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Efficient Data Valuation Approximation in Federated Learning: A Sampling-based Approach

T0 review · 5 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read This paper claims that Shapley-value data valuation in federated learning can be approximated with under 1% relative error by evaluating only combinations of at most two clients, and proposes an algorithm, IPSS, that achieves this by…

desk verdict A practically useful algorithm and extensive experiments, but the theory has load-bearing holes and the key-combinations premise is not established beyond one benchmark. read the letter →

arxiv 2504.16668 v1 pith:FYZF6GMO submitted 2025-04-23 cs.LG cs.DB

Shuyue Wei , Yongxin Tong , Zimu Zhou , Tianran He , Yi Xu This is my paper · ORCID
classification cs.LGcs.DB
keywords Shapleyvaluedatavaluationfederatedlearningstratifiedsamplingapproximationalgorithmkeycombinationsmodelutility
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper is trying to make Shapley-value-based data valuation practical in cross-silo federated learning, where the exact value requires training and evaluating models on an exponential number of client subsets. It argues that most of those combinations do not matter: because model utility saturates as more clients join, only combinations with a few clients carry significant weight. On that premise it builds IPSS, a sampling algorithm that evaluates all combinations up to a small size and samples a balanced set of the next size, and it reports relative errors under 1% with large speedups over exact computation and over existing approximation baselines. If the premise holds broadly, fair monetary or credit allocation among data providers becomes computationally feasible.

What carries the argument

The central machinery is a stratified-sampling estimator built on the marginal-contribution form of the Shapley value. Dataset combinations are grouped into strata by the number of clients they contain, and the algorithm computes average marginal contributions within each stratum, pruning all strata above a cutoff $k^*$ because those combinations are conjectured to have negligible impact. The load-bearing mechanism is the key-combinations phenomenon: marginal utilities shrink as a coalition grows, so large coalitions contribute little to the final value. The variance comparison between MC-SV and CC-SV, and the closed-form MSE analysis for linear regression, are what justify the chosen scheme and the error bound.

What would settle it

Take a federated task where accuracy keeps climbing as clients are added (for example, a small-data, high-feature-dimension regression or a task where each client contributes unique hard examples), compute the exact Shapley values by training on all subsets, run IPSS with the same sampling budget, and check whether the relative error stays under the paper's bound; if it exceeds the bound, the key-combinations premise fails.

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Extended reading notes

Core claim

The paper's central claim is that data valuation by Shapley value in federated learning can be approximated accurately using only a small group of dataset combinations, a phenomenon it calls key combinations. Concretely, on FEMNIST with ten clients, combinations involving no more than two clients yield a relative error below 1% compared with the exact marginal-contribution Shapley value. The proposed algorithm, IPSS, prunes all combinations above a cutoff size $k^*$, evaluates the remaining combinations exactly, and samples a balanced set of combinations of size $k^*+1$, then estimates each client's value by averaging marginal contributions across strata. Under a linear-regression model with negative mean squared error as utility, the paper derives a relative error bound of $O((n-k^*)/(k^* n t))$, and it argues that the marginal-contribution scheme (MC-SV) has lower estimation variance than the complementary-contribution scheme (CC-SV) when both are used inside the same stratified-sampling framework.

Load-bearing premise

The argument assumes that the value of adding one more client's data becomes negligible once enough clients are already in the coalition, so the algorithm can safely ignore all large coalitions; if that saturation fails, the pruning step discards a large share of the Shapley sum and the reported error bounds no longer apply.

Editorial extensions

If this is right

  • Data valuation cost drops from $O(2^n)$ model trainings to $O(\gamma)$ for a user-chosen sampling budget $\gamma$.
  • On FEMNIST with ten clients, combinations of size at most two give relative error under 1%, so a small budget of model trainings can produce near-exact Shapley values.
  • Within the proposed stratified-sampling framework, the marginal-contribution scheme has lower variance than the complementary-contribution scheme, so MC-SV should be preferred for FL data valuation.
  • IPSS remains efficient and accurate with up to 100 clients, and it better satisfies the no-free-rider and symmetric-fairness properties than the comparison baselines.
  • Under the linear-regression utility model, the relative error of IPSS is bounded by $O((n-k^*)/(k^* n t))$, which is small when each client holds many samples relative to the feature dimension.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference: if the key-combinations phenomenon holds beyond the linear-regression and saturated-accuracy settings tested here, the same pruning idea could apply to other cooperative-game approximation problems where utility saturates, such as model markets or data cooperatives.
  • Editorial inference: the saturation premise suggests a practical diagnostic: a server could measure how quickly marginal contributions decay from a few pilot evaluations and then decide adaptively how many strata to prune, rather than assuming a fixed cutoff.
  • Editorial inference: for tasks where utility does not saturate within the relevant number of clients, such as small-data regimes or tasks where each additional client contributes unique hard examples, IPSS would need to include larger strata; the paper's error analysis has not been shown for those cases.
  • Editorial inference: because the pruning is justified empirically on one benchmark and theoretically only under linear regression, the strongest near-term test is to reproduce the key-combinations curve on other FL benchmarks and model families before relying on IPSS for financial settlement among providers.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

