Chunked Data Shapley (C-DaSh)

• Chunked Data Shapley (C-DaSh) is a scalable method that partitions datasets into non-overlapping chunks and estimates Shapley-style contributions via one-step SGD updates.
• It achieves significant computational speedups (up to 2300×) compared to traditional tuple-level Shapley approximations by reducing the granularity of evaluation.
• C-DaSh is effective in practical settings for quality assessment on large tabular datasets, aiding in data cleaning, removal, and outlier detection in both classification and regression tasks.

Searching arXiv for recent and directly related papers on Chunked Data Shapley and adjacent structured Shapley approximations. Chunked Data Shapley (C-DaSh) is a scalable approximation to Data Shapley for large tabular datasets that changes the unit of valuation from individual tuples to chunks of tuples, with the aim of making dataset quality assessment feasible at scales where exact or existing approximate Shapley methods are too slow (Loizou et al., 22 Aug 2025). In C-DaSh, the dataset is partitioned into non-overlapping chunks, a Shapley-style contribution is estimated for each chunk rather than each tuple, and the resulting chunk scores are used as data-quality signals for identifying high-quality and low-quality regions of the data (Loizou et al., 22 Aug 2025). The method is positioned within a broader line of research on efficient data valuation that includes the original Data Shapley framework (Ghorbani et al., 2019), grouped valuation via Owen values (Bax, 2019), stability-based small-coalition approximations (Watson et al., 2022), cardinality-restricted semivalues such as $\delta$-Shapley (Watson et al., 2023), proxy formulations such as DU-Shapley (Garrido-Lucero et al., 2023), and model-specific scalable coalition evaluation for LLM alignment (Tamine et al., 12 Dec 2025). Within that landscape, C-DaSh is distinctive in that it is explicitly chunk-based, is aimed at practical dataset quality assessment, and is empirically evaluated on large tabular classification and regression tasks with reported speedups between $80\times$ and $2300\times$ over prior approximations (Loizou et al., 22 Aug 2025).

1. Conceptual basis and relation to Data Shapley

Data Shapley treats each training datum as a player in a cooperative game and assigns value according to average marginal contribution to model performance over all subsets of the dataset (Ghorbani et al., 2019). In the standard formulation used for supervised learning, a dataset $D=\{(x_i,y_i)\}_{i=1}^n$, a learning algorithm $\mathcal A$, and a performance function $V(S)$ over subsets $S\subseteq D$ define the game, and the Shapley value of datum $i$ is the expectation of $V(S_\pi^i\cup\{i\})-V(S_\pi^i)$ over permutations $\pi$ (Ghorbani et al., 2019). The same player-by-subset logic underlies dataset valuation more broadly, including owner-level games (Garrido-Lucero et al., 2023) and source-level valuation for LLM alignment (Tamine et al., 12 Dec 2025).

C-DaSh adopts this game-theoretic view but changes the granularity of the players. Instead of asking for the value of each tuple $80\times$0, it partitions the dataset into $80\times$1 distinct, non-overlapping chunks $80\times$2 and estimates a score $80\times$3 for each chunk (Loizou et al., 22 Aug 2025). The practical motivation is explicit: the paper argues that the granularity of valuation itself must be changed when exact Data Shapley, with complexity $80\times$4, and tuple-level approximations such as TMC-Shapley and G-Shapley remain too expensive for large datasets (Loizou et al., 22 Aug 2025).

This chunk-level reformulation places C-DaSh in partial continuity with earlier grouped-valuation ideas. “Computing a Data Dividend” studies individual sample valuation with Shapley values and contributor-level or coalition-structured valuation with Owen values, where pre-defined groups of samples are treated as coalitions and attribution is available both within and across groups (Bax, 2019). That paper is a conceptual precursor to chunked valuation, but its exact polynomial-time algorithms are restricted to structured frequency-based decision rules and binary $80\times$5-NN classification rather than general tabular ML pipelines (Bax, 2019). C-DaSh, by contrast, is presented as a practical approximation for large tabular datasets using chunk-level Shapley-style estimation and SGD-based model-state updates (Loizou et al., 22 Aug 2025).

