FedTSV: Fairness-Aware Federated Learning

• FedTSV is a federated learning aggregation method that uses trajectory Shapley values to dynamically evaluate each client’s long-term contribution.
• It replaces fixed client weights with adaptive, validation-based weights that reward beneficial updates and suppress harmful or infrequent contributions.
• Utilizing Monte Carlo approximations, FedTSV improves convergence and robustness in non-IID settings and under adversarial behavior.

Searching arXiv for the exact FedTSV paper and closely related federated contribution-valuation work. FedTSV is a federated learning aggregation method introduced in "Fairness-Aware Federated Learning with Trajectory Shapley Value" (Kuznetsov et al., 28 May 2026). It replaces fixed client weights with dynamic, contribution-aware weights derived from the Trajectory Shapley Value (TSV), a contribution metric designed to evaluate how each client influences the optimization trajectory of the global model over training rounds. In contrast to conventional schemes such as FedAvg, which typically use uniform or data-size-proportional aggregation weights, FedTSV values clients according to a validation-based, temporally consistent utility and then converts accumulated contribution estimates into nonnegative aggregation weights. The method is motivated by the observation that client usefulness in federated optimization is unequal, time-varying, and potentially corrupted by heterogeneity, intermittent participation, or malicious behavior (Kuznetsov et al., 28 May 2026).

1. Problem formulation and fairness objective

FedTSV is developed in the standard federated optimization setting with objective

$$\min_{\theta \in \mathbb{R}^d}\; F(\theta;\mathcal{D}) \coloneqq \sum_{i \in \mathcal{N}} \alpha_i\, F_i(\theta;\mathcal{D}_i),$$

where client $i$ has local loss $F_i(\cdot;\mathcal{D}_i)$ and $\alpha_i$ denotes its aggregation weight (Kuznetsov et al., 28 May 2026). In standard FedAvg, after local training, the server aggregates selected client models as

$$\theta^{t+1} = \sum_{i \in \mathcal{S}_t} \frac{\alpha_i}{\sum_{j \in \mathcal{S}_t}\alpha_j} \theta^{t+1}_i.$$

The core criticism advanced by FedTSV is that these weights are fixed even though client contributions are not (Kuznetsov et al., 28 May 2026).

The paper identifies four conditions under which fixed aggregation is inadequate: heterogeneous data, partial participation, time-varying utility, and adversarial or corrupted clients (Kuznetsov et al., 28 May 2026). Under such conditions, fixed weighting may produce biased and unstable learning because it fails to reflect "unequal and time-varying client contributions." A harmful or low-quality update may retain the same influence as a helpful one, while a genuinely useful but infrequently sampled client may be systematically undervalued (Kuznetsov et al., 28 May 2026).

FedTSV targets three related objectives simultaneously. First, it aims at fairness in contribution assessment by estimating each client’s actual marginal value more appropriately than static heuristics. Second, it seeks fairness in aggregation influence by translating contribution estimates into adaptive weights. Third, it treats robustness as part of the fairness problem by suppressing harmful or malicious clients through low or zero effective weight (Kuznetsov et al., 28 May 2026). This formulation makes FedTSV more than a post hoc valuation mechanism: the valuation directly modifies the training dynamics.

2. Trajectory Shapley Value

The conceptual foundation of FedTSV is the Shapley Value (SV), recalled in the paper as

$$\phi_i(u) = \sum_{\mathcal{S} \subseteq \mathcal{N} \setminus \{i\}} \frac{|\mathcal{S}|! \, (n - |\mathcal{S}| - 1)!}{n!} \big( u(\mathcal{S} \cup \{i\}) - u(\mathcal{S}) \big),$$

which measures the average marginal contribution of client $i$ over all coalition orderings (Kuznetsov et al., 28 May 2026). The paper also emphasizes the additive decomposition

$$u(\mathcal{S}) = \sum_{t=0}^{T-1} u^t(\mathcal{S}), \qquad \phi_i(u) = \sum_{t=0}^{T-1}\phi_i(u^t),$$

using the linearity of SV across rounds (Kuznetsov et al., 28 May 2026).

