5)\.

###### Theorem 4\.3\.

LetRRdenote the maximum marginal contribution\|u​\(𝒮∪\{i\}\)−u​\(𝒮\)\|\|u\(\\mathcal\{S\}\\cup\\\{i\\\}\)\-u\(\\mathcal\{S\}\)\|over all subsets in the Shapley expansion ofϕi\\phi\_\{i\}\. Then,

\|ϕi−ϕ~i\|≤n−2n​\(2​α​n2−2​α​n\+2​n\+α−1\)​L​C\+2​Rn,\\displaystyle\|\\phi\_\{i\}\-\\tilde\{\\phi\}\_\{i\}\|\\leq\\frac\{n\-2\}\{n\}\(2\\alpha n^\{2\}\-2\\alpha n\+2n\+\\alpha\-1\)\\,LC\+\\frac\{2R\}\{n\},whereLLandCCare defined as in Theorem[4\.1](https://arxiv.org/html/2607.03346#S4.Thmtheorem1)\.

Remarks\.Our privacy\-aware protocol ensures that: \(i\) The buyer never accesses individual models or raw data from the sellers; \(ii\) sellers do not receive any information about other sellers’ data or models; \(iii\) buyers cannot reconstruct individual models by differencing merged ones; and \(iv\) dishonest participants can be detected through the protocol’s mask consistency check\. A high\-level overview of DMVM framework is illustrated in Figure[1](https://arxiv.org/html/2607.03346#S2.F1)in Appendix[A\.1](https://arxiv.org/html/2607.03346#A1.SS1)\. The complete procedure for DMVM is presented in Algorithm[1](https://arxiv.org/html/2607.03346#alg1)in Appendix[A\.1](https://arxiv.org/html/2607.03346#A1.SS1)\. In the next section, we evaluate the performance of this framework by simulating data marketplaces in a multi\-task setting\.

## 5Numerical Experiments

This section investigates whether DMVM can serve as a reliable and efficient proxy for dataset valuation in practical multi\-task settings\. Since this is, to our knowledge, the first study on multi\-task dataset valuation, there exists no established benchmark or baseline for comparison\. Consequently, we design two types of realistic and controlled simulation environments that allow us to test whether our method can correctly recover the ground\-truth ranking and values of sellers with respect to their ground\-truth Shapley value\. Here, we discuss the realistic scenario and the results for the controlled setting are provided in Appendix[A\.7](https://arxiv.org/html/2607.03346#A1.SS7), where we construct a spectrum of sellers with varying degrees of relevance to the buyer’s target tasks\.

### 5\.1Experimental Setup\.

We simulate a decentralized marketplace withnnsellers, where each seller owns a dataset𝒟i\\mathcal\{D\}\_\{i\}\. The buyer is interested in a set of target tasks𝒯\\mathcal\{T\}\. For natural language processing \(NLP\) experiments, we consider three task families, each represented by three datasets:

- •Review classification:SST\-2\[[59](https://arxiv.org/html/2607.03346#bib.bib67)\], IMDb\[[39](https://arxiv.org/html/2607.03346#bib.bib57)\], and Yelp\[[71](https://arxiv.org/html/2607.03346#bib.bib58)\]\.
- •Paraphrase identification:MRPC\[[59](https://arxiv.org/html/2607.03346#bib.bib67)\], QQP\[[59](https://arxiv.org/html/2607.03346#bib.bib67)\], and PAWS\[[72](https://arxiv.org/html/2607.03346#bib.bib66)\]\.
- •Question answering:BoolQ\[[14](https://arxiv.org/html/2607.03346#bib.bib64)\], PubMedQA\[[29](https://arxiv.org/html/2607.03346#bib.bib61)\], and StrategyQA\[[20](https://arxiv.org/html/2607.03346#bib.bib63)\]\.

Each dataset is treated as a seller\. Thus, the NLP marketplace contains nine sellers in total\. The buyer’s target tasks can be any subset of these tasks, and the validation utility is computed by averaging performance across the corresponding task\-level validation sets\. This design captures a realistic setting in which multiple sellers provide data for the same task, but from different domains or distributions\. For instance, SST\-2, IMDb, and Yelp all correspond to review or sentiment classification, yet differ in text length, source domain, and linguistic style\. We usegoogle/flan\-t5\-small\[[12](https://arxiv.org/html/2607.03346#bib.bib46)\]as the shared pretrained model\.

Forvisionexperiments, we construct a marketplace using four task families:

- •Digit classification:MNIST\[[34](https://arxiv.org/html/2607.03346#bib.bib43)\], SVHN\[[44](https://arxiv.org/html/2607.03346#bib.bib15)\], USPS\[[24](https://arxiv.org/html/2607.03346#bib.bib36)\], and EMNIST\[[17](https://arxiv.org/html/2607.03346#bib.bib56)\]\.
- •Animal classification:CIFAR\-10 cat/dog\[[33](https://arxiv.org/html/2607.03346#bib.bib42)\]and Oxford\-IIIT Pets cat/dog\[[47](https://arxiv.org/html/2607.03346#bib.bib38)\]\.
- •Scene classification:SUN397\[[66](https://arxiv.org/html/2607.03346#bib.bib20)\]and MIT Indoor67\[[48](https://arxiv.org/html/2607.03346#bib.bib35)\]\.
- •Land\-use classification:EuroSAT\[[23](https://arxiv.org/html/2607.03346#bib.bib17)\]and RESISC45\[[9](https://arxiv.org/html/2607.03346#bib.bib76)\]\.

Again, each dataset corresponds to a seller\. This yields ten vision sellers in total\. The datasets within each task family share the same high\-level prediction objective, but differ in visual domain, image statistics, and label distribution\. For the shared pretrained model, we useopenai/clip\-vit\-base\-patch32\[[50](https://arxiv.org/html/2607.03346#bib.bib37)\]\. More implementation details are provided in Appendix[A\.10](https://arxiv.org/html/2607.03346#A1.SS10)\.

### 5\.2Baselines

Dataset Shapley as ground truth \(GT\)\.Dataset Shapley serves as the oracle reference\. For each sellerii, it computes the exact Shapley value using Eq\. \([2](https://arxiv.org/html/2607.03346#S3.E2)\), where each coalition utility is obtained by training on the corresponding subset of sellers’ datasets\. Since this requires access to all sellers’ raw datasets, we use it only for evaluation\.

MC\-Dataset\-Shapley\[[21](https://arxiv.org/html/2607.03346#bib.bib24)\]\.MC\-Dataset\-Shapley provides MC approximation of Dataset Shapley by samplingmmcoalitions or permutations instead of enumerating all possible coalitions\.

DU\-Shapley\[[19](https://arxiv.org/html/2607.03346#bib.bib8)\]\.DU\-Shapley is a dataset\-level Shapley approximation based on discrete\-uniform sampling\. For sellerii, it estimates the value asψi=1n​∑k=0n−1\[u​\(𝒟\(k\)∪𝒟i\)−u​\(𝒟\(k\)\)\],\\psi\_\{i\}=\\frac\{1\}\{n\}\\sum\_\{k=0\}^\{n\-1\}\\left\[u\\\!\\left\(\\mathcal\{D\}^\{\(k\)\}\\cup\\mathcal\{D\}\_\{i\}\\right\)\-u\\\!\\left\(\\mathcal\{D\}^\{\(k\)\}\\right\)\\right\],where𝒟\(k\)\\mathcal\{D\}^\{\(k\)\}is a dataset that its data samples are sampled uniformly without replacement from𝒟−i=⋃j∈\[n\]∖\{i\}𝒟j,\\mathcal\{D\}\_\{\-i\}=\\bigcup\_\{j\\in\[n\]\\setminus\\\{i\\\}\}\\mathcal\{D\}\_\{j\},with sizek​μ−ik\\mu\_\{\-i\}, andμ−i=1n−1​\|𝒟−i\|\\mu\_\{\-i\}=\\frac\{1\}\{n\-1\}\|\\mathcal\{D\}\_\{\-i\}\|\. \(\|𝒟−i\|\|\\mathcal\{D\}\_\{\-i\}\|is the size of data samples in𝒟−i\\mathcal\{D\}\_\{\-i\}\)\. In our multi\-task adaptation,u​\(⋅\)u\(\\cdot\)is the average validation performance defined in Eq\. \([1](https://arxiv.org/html/2607.03346#S3.E1)\)\.

Our methods\.We report both DMVM and MC\-DMVM as our proposed methods\. In MC\-DMVM, the MC sampling variant of DMVM introduced in Theorem[4\.2](https://arxiv.org/html/2607.03346#S4.Thmtheorem2), seller values are estimated frommmsampled marginal contributions instead of all coalitions\. Table[1](https://arxiv.org/html/2607.03346#S4.T1)illustrates the computational time and privacy aspect of each method\.

### 5\.3Evaluation Metrics

We evaluate each method from two complementary perspectives\. First, we use rank\-correlation metrics used in the literature\[[60](https://arxiv.org/html/2607.03346#bib.bib52),[68](https://arxiv.org/html/2607.03346#bib.bib49),[57](https://arxiv.org/html/2607.03346#bib.bib27),[38](https://arxiv.org/html/2607.03346#bib.bib41)\]to measure whether a method correctly identifies the relative ordering of sellers\. Second, we use mean squared error as an end\-to\-end value\-estimation metric to measure whether the estimated numerical values are close to the ground\-truth Dataset Shapley values\.