A further distinction is that C-DaSh is not merely “group Shapley” in the sense mentioned in the original Data Shapley paper, where one can compute Shapley values for groups by replacing point $80\times$6 with group $80\times$7 if the number of groups is reasonable (Ghorbani et al., 2019). It adds a specific computational recipe: optimized subset selection, one-step SGD checkpoint updates, and iterative chunk scoring geared toward large-scale dataset quality assessment (Loizou et al., 22 Aug 2025).

2. Formal formulation and chunk-level approximation

The dataset in C-DaSh is written as

$80\times$8

with learning algorithm $80\times$9 and performance metric $2300\times$0, where examples of $2300\times$1 include accuracy, F1, MSE, and RMSE (Loizou et al., 22 Aug 2025). The paper restates the standard tuple-level Data Shapley value for tuple $2300\times$2 under metric $2300\times$3 as

$2300\times$4

with $2300\times$5 serving directly as the utility function induced by training $2300\times$6 on a subset and evaluating performance (Loizou et al., 22 Aug 2025).

C-DaSh replaces tuple-level players by chunks. The dataset is partitioned into $2300\times$7 chunks of equal size $2300\times$8, with $2300\times$9 and $D=\{(x_i,y_i)\}_{i=1}^n$0, and the chunk score vector is denoted

$D=\{(x_i,y_i)\}_{i=1}^n$1

The chunks are described as distinct, non-overlapping, and usually equal-sized (Loizou et al., 22 Aug 2025). For classification experiments they are formed by fixed-size grouping, while for regression or time-sensitive datasets the paper also uses semantically meaningful temporal chunks such as daily or monthly intervals (Loizou et al., 22 Aug 2025).

The chunk-level approximation is given as

$D=\{(x_i,y_i)\}_{i=1}^n$2

where $D=\{(x_i,y_i)\}_{i=1}^n$3 is the $D=\{(x_i,y_i)\}_{i=1}^n$4-th chunk, $D=\{(x_i,y_i)\}_{i=1}^n$5 is the selected collection of subsets, $D=\{(x_i,y_i)\}_{i=1}^n$6 is a selected subset not containing chunk $D=\{(x_i,y_i)\}_{i=1}^n$7, $D=\{(x_i,y_i)\}_{i=1}^n$8 is a constant variable, and $D=\{(x_i,y_i)\}_{i=1}^n$9 denotes a model checkpoint or parameter state associated with chunk $\mathcal A$0 and subset $\mathcal A$1 (Loizou et al., 22 Aug 2025). The marginal term $\mathcal A$2 is the estimated contribution of adding chunk $\mathcal A$3 (Loizou et al., 22 Aug 2025).

The paper states explicitly that this is not the classical Shapley coefficient. The weighting term is

$\mathcal A$4

and the paper does not further derive this weighting or provide a theoretical proof for it (Loizou et al., 22 Aug 2025). This matters for interpretation. C-DaSh is therefore a chunk-level approximation introduced for practicality rather than a theorem-backed unbiased estimator of either point-level or exact chunk-level Shapley values (Loizou et al., 22 Aug 2025).

A useful comparison is with several other structured approximations. The Layered Shapley Algorithm estimates the original full-data Shapley value by stratifying coalitions by size and sampling disproportionately from small coalitions, with sample allocation $\mathcal A$5 under a diminishing-returns condition $\mathcal A$6 (Watson et al., 2022). $\mathcal A$7-Shapley similarly restricts valuation to a band of coalition sizes and treats the result as a semivalue rather than exact Shapley (Watson et al., 2023). DU-Shapley reduces the state space by approximating owner-level coalition utility through total aggregate dataset size $\mathcal A$8 and replacing combinatorial averages by a discrete uniform average over $\mathcal A$9 representative size points (Garrido-Lucero et al., 2023). These methods do not partition the data into chunks as players in the same way as C-DaSh, but they share the broader premise that valuation can be made practical by exploiting structure rather than treating all coalitions uniformly.