TSV preserves the Shapley principle but changes the utility being valued. Rather than using a static utility such as validation accuracy at a single endpoint, TSV defines utility from how well a coalition’s update follows a validation-derived descent direction at a particular round (Kuznetsov et al., 28 May 2026). The resulting notion of contribution is trajectory-based, temporally consistent, and validation-based.

The trajectory-based interpretation is motivated by viewing federated learning as a discrete approximation of gradient flow,

$\theta'(t) = -\nabla F\big(\theta(t)\big),$

so that each communication round is a step along an optimization trajectory (Kuznetsov et al., 28 May 2026). A coalition is therefore more valuable if its aggregate update aligns with the descent direction suggested by trusted validation data.

At round $t$, for a coalition $i$0, the coalition update is defined by

$i$1

The server also computes a validation reference update

$i$2

where $i$3 is obtained by running the same number of SGD steps on held-out validation data $i$4, starting from $i$5 (Kuznetsov et al., 28 May 2026).

The per-round TSV utility is then

$i$6

with $i$7 and

$i$8

for small $i$9 (Kuznetsov et al., 28 May 2026). In experiments, the distance is Euclidean,

$F_i(\cdot;\mathcal{D}_i)$0

though angular distance is explicitly mentioned as an alternative (Kuznetsov et al., 28 May 2026).

TSV is thus the Shapley value induced by the per-round utility $F_i(\cdot;\mathcal{D}_i)$1, accumulated over time through

$F_i(\cdot;\mathcal{D}_i)$2

Per round, TSV is $F_i(\cdot;\mathcal{D}_i)$3; over the optimization trajectory, it is the accumulated sum of these roundwise values (Kuznetsov et al., 28 May 2026). The paper’s use of "temporally consistent" refers precisely to this accumulation: contributions are not judged in isolation at one round or only at the end of training.

3. Algorithmic structure of FedTSV

FedTSV modifies the standard federated loop by adding server-side validation-reference computation and contribution-aware reweighting (Kuznetsov et al., 28 May 2026). At round $F_i(\cdot;\mathcal{D}_i)$4, the server samples a client subset $F_i(\cdot;\mathcal{D}_i)$5, and each selected client initializes from the current global model,

$F_i(\cdot;\mathcal{D}_i)$6

then performs $F_i(\cdot;\mathcal{D}_i)$7 local SGD steps

$F_i(\cdot;\mathcal{D}_i)$8

After $F_i(\cdot;\mathcal{D}_i)$9 steps, the client uploads $\alpha_i$0 (Kuznetsov et al., 28 May 2026).

On the server, the same number of SGD steps are run on validation data: $\alpha_i$1 and the reference update is set to

$\alpha_i$2

This construction requires the server to possess a held-out validation dataset, which the paper treats as a standard assumption in SV-based federated learning methods (Kuznetsov et al., 28 May 2026).

Using the uploaded client models, the server evaluates coalition utilities $\alpha_i$3 for subsets $\alpha_i$4 and estimates each participating client’s Shapley value $\alpha_i$5 (Kuznetsov et al., 28 May 2026). Exact enumeration is intractable, so the paper uses a Monte Carlo approximation "following the efficient estimation strategy in [Zhang et al., 2023]," but it does not provide the explicit estimator formula, permutation count, truncation rule, or stopping criterion (Kuznetsov et al., 28 May 2026).

The cumulative contribution state is updated as

$\alpha_i$6

and if client $\alpha_i$7, then $\alpha_i$8 for that round (Kuznetsov et al., 28 May 2026). FedTSV then maps contributions to aggregation weights via

$\alpha_i$9

This zero-clipping is central to the design: weights can become zero, negative cumulative contributions are clipped to zero, and effective aggregation weights are nonnegative (Kuznetsov et al., 28 May 2026). The paper states that this "suppresses clients with negative cumulative contributions, preventing them from adversely influencing the global model."