Kendall’sτ\\taumeasures the fraction of*concordant*versus*discordant*seller pairs\. For two sellersiiandjjwith predicted values\(vi,vj\)\(v\_\{i\},v\_\{j\}\)and ground\-truth values\(gi,gj\)\(g\_\{i\},g\_\{j\}\), the pair is concordant if\(vi−vj\)​\(gi−gj\)\>0\.\(v\_\{i\}\-v\_\{j\}\)\(g\_\{i\}\-g\_\{j\}\)\>0\.Letncn\_\{c\}andndn\_\{d\}denote the number of concordant and discordant pairs among all\(n2\)\\binom\{n\}\{2\}pairs\. Then,τ\\tauis defined asτ=nc−nd\(n2\)\.\\tau=\\frac\{n\_\{c\}\-n\_\{d\}\}\{\\binom\{n\}\{2\}\}\.The score ranges from−1\-1\(perfect disagreement\) to11\(perfect agreement\)\.

Spearman’sρ\\rhocomputes the Pearson correlation between the predicted and ground\-truth rank vectors\. Letrir\_\{i\}andsis\_\{i\}denote the predicted and ground\-truth ranks of sellerii, respectively\. Letr¯\\bar\{r\}ands¯\\bar\{s\}denote the corresponding average of ranks across all sellers\. Then,ρ\\rhois defined asρ=∑i=1n\(ri−r¯\)​\(si−s¯\)∑i=1n\(ri−r¯\)2​∑i=1n\(si−s¯\)2\.\\rho=\\frac\{\\sum\_\{i=1\}^\{n\}\(r\_\{i\}\-\\bar\{r\}\)\(s\_\{i\}\-\\bar\{s\}\)\}\{\\sqrt\{\\sum\_\{i=1\}^\{n\}\(r\_\{i\}\-\\bar\{r\}\)^\{2\}\}\\sqrt\{\\sum\_\{i=1\}^\{n\}\(s\_\{i\}\-\\bar\{s\}\)^\{2\}\}\}\.When no rank ties exist, this simplifies toρ=1−6​∑i=1ndi2n​\(n2−1\)\\rho=1\-\\frac\{6\\sum\_\{i=1\}^\{n\}d\_\{i\}^\{2\}\}\{n\(n^\{2\}\-1\)\}, wheredi=ri−sid\_\{i\}=r\_\{i\}\-s\_\{i\}\.ρ=1\\rho=1indicates perfect alignment,ρ=−1\\rho=\-1complete reversal, andρ=0\\rho=0no correlation\. While Kendall’sτ\\tauemphasizes pairwise ordering consistency, Spearman’sρ\\rhocaptures overall rank alignment; using both provides a comprehensive evaluation of ranking quality\.

End\-to\-end value\-estimation error\.While rank correlations evaluate whether a method orders sellers correctly, they do not measure whether the estimated seller values are numerically accurate\. Therefore, we also report the mean squared error \(MSE\) between the estimated seller values and the ground\-truth Dataset Shapley values asMSE=1n​∑i=1n\(ϕi−ϕ^i\)2,\\mathrm\{MSE\}=\\frac\{1\}\{n\}\\sum\_\{i=1\}^\{n\}\(\\phi\_\{i\}\-\\hat\{\\phi\}\_\{i\}\)^\{2\},whereϕi\\phi\_\{i\}is the ground\-truth Dataset Shapley value of sellerii, andϕ^i\\hat\{\\phi\}\_\{i\}is the value estimated by the method being evaluated\. We refer to this as an end\-to\-end metric because it evaluates the final output of the valuation pipeline: the numerical value assigned to each seller\.

Table 2:Performance of valuation methods on vision and NLP tasks\. Buyer’s target task size\|𝒯\|=3\|\\mathcal\{T\}\|=3for NLP and\|𝒯\|=4\|\\mathcal\{T\}\|=4for vision\.Table 3:Comparison of valuation methods for different numbers of buyer’s target tasks\. Number of sellersn=9n=9for NLP andn=10n=10for vision\.![Refer to caption](https://arxiv.org/html/2607.03346v1/x1.png)

![Refer to caption](https://arxiv.org/html/2607.03346v1/x2.png)

Figure 2:Efficiency\-accuracy trade\-off of valuation methods on NLP and vision tasks\. Thexx\-axis reports wall\-clock computational time in minutes, and theyy\-axis reports MSE relative to ground\-truth\.
### 5\.4Results and Discussion

We evaluate whether DMVM can approximate ground\-truth multi\-task Dataset Shapley values while avoiding the two main limitations of existing valuation baselines: repeated retraining and access to sellers’ private datasets\. We emphasize that all non\-DMVM baselines are privileged baselines: Dataset Shapley, MC\-Dataset\-Shapley, and DU\-Shapley require access to sellers’ datasets in order to train or evaluate models on seller coalitions\. Therefore, they should be interpreted as practical upper\-bound references rather than deployable baselines in decentralized data marketplaces\. In contrast, DMVM estimates seller values only through merged models and does not require access to raw seller datasets\.

Table[2](https://arxiv.org/html/2607.03346#S5.T2)evaluates DMVM as the number of sellers varies\. Across both modalities, DMVM achieves strong agreement with the ground\-truth Dataset Shapley values\. In vision, DMVM obtains Kendall’sτ\\tauvalues around0\.780\.78–0\.810\.81and Spearman’sρ\\rhovalues around0\.890\.89, with MSE on the order of10−410^\{\-4\}\. Although DU\-Shapley gives slightly higher rank correlations in vision, DMVM remains close while operating under a substantially more realistic constraint\. In NLP, DMVM consistently outperforms DU\-Shapley in both ranking and MSE across all seller counts\. For example, whenn=9n=9, DMVM improves Kendall’sτ\\taufrom0\.70300\.7030to0\.74870\.7487, Spearman’sρ\\rhofrom0\.81790\.8179to0\.86170\.8617, and reduces MSE from0\.0021220\.002122to0\.0016680\.001668\.

Table[3](https://arxiv.org/html/2607.03346#S5.T3)studies the effect of the number of buyer target tasks\. DMVM remains stable across different values of\|𝒯\|\|\\mathcal\{T\}\|for both Vision and NLP experiments, suggesting that model merging provides a reliable proxy for multi\-task valuation\. In NLP, DMVM achieves lower MSE than DU\-Shapley for all target\-task sizes, indicating that it provides accurate end\-to\-end numerical value estimates even when the ranking metrics are sometimes slightly lower\. All reported results are averaged over all possible choices ofnnsellers and target task sets of size\|𝒯\|\|\\mathcal\{T\}\|\.

Figure[2](https://arxiv.org/html/2607.03346#S5.F2)shows the efficiency\-accuracy trade\-off\. Dataset Shapley gives the exact reference values but is computationally expensive because it requires training over all coalitions\. MC\-Dataset\-Shapley reduces the number of coalitions but still requires retraining on sellers’ datasets\. In contrast, DMVM replaces coalition training with model merging and inference, making valuation substantially cheaper and compatible with decentralized settings\. For the MC\-based methods, the three plotted points correspond tom∈\{9,24,56\}m\\in\\\{9,24,56\\\}for NLP, andm∈\{10,30,77\}m\\in\\\{10,30,77\\\}for vision\. Notably, the largest\-sample MC\-DMVM point matches the wall\-clock time of DU\-Shapley, while achieving lower MSE in NLP and competitive MSE in vision\. This demonstrates that MC\-DMVM provides a practical accuracy\-efficiency trade\-off: increasingmmreduces valuation error while keeping the method computationally lighter than retraining\-based Shapley estimators\.

More ablation studies on the convergence of MC\-based methods asmmincreases are provided in Appendix[A\.2](https://arxiv.org/html/2607.03346#A1.SS2)\. Overall, these results show that DMVM provides accurate seller rankings and low end\-to\-end valuation error while avoiding the key limitations of existing baselines\. Unlike Dataset Shapley, MC\-Dataset\-Shapley, and DU\-Shapley, DMVM does not require access to sellers’ datasets, making it both computationally efficient and privacy\-aware for realistic multi\-task data marketplaces\. Additional experiments with more model architectures and datasets for a systematic and controlled simulation, where the sellers might have datasets including more than one task in both overlapping and non\-overlapping settings are provided in Appendix[A\.7](https://arxiv.org/html/2607.03346#A1.SS7)\. Scalability analysis is provided in Appendix[A\.9](https://arxiv.org/html/2607.03346#A1.SS9)\. Moreover, we further investigate whether DMVM can identify and downweight harmful data sources by simulating sellers with increasing levels of label noise; the complete setup and analysis are presented in Appendix[A\.8](https://arxiv.org/html/2607.03346#A1.SS8)\.