3. Algorithmic workflow

C-DaSh proceeds through a fixed pipeline centered on chunking, subset selection, and single-iteration SGD updates (Loizou et al., 22 Aug 2025). First, the dataset is partitioned into $V(S)$0 non-overlapping chunks of size $V(S)$1. Second, chunk scores are initialized as $V(S)$2 for $V(S)$3. Third, $V(S)$4 candidate subsets are randomly initialized and then filtered through a subset selection mechanism (Loizou et al., 22 Aug 2025).

The subset selection stage is one of the main algorithmic components emphasized in the paper. It imposes two constraints. First, each chunk should not appear in more than $V(S)$5 of the selected subsets. Second, subsets that fail a threshold test are regenerated: for classification, subsets with metric lower than $V(S)$6 are regenerated; for regression, subsets with metric higher than $V(S)$7 are regenerated (Loizou et al., 22 Aug 2025). This is intended to make the selected subset family both diverse and sufficiently informative (Loizou et al., 22 Aug 2025).

After subset selection, model parameters are initialized randomly,

$V(S)$8

Then, for each chunk $V(S)$9, C-DaSh considers subsets $S\subseteq D$0, builds a model state using subset information, estimates the performance difference when $S\subseteq D$1 is added, and accumulates a chunk-level Shapley-like contribution (Loizou et al., 22 Aug 2025). The central approximation that keeps the procedure scalable is the use of one-step SGD checkpoint updates rather than full retraining from scratch.

The model checkpoint update is

$S\subseteq D$2

where $S\subseteq D$3 is the previous checkpoint, $S\subseteq D$4 is the learning rate at chunk $S\subseteq D$5, and $S\subseteq D$6 is the optimized loss function (Loizou et al., 22 Aug 2025). The paper presents this as a single-iteration stochastic gradient descent approximation mechanism for estimating how the model would change when trained on subset $S\subseteq D$7 (Loizou et al., 22 Aug 2025). The intuition given is that model parameters encode “knowledge information” from previously included data, making marginal contribution estimation more accurate than subset-only black-box evaluation (Loizou et al., 22 Aug 2025).

The loop is repeated until a truncation criterion is satisfied, described as following “a similar approach” to Ghorbani et al. (Loizou et al., 22 Aug 2025). The paper does not provide a precise mathematical stopping rule for truncation, so that aspect is not fully specified (Loizou et al., 22 Aug 2025).

An implementation-oriented summary consistent with the paper is given below.

Stage	Operation	Key controls
Partitioning	Split $S\subseteq D$8 into non-overlapping chunks $S\subseteq D$9	chunk size $i$0
Subset selection	Generate $i$1 subsets, enforce 25% chunk-frequency cap, threshold-filter weak subsets	$i$2, $i$3
Scoring	Use one-step SGD checkpoints and accumulate chunk contributions	$i$4, $i$5, truncation criterion

This suggests that C-DaSh is best understood as a practical chunk-level valuation pipeline rather than a closed-form valuation rule. A plausible implication is that its behavior depends materially on how chunk size, subset diversity, and threshold filtering interact with the data distribution and the training dynamics.

4. Computational rationale and empirical scaling

The computational motivation for C-DaSh is direct. Exact Data Shapley has complexity

$i$6

and the paper summarizes TMC-Shapley as $i$7, with $i$8 in practice, leading to roughly $i$9 behavior, while G-Shapley is given as

$V(S_\pi^i\cup\{i\})-V(S_\pi^i)$0

where $V(S_\pi^i\cup\{i\})-V(S_\pi^i)$1 is the number of gradient descent steps or permutation passes and $V(S_\pi^i\cup\{i\})-V(S_\pi^i)$2 is model parameter dimensionality (Loizou et al., 22 Aug 2025). These approximations remain expensive on large datasets with tens or hundreds of thousands of tuples (Loizou et al., 22 Aug 2025).

For C-DaSh, no formal asymptotic complexity formula is given. The paper instead argues qualitatively that computation drops because valuation is performed over chunks and only $V(S_\pi^i\cup\{i\})-V(S_\pi^i)$3 selected subsets are used, with $V(S_\pi^i\cup\{i\})-V(S_\pi^i)$4 and $V(S_\pi^i\cup\{i\})-V(S_\pi^i)$5 (Loizou et al., 22 Aug 2025). Thus the empirical efficiency gain comes from three sources: fewer players to value, fewer subsets to evaluate, and one-step SGD updates instead of full retraining (Loizou et al., 22 Aug 2025).