Finally, the server aggregates selected clients with dynamic normalized weights,

$$\theta^{t+1} = \sum_{i \in \mathcal{S}_t} \frac{\alpha_i}{\sum_{j \in \mathcal{S}_t}\alpha_j} \theta^{t+1}_i.$$0

If all participating clients have zero adaptive weight in a round, FedTSV falls back to uniform aggregation (Kuznetsov et al., 28 May 2026). No smoothing, momentum, temperature scaling, projection, or regularization term is introduced beyond zero-clipping and usual normalization.

A concise summary of the server-side logic is useful:

Stage	FedTSV operation	Paper-stated role
Reference construction	Compute $$\theta^{t+1} = \sum_{i \in \mathcal{S}_t} \frac{\alpha_i}{\sum_{j \in \mathcal{S}_t}\alpha_j} \theta^{t+1}_i.$$1 on validation data	Trusted per-round direction
Valuation	Estimate $$\theta^{t+1} = \sum_{i \in \mathcal{S}_t} \frac{\alpha_i}{\sum_{j \in \mathcal{S}_t}\alpha_j} \theta^{t+1}_i.$$2 from coalition utilities	Contribution assessment
Weighting	Set $$\theta^{t+1} = \sum_{i \in \mathcal{S}_t} \frac{\alpha_i}{\sum_{j \in \mathcal{S}_t}\alpha_j} \theta^{t+1}_i.$$3	Nonnegative adaptive influence
Aggregation	Normalize weights over active clients	Dynamic reweighting

This structure shows that TSV is not only a diagnostic score but the control variable that determines subsequent aggregation influence (Kuznetsov et al., 28 May 2026).

4. Stability, complexity, and theoretical status

The paper attributes stability to four specific mechanisms: the bounded utility $$\theta^{t+1} = \sum_{i \in \mathcal{S}_t} \frac{\alpha_i}{\sum_{j \in \mathcal{S}_t}\alpha_j} \theta^{t+1}_i.$$4, the normalization

$$\theta^{t+1} = \sum_{i \in \mathcal{S}_t} \frac{\alpha_i}{\sum_{j \in \mathcal{S}_t}\alpha_j} \theta^{t+1}_i.$$5

the clipping rule

$$\theta^{t+1} = \sum_{i \in \mathcal{S}_t} \frac{\alpha_i}{\sum_{j \in \mathcal{S}_t}\alpha_j} \theta^{t+1}_i.$$6

and the fallback to uniform aggregation when all current adaptive weights are zero (Kuznetsov et al., 28 May 2026). The paper does not state any upper clipping or explicit variance reduction for the weights.

From a computational perspective, exact Shapley computation is exponential because it requires evaluating all $$\theta^{t+1} = \sum_{i \in \mathcal{S}_t} \frac{\alpha_i}{\sum_{j \in \mathcal{S}_t}\alpha_j} \theta^{t+1}_i.$$7 coalitions or averaging over all permutations (Kuznetsov et al., 28 May 2026). FedTSV argues for practicality through two ideas. First, round decomposition allows contribution to be estimated online, round by round, rather than from a full retraining perspective. Second, Monte Carlo approximation reduces cost from exponential complexity to approximately

$$\theta^{t+1} = \sum_{i \in \mathcal{S}_t} \frac{\alpha_i}{\sum_{j \in \mathcal{S}_t}\alpha_j} \theta^{t+1}_i.$$8

while controlling approximation error, although this complexity claim is inherited from the cited approximation literature rather than derived specifically for FedTSV (Kuznetsov et al., 28 May 2026).

The paper states that each round requires one client-training pass and one server-side validation pass, with additional server overhead for coalition evaluation and Monte Carlo SV approximation (Kuznetsov et al., 28 May 2026). It does not specify the number of sampled permutations, whether truncation is used, any incremental estimator formula, a bias-variance analysis, client subsampling within Shapley estimation, or an explicit dependence on model dimension $$\theta^{t+1} = \sum_{i \in \mathcal{S}_t} \frac{\alpha_i}{\sum_{j \in \mathcal{S}_t}\alpha_j} \theta^{t+1}_i.$$9 (Kuznetsov et al., 28 May 2026). A plausible implication is that local training cost remains comparable to standard federated learning, while server overhead scales with the number of participating clients, the number of Monte Carlo samples, and the dimensionality of the updates through repeated distance computations.