## 6Conclusion

This work reframes multi\-task dataset valuation as a model\-composition problem rather than a repeated retraining problem\. In decentralized data marketplaces, the challenge is not only estimating which sellers are useful, but doing so before purchase, without assuming access to their private datasets\. DMVM addresses this setting by using model merging as a proxy for coalition utility, enabling Shapley\-style valuation through merged models instead of training on every seller subset\. This preserves the economic interpretation of marginal contribution while making valuation more practical in multi\-task and privacy\-sensitive environments\. Our theoretical analysis characterizes the approximation gap between merging\-based and ground\-truth multi\-task valuation, while our secure aggregation protocol prevents direct exposure of raw data or individual seller models\. Empirically, DMVM and its MC variant achieve strong agreement with privileged retraining\-based baselines across vision and NLP tasks, often attaining low value\-estimation error with only inference\-time computation\. These results suggest that model merging can serve not only as a tool for multi\-task adaptation, but also as an efficient mechanism for decentralized market valuation\.

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## Appendix AAppendix

### A\.1Algorithm

Algorithm 1DMVM: Decentralized Multi\-task Valuation via Secure Model Merging1:Input:Buyer queries a potential group of sellers

\[n\]\[n\], base model weights

𝑾pre\\bm\{W\}\_\{\\text\{pre\}\}, scaling hyperparameter

α\\alpha
2:foreach subset

𝒮⊆\[n\]\\mathcal\{S\}\\subseteq\[n\]with

\|𝒮\|≥2\|\\mathcal\{S\}\|\\geq 2do

3:foreach layer pair

\(𝑾\(1\),𝑾\(2\)\)\(\\bm\{W\}^\{\(1\)\},\\bm\{W\}^\{\(2\)\}\)do

4:if\(

𝑾\(1\),𝑾\(2\)\\bm\{W\}^\{\(1\)\},\\bm\{W\}^\{\(2\)\}\) correspond to a 2\-layer MLPthen

5:Sellers agree on a random permutation matrix

𝑷\\bm\{P\}\.

6:Define the layer transformation:

\(𝚿1,𝚿2\)=\(𝑷,𝑷⊤\)\(\\bm\{\\Psi\}\_\{1\},\\bm\{\\Psi\}\_\{2\}\)=\(\\bm\{P\},\\bm\{P\}^\{\\top\}\)\.

7:elseif\(

𝑾\(1\),𝑾\(2\)\\bm\{W\}^\{\(1\)\},\\bm\{W\}^\{\(2\)\}\) are two consecutive linear layersthen

8:Sellers agree on a random scalar

ρ\\rho\.

9:Define the layer transformation:

\(𝚿1,𝚿2\)=\(ρ​𝑰,ρ−1​𝑰\)\(\\bm\{\\Psi\}\_\{1\},\\bm\{\\Psi\}\_\{2\}\)=\(\\rho\\bm\{I\},\\rho^\{\-1\}\\bm\{I\}\)\.

10:endif

11:foreach seller

i∈𝒮i\\in\\mathcal\{S\}do

12:if

α=1\|𝒮\|\\alpha=\\frac\{1\}\{\|\\mathcal\{S\|\}\}then

13:Generate masked shares:

𝒆1\(i\)=𝚿1​𝑾i\(1\)\|𝒮\|−𝒎i\(1\),𝒆2\(i\)=𝑾i\(2\)​𝚿2\|𝒮\|−𝒎i\(2\)\\quad\\bm\{e\}^\{\(i\)\}\_\{1\}=\\frac\{\\bm\{\\Psi\}\_\{1\}\\bm\{W\}^\{\(1\)\}\_\{i\}\}\{\|\\mathcal\{S\}\|\}\-\\bm\{m\}^\{\(1\)\}\_\{i\},\\ \\bm\{e\}^\{\(i\)\}\_\{2\}=\\frac\{\\bm\{W\}^\{\(2\)\}\_\{i\}\\bm\{\\Psi\}\_\{2\}\}\{\|\\mathcal\{S\}\|\}\-\\bm\{m\}^\{\(2\)\}\_\{i\}
14:else

15:Compute local secrets:

𝒔i\(1\)=𝑾pre\(1\)​\(1−\|𝒮\|​α\)\|𝒮\|\+α​𝑾i\(1\),𝒔i\(2\)=𝑾pre\(2\)​\(1−\|𝒮\|​α\)\|𝒮\|\+α​𝑾i\(2\)\\ \\bm\{s\}^\{\(1\)\}\_\{i\}=\\frac\{\\bm\{W\}^\{\(1\)\}\_\{\\text\{pre\}\}\(1\-\|\\mathcal\{S\}\|\\alpha\)\}\{\|\\mathcal\{S\}\|\}\+\\alpha\\bm\{W\}^\{\(1\)\}\_\{i\},\\ \\bm\{s\}^\{\(2\)\}\_\{i\}=\\frac\{\\bm\{W\}^\{\(2\)\}\_\{\\text\{pre\}\}\(1\-\|\\mathcal\{S\}\|\\alpha\)\}\{\|\\mathcal\{S\}\|\}\+\\alpha\\bm\{W\}^\{\(2\)\}\_\{i\}
16:Generate masked shares:

𝒆1\(i\)=𝚿1​𝒔i\(1\)−𝒎i\(1\),𝒆2\(i\)=𝒔i\(2\)​𝚿2−𝒎i\(2\)\\quad\\bm\{e\}^\{\(i\)\}\_\{1\}=\\bm\{\\Psi\}\_\{1\}\\bm\{s\}^\{\(1\)\}\_\{i\}\-\\bm\{m\}^\{\(1\)\}\_\{i\},\\ \\bm\{e\}^\{\(i\)\}\_\{2\}=\\bm\{s\}^\{\(2\)\}\_\{i\}\\bm\{\\Psi\}\_\{2\}\-\\bm\{m\}^\{\(2\)\}\_\{i\}
17:endif

18:Send

\(𝒆1\(i\),𝒆2\(i\)\)\(\\bm\{e\}^\{\(i\)\}\_\{1\},\\bm\{e\}^\{\(i\)\}\_\{2\}\)to the buyer and

\(𝒎i\(1\),𝒎i\(2\)\)\(\\bm\{m\}^\{\(1\)\}\_\{i\},\\bm\{m\}^\{\(2\)\}\_\{i\}\)to all other sellers\.

19:endfor

20:Sellers compute aggregate masks:

𝒎\(1\)=∑i∈𝒮𝒎i\(1\),𝒎\(2\)=∑i∈𝒮𝒎i\(2\)\\quad\\bm\{m\}^\{\(1\)\}=\\sum\_\{i\\in\\mathcal\{S\}\}\\bm\{m\}^\{\(1\)\}\_\{i\},\\quad\\bm\{m\}^\{\(2\)\}=\\sum\_\{i\\in\\mathcal\{S\}\}\\bm\{m\}^\{\(2\)\}\_\{i\}
21:Each seller sends

\(𝒎\(1\),𝒎\(2\)\)\(\\bm\{m\}^\{\(1\)\},\\bm\{m\}^\{\(2\)\}\)to the buyer for verification\.

22:Buyer performs sanity check on consistency of

\(𝒎\(1\),𝒎\(2\)\)\(\\bm\{m\}^\{\(1\)\},\\bm\{m\}^\{\(2\)\}\)\.

23:Buyer reconstructs the merged model weights:

𝑾merge\(1\)=𝒎\(1\)\+∑i∈𝒮𝒆1\(i\),𝑾merge\(2\)=𝒎\(2\)\+∑i∈𝒮𝒆2\(i\)\\displaystyle\\quad\\bm\{W\}^\{\(1\)\}\_\{\\text\{merge\}\}=\\bm\{\\bm\{m\}\}^\{\(1\)\}\+\\sum\_\{i\\in\\mathcal\{S\}\}\\bm\{e\}^\{\(i\)\}\_\{1\},\\ \\bm\{W\}^\{\(2\)\}\_\{\\text\{merge\}\}=\\bm\{\\bm\{m\}\}^\{\(2\)\}\+\\sum\_\{i\\in\\mathcal\{S\}\}\\bm\{e\}^\{\(i\)\}\_\{2\}
24:endfor

25:endfor

26:Output:Privacy\-aware merged models for all the subsets ready to use for valuation using Eq\. \([3](https://arxiv.org/html/2607.03346#S4.E3)\) for each of the sellers\.

![Refer to caption](https://arxiv.org/html/2607.03346v1/x3.png)

![Refer to caption](https://arxiv.org/html/2607.03346v1/x4.png)

![Refer to caption](https://arxiv.org/html/2607.03346v1/x5.png)

Figure 3:Convergence behavior of MC\-based valuation methods on vision tasks withn=10n=10sellers\. The plots report Kendall’sτ\\tau, Spearman’sρ\\rho, and MSE as the number of MC samplesmmincreases\. MC\-Dataset\-Shapley converges toward Dataset Shapley, while MC\-DMVM converges toward deterministic DMVM\.![Refer to caption](https://arxiv.org/html/2607.03346v1/)

![Refer to caption](https://arxiv.org/html/2607.03346v1/x7.png)

![Refer to caption](https://arxiv.org/html/2607.03346v1/x8.png)

Figure 4:Convergence behavior of MC\-based valuation methods on NLP tasks withn=9n=9sellers\. The plots report Kendall’sτ\\tau, Spearman’sρ\\rho, and MSE as the number of MC samplesmmincreases\. MC\-Dataset\-Shapley approaches Dataset Shapley with largermm, while MC\-DMVM approaches deterministic DMVM using only merging\-based utility estimates\.
### A\.2Convergence Analysis of MC\-based methods

We further study the convergence behavior of the MC variants as the number of sampled coalitionsmmincreases\. We focus on the setting withn=10n=10sellers for vision andn=9n=9sellers for NLP\. For vision, we evaluatem∈\{10,30,90,270,512\}m\\in\\\{10,30,90,270,512\\\}, and for NLP, we evaluatem∈\{9,23,56,140,256\}m\\in\\\{9,23,56,140,256\\\}\. The goal is to examine whether the MC estimators approach their corresponding deterministic counterparts asmmincreases, while using only a fraction of the full coalition space\.