5 major / 5 minor

Summary. The paper proposes IPSS, a sampling-based approximation algorithm for Shapley-value data valuation in cross-silo federated learning. It first introduces a unified stratified-sampling framework that can instantiate both the marginal-contribution (MC-SV) and complementary-contribution (CC-SV) schemes, claims an unbiasedness result and a variance comparison favoring MC-SV, then identifies an empirically observed 'key combinations' phenomenon: on FEMNIST with ten clients, using only coalitions of size at most two gives relative error below 1%. Building on this, IPSS prunes all coalitions above a threshold k* and samples only a small number of size-(k*+1) coalitions. The paper provides a theoretical error bound under a linear-regression MSE utility model and reports extensive experiments on synthetic partitions of MNIST and on FEMNIST and Adult, showing that IPSS achieves low relative error and competitive runtime compared with several baselines.

Significance. If the central claims were established, the paper would address an important practical problem: approximate Shapley-value data valuation in FL with only a small number of FL model trainings. The proposed framework, the systematic comparison of MC-SV and CC-SV under stratified sampling, and the extensive benchmark evaluation are useful contributions. The paper also provides a public code repository. However, the theoretical results that support the main claims have serious gaps, and the key-combinations phenomenon is shown only on one configuration and under a specialized utility model; a simple complementary utility counterexample defeats the claimed small-error guarantee. The empirical results are suggestive but do not compensate for the load-bearing theoretical defects.