The theoretical claims are deliberately limited. The paper provides the standard Shapley definition, the linear decomposition across rounds, and the qualitative argument that bounded utility aids efficient Monte Carlo estimation (Kuznetsov et al., 28 May 2026). It explicitly states that "the usual convergence proofs for FedAvg do not directly apply, and a rigorous convergence analysis for FedTSV is left for future work" (Kuznetsov et al., 28 May 2026). Accordingly, the paper does not prove convergence, fairness guarantees specific to TSV, robustness guarantees, regret bounds, or stability bounds. The fairness argument is inherited from the use of Shapley valuation on a trajectory-aware utility rather than from a new theorem.

5. Experimental design and empirical behavior

The experiments are conducted on MNIST and CIFAR-10 with 100 total clients, comprising 70 benign IID clients, 10 benign non-IID clients, and 20 malicious clients (Kuznetsov et al., 28 May 2026). Non-IID partitions are generated with a Dirichlet distribution with concentration parameter $$\phi_i(u) = \sum_{\mathcal{S} \subseteq \mathcal{N} \setminus \{i\}} \frac{|\mathcal{S}|! \, (n - |\mathcal{S}| - 1)!}{n!} \big( u(\mathcal{S} \cup \{i\}) - u(\mathcal{S}) \big),$$0, and malicious clients use a fixed label-shuffling mapping (Kuznetsov et al., 28 May 2026). The models are a single-hidden-layer neural network of width 64 on MNIST and ResNet-20 on CIFAR-10. Optimization uses SGD with batch size 64 for MNIST and 8 for CIFAR-10, learning rate 0.001 for MNIST and 0.0005 for CIFAR-10, 5 sampled clients per round, 1 local epoch per selected client, and 400 communication rounds for MNIST and 1000 for CIFAR-10, on an NVIDIA RTX 3080 GPU (Kuznetsov et al., 28 May 2026).

The baselines are FedAvg, CGSV, LOO, and FedTSV (Kuznetsov et al., 28 May 2026). The paper emphasizes two metric families: predictive performance, measured by global accuracy and global loss, and contribution quality, assessed through learned client weights and contribution separation (Kuznetsov et al., 28 May 2026). It does not define a single scalar fairness metric such as Jain’s index or equalized odds. Instead, fairness is evaluated operationally through whether IID benign clients receive high scores, non-IID benign clients receive intermediate but positive scores, and malicious clients receive low scores (Kuznetsov et al., 28 May 2026).

The experiments cover IID and non-IID data, partial participation, malicious or noisy clients, and time-varying participation through random client sampling each round (Kuznetsov et al., 28 May 2026). They do not explicitly evaluate concept drift or changing data distributions over time, though the method is motivated as applicable to time-varying contributions.

The main reported findings are that FedTSV accelerates convergence, improves final accuracy, improves robustness under heterogeneity and malicious clients, and yields fairer contribution estimates than the baselines (Kuznetsov et al., 28 May 2026). Quantitatively, the text reports that FedAvg saturates at about 0.75 on MNIST and about 0.65 on CIFAR-10, while LOO and FedTSV achieve higher accuracy, with FedTSV the most stable and consistent overall (Kuznetsov et al., 28 May 2026). No exact final numbers for FedTSV, CGSV, or LOO are printed in the text.

The qualitative comparison is sharper for contribution assessment. FedAvg gives identical weights to all clients; LOO produces scattered weights with poor separation; CGSV shows partial separation but can inflate weights for some adversarial clients; FedTSV assigns low scores to malicious clients, positive clustered scores to IID benign clients, and intermediate scores to non-IID benign clients (Kuznetsov et al., 28 May 2026). This supports the claim that trajectory-aware contribution estimation can simultaneously improve fairness of assessment and robustness of aggregation.