Figures[3](https://arxiv.org/html/2607.03346#A1.F3)and[4](https://arxiv.org/html/2607.03346#A1.F4)show that both MC\-Dataset\-Shapley and MC\-DMVM improve consistently asmmincreases\. MC\-Dataset\-Shapley converges toward Dataset Shapley: its MSE decreases substantially, and both Kendall’sτ\\tauand Spearman’sρ\\rhoimprove with more samples\. This is expected, since increasingmmgives a better MC approximation to the exact Shapley value\. However, MC\-Dataset\-Shapley still requires access to sellers’ datasets and repeated coalition training\.

MC\-DMVM converges toward deterministic DMVM, since it samples marginal contributions using the merging\-based utility proxy rather than the retraining\-based utility\. In both vision and NLP, increasingmmreduces the gap between MC\-DMVM and DMVM, especially in terms of MSE\. The ranking metrics show small non\-monotonic fluctuations, which are expected for sampling\-based estimators, but they generally stabilize near the deterministic DMVM values asmmbecomes larger\.

These results support the error decomposition in Theorem[4\.2](https://arxiv.org/html/2607.03346#S4.Thmtheorem2): increasingmmreduces the statistical sampling error, while the remaining gap is governed by the deterministic merging approximation\. Importantly, MC\-DMVM provides two levels of computational savings\. First, it replaces coalition training with model merging and inference\. Second, it reduces the number of required coalition evaluations from exponential in the number of sellers, i\.e\.,2n−12^\{n\-1\}, to onlymmsampled evaluations\. Therefore, MC\-DMVM provides a practical accuracy\-efficiency trade\-off for decentralized valuation, achieving stable seller rankings and low value\-estimation error without raw\-data access or repeated retraining\.

### A\.3Lemma

###### Lemma A\.1\(Convex Combination Upper Bound\)\.

Leta1,a2,…,an∈ℝa\_\{1\},a\_\{2\},\\dots,a\_\{n\}\\in\\mathbb\{R\}and letw1,w2,…,wn≥0w\_\{1\},w\_\{2\},\\dots,w\_\{n\}\\geq 0satisfy∑i=1nwi=b\.\\sum\_\{i=1\}^\{n\}w\_\{i\}=b\.Then the weighted sum is bounded by the largest element, i\.e\.,

∑i=1nwi​ai≤b​max1≤i≤n⁡ai≤b​sup1≤i≤nai\.\\displaystyle\\sum\_\{i=1\}^\{n\}w\_\{i\}a\_\{i\}\\;\\leq\\;b\\max\_\{1\\leq i\\leq n\}a\_\{i\}\\;\\leq\\;b\\sup\_\{1\\leq i\\leq n\}a\_\{i\}\.

### A\.4Error bound of our method

Zhouet al\.\[[74](https://arxiv.org/html/2607.03346#bib.bib51)\]proposed the following error bound between multi\-task learning and task\-arithmetic merging:

‖𝜽𝒮merge−𝜽𝒮MTL‖2≤s​\(α​s\+1\)​C,\\displaystyle\\\|\\bm\{\\theta\}^\{\\text\{merge\}\}\_\{\\mathcal\{S\}\}\-\\bm\{\\theta\}^\{\\text\{MTL\}\}\_\{\\mathcal\{S\}\}\\\|\_\{2\}\\leq s\(\\alpha s\+1\)C,wheressis the cardinality of𝒮\\mathcal\{S\}, i\.e\.,s=\|𝒮\|s=\|\\mathcal\{S\}\|, andC=\(h\+22\)​Hmax​GmaxC=\\binom\{h\+2\}\{2\}H\_\{\\max\}G\_\{\\max\}is a constant, wherehhdenotes the number of optimization steps \(i\.e\., epochs\), andGmaxG\_\{\\max\}andHmaxH\_\{\\max\}are upper bounds on the gradient and Hessian norms, respectively andα\\alphais the hyperparameter for task\-arithmetic\. Assuming the loss function is L\-Lipschitz continuous, we have:

\|ℒ​\(𝜽𝒮merge\)−ℒ​\(𝜽𝒮MTL\)\|≤L​‖𝜽𝒮merge−𝜽𝒮MTL‖2≤L​s​\(α​s\+1\)​C\.\\displaystyle\|\\mathcal\{L\}\(\\bm\{\\theta\}^\{\\text\{merge\}\}\_\{\\mathcal\{S\}\}\)\-\\mathcal\{L\}\(\\bm\{\\theta\}^\{\\text\{MTL\}\}\_\{\\mathcal\{S\}\}\)\|\\leq L\\\|\\bm\{\\theta\}^\{\\text\{merge\}\}\_\{\\mathcal\{S\}\}\-\\bm\{\\theta\}^\{\\text\{MTL\}\}\_\{\\mathcal\{S\}\}\\\|\_\{2\}\\leq Ls\(\\alpha s\+1\)C\.\(4\)
Assuming that utilityu​\(⋅\)u\(\\cdot\)has a negative relation with loss function, we setu=−ℒu=\-\\mathcal\{L\}, then the error between our method and ground\-truth Shapely will be:

\|ϕi−ϕ^i\|\\displaystyle\|\\phi\_\{i\}\-\\hat\{\\phi\}\_\{i\}\|=1n​\|∑𝒮⊆\[n\]∖\{i\}u​\(𝒮∪\{i\}\)−u​\(𝒮\)−u^​\(𝒮∪\{i\}\)\+u^​\(𝒮\)\(n−1\|𝒮\|\)\|\\displaystyle=\\frac\{1\}\{n\}\\left\|\\sum\_\{\\mathcal\{S\}\\subseteq\[n\]\\setminus\\\{i\\\}\}\\frac\{u\(\\mathcal\{S\}\\cup\\\{i\\\}\)\-u\(\\mathcal\{S\)\}\-\\hat\{u\}\(\\mathcal\{S\}\\cup\\\{i\\\}\)\+\\hat\{u\}\(\\mathcal\{S\)\}\}\{\\binom\{n\-1\}\{\|\\mathcal\{S\}\|\}\}\\right\|≤1n​∑𝒮⊆\[n\]∖\{i\}\|u​\(𝒮∪\{i\}\)−u​\(𝒮\)−u^​\(𝒮∪\{i\}\)\+u^​\(𝒮\)\|\(n−1\|𝒮\|\)\\displaystyle\\leq\\frac\{1\}\{n\}\\sum\_\{\\mathcal\{S\}\\subseteq\[n\]\\setminus\\\{i\\\}\}\\frac\{\\left\|u\(\\mathcal\{S\}\\cup\\\{i\\\}\)\-u\(\\mathcal\{S\)\}\-\\hat\{u\}\(\\mathcal\{S\}\\cup\\\{i\\\}\)\+\\hat\{u\}\(\\mathcal\{S\)\}\\right\|\}\{\\binom\{n\-1\}\{\|\\mathcal\{S\}\|\}\}≤1n​∑𝒮⊆\[n\]∖\{i\}\|−ℒ​\(𝜽𝒮∪\{i\}MTL\)\+ℒ​\(𝜽𝒮MTL\)\+ℒ​\(𝜽𝒮∪\{i\}merge\)−ℒ​\(𝜽𝒮merge\)\|\(n−1\|𝒮\|\)\\displaystyle\\leq\\frac\{1\}\{n\}\\sum\_\{\\mathcal\{S\}\\subseteq\[n\]\\setminus\\\{i\\\}\}\\frac\{\\left\|\-\\mathcal\{L\}\(\{\\bm\{\\theta\}\}\_\{\\mathcal\{S\}\\cup\\\{i\\\}\}^\{\\text\{MTL\}\}\)\+\\mathcal\{L\}\(\{\\bm\{\\theta\}\}\_\{\\mathcal\{S\}\}^\{\\text\{MTL\}\}\)\+\{\\mathcal\{L\}\}\(\{\\bm\{\\theta\}\}\_\{\{\\mathcal\{S\}\\cup\\\{i\\\}\}\}^\{\\text\{merge\}\}\)\-\\mathcal\{L\}\(\{\\bm\{\\theta\}\}\_\{\\mathcal\{S\}\}^\{\\text\{merge\}\}\)\\right\|\}\{\\binom\{n\-1\}\{\|\\mathcal\{S\}\|\}\}≤1n​∑𝒮⊆\[n\]∖\{i\}\|ℒ​\(𝜽𝒮∪\{i\}merge\)−ℒ​\(𝜽𝒮∪\{i\}MTL\)\|\+\|ℒ​\(𝜽𝒮merge\)−ℒ​\(𝜽𝒮MTL\)\|\(n−1\|𝒮\|\)\\displaystyle\\leq\\frac\{1\}\{n\}\\sum\_\{\\mathcal\{S\}\\subseteq\[n\]\\setminus\\\{i\\\}\}\\frac\{\\left\|\{\\mathcal\{L\}\}\(\{\\bm\{\\theta\}\}\_\{\{\\mathcal\{S\}\\cup\\\{i\\\}\}\}^\{\\text\{merge\}\}\)\-\\mathcal\{L\}\(\{\\bm\{\\theta\}\}\_\{\\mathcal\{S\}\\cup\\\{i\\\}\}^\{\\text\{MTL\}\}\)\\right\|\+\\left\|\\mathcal\{L\}\(\{\\bm\{\\theta\}\}\_\{\\mathcal\{S\}\}^\{\\text\{merge\}\}\)\-\\mathcal\{L\}\(\{\\bm\{\\theta\}\}\_\{\\mathcal\{S\}\}^\{\\text\{MTL\}\}\)\\right\|\}\{\\binom\{n\-1\}\{\|\\mathcal\{S\}\|\}\}≤\(a\)​L​n​\(α​n\+1\)​C\+L​\(n−1\)​\(α​\(n−1\)\+1\)​C=\(2​α​n2−2​α​n\+2​n\+α−1\)​L​C\.\\displaystyle\\overset\{\(a\)\}\{\\leq\}Ln\(\\alpha n\+1\)C\+L\(n\-1\)\(\\alpha\(n\-1\)\+1\)C=\(2\\alpha n^\{2\}\-2\\alpha n\+2n\+\\alpha\-1\)LC\.Inequality \(a\) follows from Lemma[A\.1](https://arxiv.org/html/2607.03346#A1.Thmtheorem1)\. Since𝒮⊆\[n\]∖\{i\}\\mathcal\{S\}\\subseteq\[n\]\\setminus\\\{i\\\}, we have\|𝒮∪\{i\}\|≤n\|\\mathcal\{S\}\\cup\\\{i\\\}\|\\leq nand\|𝒮\|≤n−1\|\\mathcal\{S\}\|\\leq n\-1\. Hence, the two loss\-difference terms are uniformly bounded byL​n​\(α​n\+1\)​CLn\(\\alpha n\+1\)CandL​\(n−1\)​\(α​\(n−1\)\+1\)​CL\(n\-1\)\(\\alpha\(n\-1\)\+1\)C, respectively, using inequality \([4](https://arxiv.org/html/2607.03346#A1.E4)\)\. Moreover, the Shapley coefficients form a convex combination, i\.e\.,1n​∑𝒮⊆\[n\]∖\{i\}1\(n−1\|𝒮\|\)=1\.\\frac\{1\}\{n\}\\sum\_\{\\mathcal\{S\}\\subseteq\[n\]\\setminus\\\{i\\\}\}\\frac\{1\}\{\\binom\{n\-1\}\{\|\\mathcal\{S\}\|\}\}=1\.Therefore, applying Lemma[A\.1](https://arxiv.org/html/2607.03346#A1.Thmtheorem1)withb=1b=1yields inequality \(a\)\.