major comments (5)
  1. [Sec. III-A, Theorem 1 (Alg. 1, Eqs. (6)-(7))] The unbiasedness proof is not valid as written. In Algorithm 1, m_{i,k} is a random variable counting how many sampled coalitions containing i have a sampled paired coalition, and the estimator in line 17 divides φ̂_{i,k} by m_{i,k}. The proof in Eq. (6) treats m_{i,k} as a fixed denominator, writing E[φ̂_{i,k}/m_{i,k}] = E[φ̂_{i,k}]/m_{i,k}, which is not justified for a ratio estimator. Moreover, m_{i,k}=0 occurs with positive probability, in which case the estimator is undefined. Thus Theorem 1 does not establish that Algorithm 1 gives an unbiased estimate of the Shapley value.
  2. [Sec. III-B, Theorem 2 (Eqs. (8)-(11))] The variance comparison rests on an unjustified and very strong assumption. Eq. (8) states Var[Σ_{j=1}^t e_j] = t^2 σ^2, which treats the per-sample prediction errors as perfectly correlated across all t training samples. This is not derived from a standard linear-regression noise model and is implausible as a general assumption. In addition, the proof compares MC-SV and CC-SV only when both use the exact same m_{i,k} values, whereas the theorem statement claims a comparison 'for any sampling strategy' of CC-SV. Therefore the conclusion that MC-SV always has lower variance in the stratified framework is not established.
  3. [Sec. IV-C, Theorem 3 (Alg. 3, Eqs. (16)-(19))] Theorem 3's proof bounds a deterministic truncation error, not the stochastic estimator produced by Algorithm 3. The quantity E[φ̂_i^{k*}] in Eq. (16) is the expected value obtained when all coalitions of size at most k* are fully evaluated, i.e., when strata above k* are simply discarded. However, Algorithm 3 lines 8-14 do not discard stratum k*; they randomly sample a set P of size-(k*+1) coalitions, and line 16 uses only the sampled P in the second sum. The variance and sampling error introduced by P are never analyzed. Hence the claimed error bound O((n-k*)/(k* n t)) does not apply to the random output of Algorithm 3.
  4. [Sec. IV-A (Fig. 4, 'Key Combinations Phenomenon')] The key-combinations premise is not established for general FL utility functions. The empirical evidence is limited to one benchmark (FEMNIST) with ten clients, and the theoretical support in Lemma 1 and Theorem 3 assumes a linear-regression MSE utility whose expected value depends only on the total sample size and decays as O(1/(t|S|)). No argument rules out FL tasks with complementary or near-additive utilities. For example, consider n=10 clients where each holds one disjoint digit class and U(S) is the test accuracy on all ten classes, so U(S)=|S|/10. Then every marginal contribution is exactly 1/10, each stratum contributes 1/10 of the total Shapley value, and truncating at |S|≤1 (the K=2 regime of Fig. 4) discards 80% of the Shapley mass, not <1%. The pruning step in IPSS therefore relies on an unproven saturation assumption, and the paper's central claim of 'minor approximation error' for arbitrary FL settings is not supported.
  5. [Sec. IV-C, Lemma 1 and Theorem 3 (utility model)] Even within the linear-regression MSE model, the analysis only covers a very narrow class of utilities: the expected MSE in Eq. (12) is taken from Donahue and Kleinberg [24] and depends only on the total number of training samples, not on which clients or classes are included. The paper does not state this restriction clearly in the main text or in the abstract, and the algorithm is advocated for general FL utility functions such as model accuracy on real benchmarks. The theoretical analysis should be explicitly framed as a case study, and the general claim that IPSS provably achieves small error should be withdrawn or substantially qualified.
minor comments (5)
  1. [Sec. V-A, Algorithm 2 line 7] In K-Greedy, the denominator is written as n·Δ{n \choose |S|}, but the MC-SV formula in Definition 3 requires n·Δ{n-1 \choose |S|}; this appears to be a typo and should be corrected.
  2. [Sec. V-A, 'Compared Algorithms'] The name 'Exteneded-TMC' is misspelled; it should be 'Extended-TMC'.
  3. [Sec. V-C, datasets] The text says 'validate our approximation algorithm on two real dataset' (and earlier mentions Sent-140), but only FEMNIST and Adult are reported. Please clarify whether Sent-140 results are omitted or whether the sentence is inaccurate.
  4. [Fig. 4 and Fig. 10] Figure 4 reports the key empirical phenomenon supporting the main pruning idea, but the figure caption and text do not report error bars, number of runs, or the variance across runs. Adding this information would help assess the robustness of the '<1%' claim.
  5. [Sec. IV-B, Example 3] The example does not explain how the sampled set P is chosen to satisfy the equal-frequency constraint (line 11) when n=4 and |P|=5; a brief explanation or a different illustrative choice would improve clarity.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper's derivation is self-contained, and the key-combinations phenomenon is an empirical observation plus a model-based error bound with external assumptions, not a fitted input or self-citation chain.

full rationale

The paper's central claim is that the Shapley value for federated-learning data valuation can be approximated accurately by pruning large dataset combinations. This claim rests on the key-combinations phenomenon observed in Sec. IV-A on FEMNIST and on the theoretical error bound in Theorem 3 derived under a specialized linear-regression MSE model from the external reference [24]. Neither of these is circular: the empirical observation is tested against exact MC-SV values, and the theoretical bound follows from an externally stated expected-MSE formula rather than from a parameter fitted to the target result. The algorithm IPSS (Alg. 3) does not fit constants to force low error; it simply enumerates all combinations up to a budget-determined size k* and samples a few combinations of size k*+1, and its reported approximation error is then evaluated against exact Shapley values. The self-citations to the authors' prior work [6], [9] appear as baseline algorithms and related-work references, not as load-bearing justifications for the pruning step or the error bound. The comparison of MC-SV versus CC-SV in Theorem 2 likewise uses an external variance model [22] and is not a reduction of the conclusion to an input assumption. The main weakness of the paper is generality: the key-combinations phenomenon is only empirically demonstrated on one benchmark with ten clients and theoretically justified under a restrictive MSE model, so the pruning step may fail for other FL utility functions. That is a correctness and scope risk, not a circularity, and per the review rules it should not raise the circularity score.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The central algorithm rests on several unverified assumptions: saturation of FL utility (key combinations), a specific linear-regression MSE model, and a variance formula implying perfectly correlated errors. These are not fitted to data, but they are not independently established either. The experimental budget gamma is hand-chosen. No new entities are introduced.