The reported speedups are substantial. For the best chunk size of 250, the paper reports:

Dataset	Speedup over G-Shapley	Speedup over TMC-Shapley
Adult	$V(S_\pi^i\cup\{i\})-V(S_\pi^i)$6	$V(S_\pi^i\cup\{i\})-V(S_\pi^i)$7
Bank Marketing	$V(S_\pi^i\cup\{i\})-V(S_\pi^i)$8	$V(S_\pi^i\cup\{i\})-V(S_\pi^i)$9
MIMIC-III	$\pi$0	$\pi$1

These values are drawn from Table 3 and are summarized in the abstract as speedups between $\pi$2 and $\pi$3 (Loizou et al., 22 Aug 2025). The speedup metric is

$\pi$4

where $\pi$5 is baseline runtime and $\pi$6 is C-DaSh runtime with subset selection (Loizou et al., 22 Aug 2025).

The paper also reports a chunk-size tradeoff. Speed improves as chunk size increases up to around 250, then degrades beyond that because larger chunks mix high- and low-quality tuples and make subset selection less effective (Loizou et al., 22 Aug 2025). This suggests a nontrivial bias-variance-style compromise: smaller chunks preserve finer resolution but reduce scalability; larger chunks increase efficiency but blur localized quality signals. The paper’s own recommended operating point is around 250 tuples per chunk (Loizou et al., 22 Aug 2025).

This tradeoff has close analogues in related literature. The stability-based Layered Shapley work argues that large coalitions carry little incremental signal under $\pi$7 marginal bounds and can therefore be sampled only rarely (Watson et al., 2022). $\pi$8-Shapley similarly argues that small-to-medium coalitions often carry most of the useful valuation signal per unit compute, with reported speedup of up to $\pi$9 while preserving rank correlation with Monte Carlo Shapley in the range $80\times$00 across several datasets (Watson et al., 2023). DU-Shapley reduces combinatorial support from $80\times$01 to $80\times$02 under a size-based utility assumption (Garrido-Lucero et al., 2023). C-DaSh differs in mechanism, but the same general principle is visible: coarse-graining the coalition problem is what makes large-scale valuation tractable.

5. Empirical use for dataset quality assessment

The paper evaluates C-DaSh on five real-world tabular datasets. The classification datasets are Adult, Bank Marketing, and MIMIC-III; the regression datasets are Air Quality and Household Power Consumption (HPC) (Loizou et al., 22 Aug 2025). For classification, the model is an MLP with two hidden layers; for regression, a similarly structured MLP for regression is used (Loizou et al., 22 Aug 2025). Categorical variables are one-hot encoded, and the main baselines are G-Shapley and TMC-Shapley from Ghorbani et al. (Loizou et al., 22 Aug 2025).

The core evaluation protocol is valuation-guided removal. Chunk or data scores are computed, the lowest-scoring fraction $80\times$03 is removed, the predictor is retrained, and downstream performance is measured. For classification the metric is accuracy; for regression it is RMSE. The MLP is trained five times per experiment and average performance is reported (Loizou et al., 22 Aug 2025). The rationale is standard in data valuation: if removing low-valued data improves performance, the valuation method has identified harmful or low-quality regions.

Across the three classification datasets, the paper reports that C-DaSh yields slightly better or similar accuracy to G-Shapley on Adult, is competitive with or slightly better than G-Shapley on Bank Marketing, and shows the clearest advantage on MIMIC-III, outperforming G-Shapley by about $80\times$04 and TMC-Shapley by about $80\times$05 in some removal settings (Loizou et al., 22 Aug 2025). These claims are specifically tied to the removal-based evaluation used in the paper.