### A\.5Error bound of our privacy\-aware method

Assuming that we want to exclude subsets of size less thankk\. LetRRbe the maximum marginal utility, i\.e\.,∀𝒮⊆\[n\]∖\{i\},\|u​\(𝒮∪\{i\}\)−u​\(𝒮\)\|≤R\\forall\\mathcal\{S\}\\subseteq\[n\]\\setminus\\\{i\\\},\\quad\|u\(\\mathcal\{S\}\\cup\\\{i\\\}\)\-u\(\\mathcal\{S\)\}\|\\leq R\. Then, the error will be:

\|ϕi−ϕ~i\|\\displaystyle\|\\phi\_\{i\}\-\\tilde\{\\phi\}\_\{i\}\|=1n\|∑𝒮⊆\[n\]∖\{i\},\|S\|≥ku​\(𝒮∪\{i\}\)−u​\(𝒮\)−u^​\(𝒮∪\{i\}\)\+u^​\(𝒮\)\(n−1\|𝒮\|\)\\displaystyle=\\frac\{1\}\{n\}\\Bigg\|\\sum\_\{\\mathcal\{S\}\\subseteq\[n\]\\setminus\\\{i\\\},\|S\|\\geq k\}\\frac\{u\(\\mathcal\{S\}\\cup\\\{i\\\}\)\-u\(\\mathcal\{S\)\}\-\\hat\{u\}\(\\mathcal\{S\}\\cup\\\{i\\\}\)\+\\hat\{u\}\(\\mathcal\{S\)\}\}\{\\binom\{n\-1\}\{\|\\mathcal\{S\}\|\}\}\+∑𝒮⊆\[n\]∖\{i\},\|S\|<ku​\(𝒮∪\{i\}\)−u​\(𝒮\)\(n−1\|𝒮\|\)\|\\displaystyle\\quad\+\\sum\_\{\\mathcal\{S\}\\subseteq\[n\]\\setminus\\\{i\\\},\|S\|<k\}\\frac\{u\(\\mathcal\{S\}\\cup\\\{i\\\}\)\-u\(\\mathcal\{S\)\}\}\{\\binom\{n\-1\}\{\|\\mathcal\{S\}\|\}\}\\Bigg\|≤1n​∑𝒮⊆\[n\]∖\{i\},\|S\|≥k\|u​\(𝒮∪\{i\}\)−u​\(𝒮\)−u^​\(𝒮∪\{i\}\)\+u^​\(𝒮\)\(n−1\|𝒮\|\)\|\\displaystyle\\leq\\frac\{1\}\{n\}\\sum\_\{\\mathcal\{S\}\\subseteq\[n\]\\setminus\\\{i\\\},\|S\|\\geq k\}\\Big\|\\frac\{u\(\\mathcal\{S\}\\cup\\\{i\\\}\)\-u\(\\mathcal\{S\)\}\-\\hat\{u\}\(\\mathcal\{S\}\\cup\\\{i\\\}\)\+\\hat\{u\}\(\\mathcal\{S\)\}\}\{\\binom\{n\-1\}\{\|\\mathcal\{S\}\|\}\}\\Big\|\+1n​∑𝒮⊆\[n\]∖\{i\},\|S\|<k\|u​\(𝒮∪\{i\}\)−u​\(𝒮\)\(n−1\|𝒮\|\)\|\\displaystyle\\quad\+\\frac\{1\}\{n\}\\sum\_\{\\mathcal\{S\}\\subseteq\[n\]\\setminus\\\{i\\\},\|S\|<k\}\\Big\|\\frac\{u\(\\mathcal\{S\}\\cup\\\{i\\\}\)\-u\(\\mathcal\{S\)\}\}\{\\binom\{n\-1\}\{\|\\mathcal\{S\}\|\}\}\\Big\|≤1n​∑𝒮⊆\[n\]∖\{i\},\|S\|≥k\|ℒ​\(𝜽𝒮∪\{i\}merge\)−ℒ​\(𝜽𝒮∪\{i\}MTL\)\|\+\|ℒ​\(𝜽𝒮merge\)−ℒ​\(𝜽𝒮MTL\)\|\(n−1\|𝒮\|\)\\displaystyle\\leq\\frac\{1\}\{n\}\\sum\_\{\\mathcal\{S\}\\subseteq\[n\]\\setminus\\\{i\\\},\|S\|\\geq k\}\\frac\{\\left\|\{\\mathcal\{L\}\}\(\{\\bm\{\\theta\}\}\_\{\{\\mathcal\{S\}\\cup\\\{i\\\}\}\}^\{\\text\{merge\}\}\)\-\\mathcal\{L\}\(\{\\bm\{\\theta\}\}\_\{\\mathcal\{S\}\\cup\\\{i\\\}\}^\{\\text\{MTL\}\}\)\\right\|\+\\left\|\\mathcal\{L\}\(\{\\bm\{\\theta\}\}\_\{\\mathcal\{S\}\}^\{\\text\{merge\}\}\)\-\\mathcal\{L\}\(\{\\bm\{\\theta\}\}\_\{\\mathcal\{S\}\}^\{\\text\{MTL\}\}\)\\right\|\}\{\\binom\{n\-1\}\{\|\\mathcal\{S\}\|\}\}\+1n​∑𝒮⊆\[n\]∖\{i\},\|S\|<k\|ℒ​\(𝜽𝒮MTL\)−ℒ​\(𝜽𝒮∪\{i\}MTL\)\|\(n−1\|𝒮\|\)\\displaystyle\\quad\+\\frac\{1\}\{n\}\\sum\_\{\\mathcal\{S\}\\subseteq\[n\]\\setminus\\\{i\\\},\|S\|<k\}\\frac\{\|\\mathcal\{L\}\(\{\\bm\{\\theta\}\}\_\{\\mathcal\{S\}\}^\{\\text\{MTL\}\}\)\-\\mathcal\{L\}\(\{\\bm\{\\theta\}\}\_\{\\mathcal\{S\}\\cup\\\{i\\\}\}^\{\\text\{MTL\}\}\)\|\}\{\\binom\{n\-1\}\{\|\\mathcal\{S\}\|\}\}≤\(a\)​\(n−k\)n​\(2​α​n2−2​α​n\+2​n\+α−1\)​L​C\+kn​R\.\\displaystyle\\overset\{\(a\)\}\{\\leq\}\\frac\{\(n\-k\)\}\{n\}\(2\\alpha n^\{2\}\-2\\alpha n\+2n\+\\alpha\-1\)LC\+\\frac\{k\}\{n\}R\.Inequality \(a\) follows by applying Lemma[A\.1](https://arxiv.org/html/2607.03346#A1.Thmtheorem1)separately to the two sums\. For the first sum, the same argument as in Appendix[A\.4](https://arxiv.org/html/2607.03346#A1.SS4)gives the uniform bound:

\|ℒ​\(𝜽𝒮∪\{i\}merge\)−ℒ​\(𝜽𝒮∪\{i\}MTL\)\|\+\|ℒ​\(𝜽𝒮merge\)−ℒ​\(𝜽𝒮MTL\)\|≤\(2​α​n2−2​α​n\+2​n\+α−1\)​L​C\.\\left\|\{\\mathcal\{L\}\}\(\{\\bm\{\\theta\}\}\_\{\{\\mathcal\{S\}\\cup\\\{i\\\}\}\}^\{\\text\{merge\}\}\)\-\\mathcal\{L\}\(\{\\bm\{\\theta\}\}\_\{\\mathcal\{S\}\\cup\\\{i\\\}\}^\{\\text\{MTL\}\}\)\\right\|\+\\left\|\\mathcal\{L\}\(\{\\bm\{\\theta\}\}\_\{\\mathcal\{S\}\}^\{\\text\{merge\}\}\)\-\\mathcal\{L\}\(\{\\bm\{\\theta\}\}\_\{\\mathcal\{S\}\}^\{\\text\{MTL\}\}\)\\right\|\\leq\\left\(2\\alpha n^\{2\}\-2\\alpha n\+2n\+\\alpha\-1\\right\)LC\.The total weight of coalitions with\|𝒮\|≥k\|\\mathcal\{S\}\|\\geq kis1n​∑𝒮⊆\[n\]∖\{i\}\|𝒮\|≥k1\(n−1\|𝒮\|\)=1n​∑s=kn−11=n−kn\.\\frac\{1\}\{n\}\\sum\_\{\\begin\{subarray\}\{c\}\\mathcal\{S\}\\subseteq\[n\]\\setminus\\\{i\\\}\\\\ \|\\mathcal\{S\}\|\\geq k\\end\{subarray\}\}\\frac\{1\}\{\\binom\{n\-1\}\{\|\\mathcal\{S\}\|\}\}=\\frac\{1\}\{n\}\\sum\_\{s=k\}^\{n\-1\}1=\\frac\{n\-k\}\{n\}\.For the excluded coalitions with\|𝒮\|<k\|\\mathcal\{S\}\|<k, the marginal utility is bounded byRR, and their total weight is:1n​∑𝒮⊆\[n\]∖\{i\}\|𝒮\|<k1\(n−1\|𝒮\|\)=1n​∑s=0k−11=kn\.\\frac\{1\}\{n\}\\sum\_\{\\begin\{subarray\}\{c\}\\mathcal\{S\}\\subseteq\[n\]\\setminus\\\{i\\\}\\\\ \|\\mathcal\{S\}\|<k\\end\{subarray\}\}\\frac\{1\}\{\\binom\{n\-1\}\{\|\\mathcal\{S\}\|\}\}=\\frac\{1\}\{n\}\\sum\_\{s=0\}^\{k\-1\}1=\\frac\{k\}\{n\}\.Applying Lemma[A\.1](https://arxiv.org/html/2607.03346#A1.Thmtheorem1)with these two values ofbbgives inequality \(a\)\.

### A\.6MC Sampling approximation proof

Letϕi\\phi\_\{i\}denote the true Shapley value,ϕ^i\\hat\{\\phi\}\_\{i\}the deterministic approximation with error

\|ϕ^i−ϕi\|≤\(2​α​n2−2​α​n\+2​n\+α−1\)​L​C,\|\\hat\{\\phi\}\_\{i\}\-\\phi\_\{i\}\|\\leq\(2\\alpha n^\{2\}\-2\\alpha n\+2n\+\\alpha\-1\)\\,LC,andϕ¯i\(m\)\\bar\{\\phi\}\_\{i\}^\{\(m\)\}the MC estimator ofϕ^i\\hat\{\\phi\}\_\{i\}based onmmsamples\. Assume each sampled marginal contribution lies in an interval of widthRR, i\.e\.,∀𝒮⊆\[n\]∖\{i\},\|u^​\(𝒮∪\{i\}\)−u^​\(𝒮\)\|≤R\\forall\\mathcal\{S\}\\subseteq\[n\]\\setminus\\\{i\\\},\\quad\|\\hat\{u\}\(\\mathcal\{S\}\\cup\\\{i\\\}\)\-\\hat\{u\}\(\\mathcal\{S\)\}\|\\leq R\.

By Hoeffding’s inequality, for anyε\>0\\varepsilon\>0,

Pr⁡\(\|ϕ¯i\(m\)−ϕ^i\|≥ε\)≤2​exp⁡\(−2​m​ε2R2\)\.\\Pr\\\!\\left\(\|\\bar\{\\phi\}\_\{i\}^\{\(m\)\}\-\\hat\{\\phi\}\_\{i\}\|\\geq\\varepsilon\\right\)\\leq 2\\exp\\\!\\left\(\-\\frac\{2m\\varepsilon^\{2\}\}\{R^\{2\}\}\\right\)\.
Using the triangle inequality,

\|ϕ¯i\(m\)−ϕi\|≤\|ϕ¯i\(m\)−ϕ^i\|\+\|ϕ^i−ϕi\|≤\|ϕ¯i\(m\)−ϕ^i\|\+\(2​α​n2−2​α​n\+2​n\+α−1\)​L​C\.\|\\bar\{\\phi\}\_\{i\}^\{\(m\)\}\-\\phi\_\{i\}\|\\leq\|\\bar\{\\phi\}\_\{i\}^\{\(m\)\}\-\\hat\{\\phi\}\_\{i\}\|\+\|\\hat\{\\phi\}\_\{i\}\-\\phi\_\{i\}\|\\leq\|\\bar\{\\phi\}\_\{i\}^\{\(m\)\}\-\\hat\{\\phi\}\_\{i\}\|\+\(2\\alpha n^\{2\}\-2\\alpha n\+2n\+\\alpha\-1\)\\,LC\.
Thus, with probability at least1−δ1\-\\delta, we have:

\|ϕ¯i\(m\)−ϕi\|≤\(2​α​n2−2​α​n\+2​n\+α−1\)​L​C\+R22​m​log⁡\(2δ\)\.\|\\bar\{\\phi\}\_\{i\}^\{\(m\)\}\-\\phi\_\{i\}\|\\leq\(2\\alpha n^\{2\}\-2\\alpha n\+2n\+\\alpha\-1\)\\,LC\+\\sqrt\{\\frac\{R^\{2\}\}\{2m\}\\,\\log\\\!\\left\(\\frac\{2\}\{\\delta\}\\right\)\}\.

### A\.7Controlled Simulations

The experiments in Section[5](https://arxiv.org/html/2607.03346#S5)evaluate DMVM in practical marketplace scenarios where sellers correspond to different datasets and domains\. However, in such settings, the true relevance of each seller to the buyer’s target tasks is not directly controllable, since datasets may differ along several factors simultaneously, including domain, distribution, label space, and sample complexity\. Therefore, we further conduct controlled simulations in which the relationship between each seller’s data and the buyer’s target tasks is explicitly designed\. These experiments allow us to test whether DMVM can recover known relevance patterns, including highly useful, partially useful, irrelevant, and harmful sellers\.

We consider two complementary experimental regimes\. First, we study*single\-domain multi\-task*vision benchmarks, which provide an intuitive and interpretable setting for analyzing seller relevance\. Second, we extend our evaluation to*multi\-domain multi\-task*experiments in both vision and NLP, where tasks may correspond to distinct domains\. Detailed task distribution schemes for different buyer task set sizes and seller constructions are provided in Appendix[A\.11](https://arxiv.org/html/2607.03346#A1.SS11)\. Scalability analysis is provided in Appendix[A\.9](https://arxiv.org/html/2607.03346#A1.SS9)\.

#### A\.7\.1Single\-Domain Multi\-Task Vision Experiments

To establish intuition, we begin with vision experiments where all tasks belong to a single domain\. We consider MNIST and CIFAR\-10, and define the buyer’s objective as learning a subset of classes, treated as multiple target tasks\. Multiple sellers provide datasets with varying relevance to the buyer’s task set\. For these experiments, we compute exact multi\-task Shapley values by explicitly training models on all subsets of sellers\. The resulting Shapley values serve as the reference ordering against which DMVM\-induced rankings are compared\.

##### Non\-Overlapping Sellers\.

We first consider a non\-overlapping seller setting, where each seller provides data for a disjoint subset of classes\. As an illustrative MNIST example, we define the buyer’s target tasks as learning to recognize digits 0\-6, and construct four sellers as follows:

- •Seller 1: classes \(0, 1, 2, 3\)
- •Seller 2: classes \(4, 5, 6\)
- •Seller 3: classes \(7, 8, 9\)
- •Seller 4: mislabeled or noisy samples

Sellers 1 and 2 contribute directly to the buyer’s tasks, with Seller 1 expected to have higher Shapley value due to broader task coverage\. Seller 3 provides irrelevant data, while Seller 4 is harmful\. The reference ranking is therefore induced by the corresponding Shapley values\.