free parameters (1)
  • Sampling budget gamma per client count = gamma=5 (n=3), 8 (n=6), 32 (n=10)
    These budgets are fixed by hand for the experiments; they determine the truncation level k* and hence the error bound, so the reported efficiency/accuracy trade-off is contingent on these arbitrary values.
assumptions (4)
  • domain assumption FL utility is monotone and saturating: marginal utility of adding a client decreases as coalition size grows (key combinations phenomenon).
    Used in Sec. IV-A to justify pruning large coalitions; empirically observed but only theoretically derived under a specialized linear-regression MSE model (Sec. IV-C).
  • domain assumption For a linear regression model with |D| samples, expected MSE equals μ_e |x|/(|D|-|x|-1) (from Donahue and Kleinberg [24]).
    Invoked in Lemma 1 and Theorem 3 to derive the truncation error bound; assumes Gaussian features and a specific noise model.
  • ad hoc to paper Variance of the sum of per-sample regression errors is t^2 σ^2 (perfectly correlated errors).
    Introduced in Eq. (8) to prove MC-SV has lower variance than CC-SV; this is not a standard statistical assumption and is not justified.
  • domain assumption Each client holds a dataset of size t and all data are drawn from the same distribution.
    Used in Theorem 2 to derive variance formulas; not stated as limiting the practical claims.

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Cite this review

Pith. "Pith review of Efficient Data Valuation Approximation in Federated Learning: A Sampling-based Approach." pith.science (2026). https://pith.science/paper/FYZF6GMO

@misc{pith2026250416668,
  author       = {Pith},
  title        = {Pith review of: Efficient Data Valuation Approximation in Federated Learning: A Sampling-based Approach},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FYZF6GMO}},
  note         = {Machine review of arXiv:2504.16668}
}
read the original abstract

Federated learning paradigm to utilize datasets across multiple data providers. In FL, cross-silo data providers often hesitate to share their high-quality dataset unless their data value can be fairly assessed. Shapley value (SV) has been advocated as the standard metric for data valuation in FL due to its desirable properties. However, the computational overhead of SV is prohibitive in practice, as it inherently requires training and evaluating an FL model across an exponential number of dataset combinations. Furthermore, existing solutions fail to achieve high accuracy and efficiency, making practical use of SV still out of reach, because they ignore choosing suitable computation scheme for approximation framework and overlook the property of utility function in FL. We first propose a unified stratified-sampling framework for two widely-used schemes. Then, we analyze and choose the more promising scheme under the FL linear regression assumption. After that, we identify a phenomenon termed key combinations, where only limited dataset combinations have a high-impact on final data value. Building on these insights, we propose a practical approximation algorithm, IPSS, which strategically selects high-impact dataset combinations rather than evaluating all possible combinations, thus substantially reducing time cost with minor approximation error. Furthermore, we conduct extensive evaluations on the FL benchmark datasets to demonstrate that our proposed algorithm outperforms a series of representative baselines in terms of efficiency and effectiveness.

Figures

Figures reproduced from arXiv: 2504.16668 by the authors.

Figure 1
Figure 1. (a):Three hospitals collaborate to train the FL model and aim to identify each hospital’s data value. The SV-based data valuation requires training and evaluating FL models across all possible hospital combinations (①∼⑦), i.e., it needs to tackle seven FL processes. As client number increases, the number of required combinations grows exponentially. (b):Evaluations on the FL benchmark dataset FEMNIST with ten FL cli… view at source ↗
Figure 2
Figure 2. Example for the unified stratified sampling framework: [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Observations when using the MC-SV-based scheme. A. Identifying the Key Combinations Phenomenon Observations. It is essential to utilize the inherent properties of the MC-SV for effective and efficient data valuation in FL. To this end, we recall and examine the MC-SV-based computation scheme as depicted in [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: Results under combinations with size no more than [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: Example of Alg. 3 with the same setup as [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 6
Figure 6. Figure 6: Experimental results on the synthetic datasets with five different setups varying in size, distribution and quality. [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]
Figure 7
Figure 7. Figure 7: Results on FEMNIST when varying sampling rounds γ. (a) Client#3+MLP (b) Client#6+MLP (c) Client#10+MLP (d) Client#3+CNN (e) Client#6+CNN (f) Client#10+CNN [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]
Figure 8
Figure 8. Figure 8: Pareto curves for trade-off in efficiency and effectiveness. across three, six, and ten FL clients are shown in [PITH_FULL_IMAGE:figures/full_fig_p011_8.png]
Figure 9
Figure 9. Figure 9: Varying client number on FEMNIST using MLP model. 3) Scalability test for larger FL clients: We conduct ex￾periments with up to 100 FL clients, a large-scale scenario for cross-silo FL [1]–[3], where more than 1030 dataset combinations must be assessed by SV definition…

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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.