The paper also performs controlled corruption experiments on Adult by injecting Gaussian noise into $80\times$06 of tuples and label corruption into $80\times$07 of labels (Loizou et al., 22 Aug 2025). For Gaussian noise, at $80\times$08, C-DaSh reportedly outperforms G-Shapley by about $80\times$09 and TMC-Shapley by about $80\times$10 (Loizou et al., 22 Aug 2025). For label corruption, TMC-Shapley is better at very small removal $80\times$11, but C-DaSh is better at larger $80\times$12 (Loizou et al., 22 Aug 2025). This suggests that chunk-level valuation is especially effective when low-quality data are regionally concentrated enough for chunking to capture them (Loizou et al., 22 Aug 2025).

For outlier detection, the paper uses Local Outlier Factor after data removal and reports that C-DaSh achieves the lowest LOF scores across all three classification datasets and tested $80\times$13 values, especially strong at $80\times$14 (Loizou et al., 22 Aug 2025). The intended interpretation is that C-DaSh captures many outliers early (Loizou et al., 22 Aug 2025).

The paper also compares C-DaSh with a “chunk average” baseline, where tuple-level Shapley scores from G-Shapley or TMC-Shapley are averaged within chunks. The text notes wording inconsistencies, but the intended conclusion is that native chunk evaluation in C-DaSh can identify low-quality chunks more precisely than simple post-hoc averaging, especially at higher removal percentages (Loizou et al., 22 Aug 2025). This is an important distinction: chunking is not just an aggregation of tuple-level values after the fact, but part of the valuation mechanism itself.

For regression, the paper evaluates C-DaSh on Air Quality and HPC and states that daily chunks outperform monthly chunks because monthly chunks are too coarse and mix good and bad temporal segments (Loizou et al., 22 Aug 2025). This reinforces the chunk-granularity theme: semantically meaningful chunking can be beneficial, but chunks that are too coarse degrade discriminatory power.

6. Position within the broader research landscape

C-DaSh sits within a broader family of scalable data valuation methods, but its method of scaling is distinct. The original Data Shapley framework supplies the normative and formal baseline: value is expected marginal contribution under retraining-and-evaluation utility, with axiomatic motivation from null-player, symmetry, and additivity (Ghorbani et al., 2019). Its major practical bottlenecks are the combinatorial number of subsets and the cost of repeated model retraining (Ghorbani et al., 2019).

Several later works preserve the original player definition while exploiting structure in coalition space. “Differentially Private Shapley Values for Data Evaluation” shows that under empirical risk minimization and uniform stability, marginal contribution can decay as $80\times$15, motivating the Layered Shapley Algorithm, which stratifies coalitions by size and allocates expected samples as

$80\times$16

to estimate the full-data Shapley value while touching only a small fraction of the dataset per query (Watson et al., 2022). That work is not chunk-based in the sense of prepartitioned groups, but it is conceptually close to small-chunk or random-chunk Shapley estimation (Watson et al., 2022).

“Accelerated Shapley Value Approximation for Data Evaluation” reaches a similar conclusion through $80\times$17-Shapley, a semivalue that truncates coalition sizes to a band $80\times$18 and reports preserved ranking behavior with up to $80\times$19 speedup (Watson et al., 2023). Again, the restriction is by coalition cardinality rather than by fixed chunks.

DU-Shapley follows a different route. It assumes coalition utility depends primarily on aggregate sample size $80\times$20, replacing combinatorial averages by the deterministic proxy

$80\times$21

with support size $80\times$22 rather than $80\times$23 (Garrido-Lucero et al., 2023). This is not chunking either, but it is another example of structured reduction of the valuation problem.

At the opposite end, CHG Shapley changes the utility function itself. Rather than approximating the original retraining-based Shapley game through coalition sampling or chunking, it defines a gradient-and-hardness surrogate utility with closed-form Shapley values computable during a single training trajectory (Cai, 2024). That method is not chunked, but it is relevant as a methodological contrast: C-DaSh approximates valuation by changing the player granularity, whereas CHG Shapley approximates valuation by changing the game (Cai, 2024).

For grouped or source-level settings, there are also complementary lines of work. “Computing a Data Dividend” provides an Owen-value foundation for pre-defined contributor groups (Bax, 2019). For LLM alignment under Direct Preference Optimization, coalition utilities can be synthesized by arithmetic on singleton fine-tuned models rather than retraining each coalition, reducing the number of DPO fine-tunings from $80\times$24 to $80\times$25 in the exact coalition-enumeration setting (Tamine et al., 12 Dec 2025). That work is not chunking examples, but it is naturally compatible with chunk-level players if chunks are treated as data sources (Tamine et al., 12 Dec 2025).