##### Overlapping Sellers\.

We next consider an overlapping seller setting, where sellers provide partially intersecting subsets of the buyer’s tasks\. For MNIST, the buyer’s tasks correspond to recognizing digits 0\-3, and we simulate five sellers with progressively decreasing overlap:

- •Seller 1: classes \(0, 1, 2, 3\)
- •Seller 2: classes \(0, 1, 2, 4\)
- •Seller 3: classes \(0, 1, 4, 5\)
- •Seller 4: classes \(0, 4, 5, 6\)
- •Seller 5: classes \(4, 5, 6, 7\)

This setting induces a smooth spectrum of Shapley values, allowing us to evaluate whether DMVM preserves relative ordering as task relevance decreases\.

#### A\.7\.2Multi\-Domain Multi\-Task Language Modeling

We next evaluate DMVM in a multi\-domain multi\-task language modeling setting, where each task corresponds to a distinct reasoning domain and models are generative\. We consider the following benchmarks:CommonsenseQA\[[53](https://arxiv.org/html/2607.03346#bib.bib74)\]\(commonsense multiple\-choice reasoning\),OpenBookQA\[[42](https://arxiv.org/html/2607.03346#bib.bib69)\]\(open\-book science QA with multi\-step reasoning\),MedMCQA\[[46](https://arxiv.org/html/2607.03346#bib.bib73)\]\(medical multiple\-choice question answering\),LogiQA\[[36](https://arxiv.org/html/2607.03346#bib.bib70)\]\(logical reasoning over short passages\),GSM8K\[[16](https://arxiv.org/html/2607.03346#bib.bib72)\]\(grade\-school mathematical problem solving\), andARC\-Easy\[[15](https://arxiv.org/html/2607.03346#bib.bib71)\]\(elementary\-level science question answering\)\. Each task is treated as a separate domain\-specific learning objective\. We again adopt non\-overlapping and overlapping seller designs\. For non\-overlapping, given a buyer with\|𝒯\|\|\\mathcal\{T\}\|target tasks, sellers partition the task set disjointly\. For example, when the buyer has six target tasks, Seller 1 provides data for three tasks, Seller 2 for two tasks, and Seller 3 for the remaining task\. An additional seller provides data from an irrelevant task outside the buyer’s target set\. This design induces a clear hierarchy of Shapley values based on task coverage, enabling quantitative comparison with DMVM\-induced rankings\.

##### Results and Discussion\.

For single\-domain multi\-task vision experiments, we employed ViT\-B/16\[[64](https://arxiv.org/html/2607.03346#bib.bib34)\]as the shared pretrained model initialized from the HuggingFace checkpointgoogle/vit\-base\-patch16\-224\-in21k\. Each seller fine\-tunes this model for two epochs on its own dataset, after which we compute the privacy\-aware valuation scores usingϕ~\\tilde\{\\phi\}in Eq\. \([3](https://arxiv.org/html/2607.03346#S4.E3)\)\. Fine\-tuning is performed with AdamW using a learning rate of1×10−41\\times 10^\{\-4\}and weight decay0\.010\.01, providing stable adaptation across all seller datasets\. Experiments for multi\-domain multi\-task vision are provided in Appendix[A\.7\.3](https://arxiv.org/html/2607.03346#A1.SS7.SSS3)\. For multi\-domain multi\-task language experiments, we used two pretrained large language models with different scales and architectures:LLaMA\-3\.2\-1B\[[22](https://arxiv.org/html/2607.03346#bib.bib80)\]andQwen2\.5\-3B\[[49](https://arxiv.org/html/2607.03346#bib.bib78)\]\. Tables[4](https://arxiv.org/html/2607.03346#A1.T4),[5](https://arxiv.org/html/2607.03346#A1.T5), and[6](https://arxiv.org/html/2607.03346#A1.T6)collectively demonstrate that DMVM induces seller rankings closely aligned with ground\-truth Shapley\-value\-based orderings across both vision and language modalities\. Performance is strongest in non\-overlapping settings, while remaining robust under partial overlap and across diverse language domains\.

In vision experiments, DMVM exactly recovers the Shapley ordering in non\-overlapping settings \(Table[4](https://arxiv.org/html/2607.03346#A1.T4)\), and remains highly correlated under partial overlap \(Table[5](https://arxiv.org/html/2607.03346#A1.T5)\)\. In language modeling, DMVM achieves strong rank correlation for both LLaMA\-3\.2\-1B and Qwen2\.5\-3B as the shared model, demonstrating that merging\-based valuation extends naturally to generative, multi\-domain scenarios \(Table[6](https://arxiv.org/html/2607.03346#A1.T6)\)\. Table[8](https://arxiv.org/html/2607.03346#A1.T8)compares the wall\-clock time required to compute dataset Shapley values using DMVM and exact ground\-truth Shapley in the language domain, highlighting the computational efficiency of DMVM\.

Table 4:Rank correlation between DMVM\-induced rankings and multi\-task Shapley\-value\-based rankings for single\-domain multi\-task vision experiments with non\-overlapping sellers\.Table 5:Rank correlation between DMVM\-induced rankings and multi\-task Shapley\-value\-based rankings for single\-domain multi\-task vision experiments with overlapping sellers\.Table 6:Rank correlation between DMVM\-induced rankings and Shapley\-value\-based rankings for multi\-domain multi\-task language modeling\.Taken together, these results demonstrate that DMVM remains effective across a wide range of shared model architectures, model scales, data modalities, and marketplace seller topologies, including both vision and language models\.

The consistent recovery of seller rankings indicates that DMVM is not tied to a specific architecture or parameter regime, but instead leverages transferable representation shifts induced by seller\-specific fine\-tuning\. Importantly, DMVM achieves this while requiring only a single fine\-tuning per seller, avoiding the combinatorial retraining costs associated with Shapley\-value estimation and enabling scalable deployment in multi\-seller settings\. Moreover, valuation is performed through model merging rather than direct access to seller data, preserving data locality and supporting privacy\-aware evaluation\.

#### A\.7\.3Multi\-Domain Multi\-Task Vision

Table 7:DMVM ranking performance in the multi\-domain multi\-task vision setting for different numbers of target tasks under non\-overlapping and overlapping task distributions\.We evaluate DMVM in a multi\-domain multi\-task vision setting usingCLIP\-ViT\-B/32\[[50](https://arxiv.org/html/2607.03346#bib.bib37)\]as the shared pretrained model\. To construct a set of heterogeneous vision tasks, we follow the dataset selection used in the Fusion benchmark\[[54](https://arxiv.org/html/2607.03346#bib.bib19)\], which was designed for evaluating model merging across diverse visual domains\.

Specifically, we consider: MNIST\[[34](https://arxiv.org/html/2607.03346#bib.bib43)\]\(handwritten digit classification\), CIFAR\-10\[[33](https://arxiv.org/html/2607.03346#bib.bib42)\]\(natural object classification\), GTSRB\[[52](https://arxiv.org/html/2607.03346#bib.bib14)\]\(traffic sign recognition\), SVHN\[[44](https://arxiv.org/html/2607.03346#bib.bib15)\]\(street\-view house number digit classification\), Stanford Cars\[[32](https://arxiv.org/html/2607.03346#bib.bib16)\]\(fine\-grained car model classification\), EuroSAT\[[23](https://arxiv.org/html/2607.03346#bib.bib17)\]\(satellite land\-use and land\-cover classification\), Oxford\-IIIT\[[47](https://arxiv.org/html/2607.03346#bib.bib38)\]\(pet category\), DTD\[[13](https://arxiv.org/html/2607.03346#bib.bib18)\]\(describable texture classification\), SUN397\[[66](https://arxiv.org/html/2607.03346#bib.bib20)\]\(scene recognition across diverse environments\), RESISC45\[[9](https://arxiv.org/html/2607.03346#bib.bib76)\]\(Remote sensing image scene classification\), Fashion\-MNIST\[[65](https://arxiv.org/html/2607.03346#bib.bib77)\]\(clothing category\), and Food101\[[5](https://arxiv.org/html/2607.03346#bib.bib75)\]\(food categories\)\. These datasets span multiple visual domains and data distributions, forming a heterogeneous multi\-task setting suitable for evaluating multi\-domain multi\-task dataset valuation\.

##### Results And Discussion\.

As summarized in Table[7](https://arxiv.org/html/2607.03346#A1.T7), DMVM consistently recovers the ground\-truth seller ranking with perfect agreement across all multi\-domain settings we consider\. Both correlations remain equal to 1\.0, under both non\-overlapping and overlapping task\-distribution schemes\. This indicates that, even in the presence of substantial domain heterogeneity—ranging from digits and traffic signs to satellite imagery, textures, scenes, and fine\-grained object categories—DMVM preserves the relative importance ordering of sellers exactly\. The results demonstrate that our valuation proxy remains stable as the number of target tasks grows and as inter\-seller task overlap increases, highlighting its robustness in realistic multi\-domain multi\-task vision scenarios\.

Table 8:Total wall\-clock time required to compute dataset Shapley values using MTL \(ground\-truth\) versus DMVM on language tasks\. Measured averages: training = 1 hour, evaluation = 0\.1 hour \(6 minutes\), merging \(negligible≈\\approx0\)\.

### A\.8Harmful Dataset Detection

![Refer to caption](https://arxiv.org/html/2607.03346v1/x9.png)Figure 5:DMVM\-induced shapley values of sellers with different percentages of mislabeled data\. As label noise increases, the seller’s contribution decreases and eventually becomes negative, showing that DMVM successfully identifies and penalizes harmful data providers\.To examine whether DMVM can distinguish between helpful and harmful data providers, we design a controlled synthetic experiment in which a seller’s dataset is progressively degraded by randomly flipping an increasing fraction of its labels\. This setup allows us to systematically vary data quality while keeping all other factors fixed\. Figure[5](https://arxiv.org/html/2607.03346#A1.F5)illustrates the resulting Shapley value of the seller as a function of the mislabeling rate\. We observe a clear and consistent monotonic decline in valuation as the noise level increases, indicating that DMVM is sensitive to degradation in data quality\. This behavior highlights an important practical property of DMVM: beyond ranking relevant contributors highly, it can also reliably detect and penalize harmful or low\-quality data sources in a multi\-task learning setting\.

![Refer to caption](https://arxiv.org/html/2607.03346v1/x10.png)Figure 6:Projected wall\-clock cost of each valuation method as the marketplace sizenngrows, computed from the asymptotic formulas in Table[1](https://arxiv.org/html/2607.03346#S4.T1)withTtrain=20T\_\{\\mathrm\{train\}\}=20min andTinference=1T\_\{\\mathrm\{inference\}\}=1min\.
### A\.9Scalability Analysis

To assess how DMVM scales beyond the regimes we measured directly, we project the wall\-clock cost of each method in Table[1](https://arxiv.org/html/2607.03346#S4.T1)usingTtrain=20T\_\{\\mathrm\{train\}\}=20minutes andTinference=1T\_\{\\mathrm\{inference\}\}=1minute, a reasonable assumption for regular downstream tasks and0\.50\.5–11B parameter models\. Figure[6](https://arxiv.org/html/2607.03346#A1.F6)reports the projected runtime as the marketplace size grows fromn=10n=10ton=30n=30sellers\. For the MC variants, we sweepm∈\{n,5​n,10​n\}m\\in\\\{n,5n,10n\\\}to span different possible sample budgets\. The figure shows two qualitatively different regimes\. Exact Dataset Shapley and deterministic DMVM both scale as𝒪​\(2n\)\\mathcal\{O\}\(2^\{n\}\)and become infeasible pastn≈11n\\\!\\approx\\\!11: atn=30n=30, exact Shapley requires more than10810^\{8\}hours of computation, while deterministic DMVM, despite avoiding all training, still exceeds10710^\{7\}hours\. The polynomial methods are tractable in this regime, but DU\-Shapley and MC\-Dataset\-Shapley both require retraining on sellers’ raw data, which is incompatible with the decentralized setting we target\. MC\-DMVM is the only method that is simultaneously polynomial innn, dependent only onTinferenceT\_\{\\mathrm\{inference\}\}, and privacy\-aware: atn=30n=30it completes in approximately1515,7575, and150150hours form=nm=n,5​n5n, and10​n10n, respectively\. Real\-world data marketplaces could involve tens of sellers per valuation round, and Figure[6](https://arxiv.org/html/2607.03346#A1.F6)shows that this is precisely the regime where MC\-DMVM remains practical while requiring a single local fine\-tuning per seller and no raw\-data exchange\.

Moreover, to better study the effect of scaling number of sellers,nn, and number of buyer’s target tasks,\|𝒯\|\|\\mathcal\{T\}\|, on the valuation performance, we have provided an extended version of Table[2](https://arxiv.org/html/2607.03346#S5.T2)and Table[3](https://arxiv.org/html/2607.03346#S5.T3)\. Figures[7](https://arxiv.org/html/2607.03346#A1.F7)and[8](https://arxiv.org/html/2607.03346#A1.F8)report DMVM’s empirical performance across the full\(n,\|𝒯\|\)\(n,\|\\mathcal\{T\}\|\)grid, for vision and NLP, respectively\. Across all configurations, Kendall’sτ\\tauremains above0\.690\.69, Spearman’sρ\\rhoabove0\.800\.80, and MSE stays on the order of10−310^\{\-3\}in both modalities\. MSE decreases as\|𝒯\|\|\\mathcal\{T\}\|grows, since averaging utility over more validation tasks reduces variance in the estimated coalition values, while rank\-correlation metrics decline mildly with bothnnand\|𝒯\|\|\\mathcal\{T\}\|: an expected effect, since larger marketplaces introduce more pairwise orderings to recover\. The decline is gradual rather than abrupt, indicating that DMVM remains stable across the parameter range relevant to multi\-task data marketplaces\.

![Refer to caption](https://arxiv.org/html/2607.03346v1/x11.png)

![Refer to caption](https://arxiv.org/html/2607.03346v1/x12.png)

![Refer to caption](https://arxiv.org/html/2607.03346v1/x13.png)

Figure 7:DMVM performance on vision tasks across the full\(n,\|𝒯\|\)\(n,\|\\mathcal\{T\}\|\)grid\. Left to right: Kendall’sτ\\tau, Spearman’sρ\\rho, and MSE relative to ground\-truth Dataset Shapley\.![Refer to caption](https://arxiv.org/html/2607.03346v1/x14.png)

![Refer to caption](https://arxiv.org/html/2607.03346v1/x15.png)

![Refer to caption](https://arxiv.org/html/2607.03346v1/x16.png)

Figure 8:DMVM performance on NLP tasks across the full\(n,\|𝒯\|\)\(n,\|\\mathcal\{T\}\|\)grid\. Left to right: Kendall’sτ\\tau, Spearman’sρ\\rho, and MSE relative to ground\-truth Dataset Shapley\.
### A\.10Implementation Details

All experiments are implemented in PyTorch\. For both vision and NLP experiments, seller models are fine\-tuned from a shared pretrained initialization using the Adam optimizer\. Since computing the ground\-truth Dataset Shapley values requires training models for an exponential number of seller coalitions, we use a lightweight fine\-tuning setup and train most coalition models for only22–33epochs\. This makes the ground\-truth computation feasible while still providing a consistent reference for comparing valuation methods\.

For the NLP experiments, we usegoogle/flan\-t5\-smallas the shared pretrained model\. The main NLP results are obtained using a batch size of3232and learning rate3×10−43\\times 10^\{\-4\}\. For the vision experiments, we useopenai/clip\-vit\-base\-patch32as the shared pretrained model\. The main vision results are obtained using a batch size of6464, learning rate10−510^\{\-5\}, and weight decay10−410^\{\-4\}\. For merging the seller models in each subset𝒮\\mathcal\{S\}, we use task arithmetic with scaling coefficientα=1\|𝒮\|\\alpha=\\frac\{1\}\{\|\\mathcal\{S\}\|\}, corresponding to averaging the task vectors within the subset\. For each buyer’s target task, we evaluate utility on at most20002000validation examples, using the full test set when it contains fewer than20002000examples\. The buyer’s utility is then computed by averaging validation performance across the target tasks\. All main experiments can be run on a single NVIDIA H100 GPU\. The main computational bottleneck is the retraining\-based ground\-truth and baseline computation, whereas DMVM only requires model merging over sellers and inference over the buyer’s validation tasks\.

### A\.11Task Distribution Schemes

In the controlled simulation experiments, for buyers with target task sets of varying cardinalities, we design task allocation schemes for both non\-overlapping and overlapping seller constructions\. In the non\-overlapping schemes, each fully relevant seller is assigned a distinct number of tasks from𝒯\\mathcal\{T\}\. This avoids ties or near\-ties in Shapley rankings and ensures a clear separation in seller relevance\. Sellers with irrelevant data may contain data from multiple irrelevant tasks or domains and can represent different combinations of such irrelevant sources\. In addition, a set of irrelevant data may come from only a subset of the irrelevant domains\.[Tables˜9](https://arxiv.org/html/2607.03346#A1.T9),[10](https://arxiv.org/html/2607.03346#A1.T10),[11](https://arxiv.org/html/2607.03346#A1.T11),[12](https://arxiv.org/html/2607.03346#A1.T12)and[13](https://arxiv.org/html/2607.03346#A1.T13)illustrate the high\-level structure of these task distribution schemes\.

Table 9:Task distribution for\|𝒯\|=3\|\\mathcal\{T\}\|=3\.Table 10:Task distribution for\|𝒯\|=4\|\\mathcal\{T\}\|=4\.Table 11:Task distribution for\|𝒯\|=5\|\\mathcal\{T\}\|=5\.Table 12:Task distribution for\|𝒯\|=6\|\\mathcal\{T\}\|=6\.Table 13:Task distribution for\|𝒯\|=7\|\\mathcal\{T\}\|=7\.
### A\.12Impact Statement

This work introduces DMVM, a framework for decentralized multi\-task dataset valuation using model merging and secure multiparty computation\. By enabling estimation of dataset utility without sharing raw data, our method supports fairer data marketplaces, incentivizes high\-quality data contribution, and facilitates safer collaboration in sensitive domains such as healthcare and finance\. At the same time, valuation mechanisms can be strategically manipulated or used to prioritize short\-term performance over broader societal benefit\. As with many advances in machine learning infrastructure, we encourage future work on governance, auditing, and fairness\-aware valuation mechanisms to mitigate these risks\.