A plausible synthesis is that C-DaSh represents one specific answer to the scalability problem: change the granularity of the cooperative game itself. Other methods preserve the original point-level game but restrict coalition sizes, exploit diminishing returns, use deterministic proxies, or exploit model-specific algebraic structure. These approaches are complementary rather than interchangeable.

7. Assumptions, limitations, and interpretive cautions

The paper presents C-DaSh as a practical method for dataset quality assessment on large tabular data, but it also leaves several points formally unresolved. First, no formal approximation error bound, consistency theorem, or variance analysis is given (Loizou et al., 22 Aug 2025). The chunk-level weighting term and the one-step SGD update are heuristic approximations rather than theorem-backed estimates of exact Shapley values (Loizou et al., 22 Aug 2025). This should be distinguished from methods such as the Layered Shapley Algorithm, which provides $80\times$26-style guarantees under diminishing marginal contribution assumptions (Watson et al., 2022), or DU-Shapley, which provides an explicit finite-sample approximation bound under its size-based utility model (Garrido-Lucero et al., 2023).

Second, chunk granularity is a central tradeoff. If chunks are too large, they mix high- and low-quality tuples and blur the score. The paper explicitly reports degradation beyond chunk size about 250 (Loizou et al., 22 Aug 2025). If chunks are too small, scalability gains shrink. This suggests that chunk size is not just an engineering parameter but part of the valuation model itself.

Third, C-DaSh values chunks, not individual points. It therefore identifies problematic regions rather than necessarily isolating the exact tuple responsible for low quality (Loizou et al., 22 Aug 2025). This is advantageous for workflows such as dataset debugging, cleaning, and regional inspection, but it changes the semantics relative to point-level Data Shapley (Loizou et al., 22 Aug 2025).

Fourth, the method depends on how corruption is distributed. The paper’s favorable results in Gaussian noise, label corruption, and time-sensitive regression settings suggest that chunk-level scoring is especially helpful when harmful data are spatially or temporally concentrated enough that chunks can localize them (Loizou et al., 22 Aug 2025). A plausible implication is that uniformly scattered corruption may be harder to detect with chunk-level valuation, because every chunk may contain a mixture of clean and noisy tuples.

Fifth, the subset-selection mechanism favors subsets that already perform reasonably well. The paper uses threshold-based regeneration of weak subsets and limits chunk repetition across subsets (Loizou et al., 22 Aug 2025). This may improve efficiency, but it may also bias the evaluation toward “good coalitions” and underrepresent difficult or atypical regions, a caveat the paper itself notes indirectly in discussing the subset selection design (Loizou et al., 22 Aug 2025).

Sixth, the method is evaluated only on tabular datasets. The paper explicitly positions C-DaSh as intended and validated primarily for large tabular datasets and does not experimentally validate image, text, or multimodal settings (Loizou et al., 22 Aug 2025). This distinguishes it from broader data valuation literature, much of which includes image and biomedical tasks (Ghorbani et al., 2019) or deep-learning selection settings (Cai, 2024).

Finally, reproducibility is only partial. The paper specifies some hyperparameters, including $80\times$27, $80\times$28 for classification, $80\times$29 for regression, $80\times$30 as a common subset count, and tested chunk sizes $80\times$31, while also noting an inconsistency between 250 and 256 in different sections (Loizou et al., 22 Aug 2025). It does not provide full architecture details, exact train/validation split details, random seed protocol, or exact truncation implementation (Loizou et al., 22 Aug 2025).

Taken together, these points indicate that C-DaSh is best read as a practically motivated region-level valuation heuristic with strong empirical runtime gains and useful dataset-quality signals, rather than as a formally characterized estimator of exact Data Shapley (Loizou et al., 22 Aug 2025). Within its validated scope—large tabular classification and regression pipelines—it offers a direct and operationally meaningful way to move from tuple-level valuation to chunk-level dataset quality assessment.