The paper does not report ablations for the number of Monte Carlo samples, validation set size, smoothing hyperparameters, clipping thresholds beyond zero truncation, adversarial fraction, alternative distance metrics, or sensitivity to $$\phi_i(u) = \sum_{\mathcal{S} \subseteq \mathcal{N} \setminus \{i\}} \frac{|\mathcal{S}|! \, (n - |\mathcal{S}| - 1)!}{n!} \big( u(\mathcal{S} \cup \{i\}) - u(\mathcal{S}) \big),$$1 and $$\phi_i(u) = \sum_{\mathcal{S} \subseteq \mathcal{N} \setminus \{i\}} \frac{|\mathcal{S}|! \, (n - |\mathcal{S}| - 1)!}{n!} \big( u(\mathcal{S} \cup \{i\}) - u(\mathcal{S}) \big),$$2 design (Kuznetsov et al., 28 May 2026). These omissions limit the precision with which the source of empirical gains can be isolated.

6. Interpretation, limitations, and relation to nearby federated methods

The central intuition of FedTSV is that a client should be valued by whether its update helps the global model follow a direction consistent with trustworthy descent, not merely by immediate validation gain or update magnitude (Kuznetsov et al., 28 May 2026). The paper argues that one-shot or static utilities can be misleading because an update may help accidentally in one round, a useful client may appear unimportant if it participates rarely, non-IID clients may be undervalued by simplistic metrics, and malicious clients may mimic local progress while deviating from the true global descent direction (Kuznetsov et al., 28 May 2026). TSV addresses this by comparing coalition updates with a validation-derived reference direction.

Accumulation across rounds,

$$\phi_i(u) = \sum_{\mathcal{S} \subseteq \mathcal{N} \setminus \{i\}} \frac{|\mathcal{S}|! \, (n - |\mathcal{S}| - 1)!}{n!} \big( u(\mathcal{S} \cup \{i\}) - u(\mathcal{S}) \big),$$3

makes the method less myopic by tying aggregation influence to long-term reliability rather than only recent utility (Kuznetsov et al., 28 May 2026). This suggests that FedTSV is especially suited to settings with significant non-IIDness, partial and intermittent participation, malicious or corrupted clients, client quality variation over time, and access to a small representative server-side validation set.

Several limitations are explicit or directly implied. The method depends on server validation data; if $$\phi_i(u) = \sum_{\mathcal{S} \subseteq \mathcal{N} \setminus \{i\}} \frac{|\mathcal{S}|! \, (n - |\mathcal{S}| - 1)!}{n!} \big( u(\mathcal{S} \cup \{i\}) - u(\mathcal{S}) \big),$$4 is unavailable, too small, or unrepresentative, the validation reference may be misleading (Kuznetsov et al., 28 May 2026). It incurs more server-side computation than FedAvg because Shapley approximation is intrinsically more expensive (Kuznetsov et al., 28 May 2026). It lacks a rigorous convergence theorem, and the effects of normalization design, choice of distance metric, and Monte Carlo budget are not systematically studied (Kuznetsov et al., 28 May 2026). The paper also implies a possible reference-mismatch problem: if validation data encode a biased target distribution, trajectory alignment may unfairly down-weight clients whose local data represent underrepresented modes. This suggests that FedTSV’s fairness is contingent on the trustworthiness and representativeness of the server-held validation set.

In the broader federated literature, the same source block mentions two nearby but distinct developments. "FedGTST: Boosting Global Transferability of Federated Models via Statistics Tuning" (Ma et al., 2024) is not FedTSV, but it is relevant as another method that replaces myopic client treatment with adaptive cross-client statistics, specifically Jacobian norm mean and variance, to improve transferability. "FeDaL: Federated Dataset Learning for Time Series Foundation Models" (Chen et al., 6 Aug 2025) addresses heterogeneous multi-dataset time-series learning through explicit bias decomposition rather than Shapley-based valuation. These comparisons indicate that FedTSV belongs to a broader line of work in which aggregation is no longer fixed but informed by estimated client utility, reliability, or compatibility. What distinguishes FedTSV within that space is its use of Shapley valuation on a validation-induced optimization trajectory (Kuznetsov et al., 28 May 2026).

FedTSV is therefore best understood as a fairness-aware federated optimization method in which contribution assessment and aggregation are unified through trajectory-based Shapley valuation. Its principal contribution is not a new convergence theory but a new utility definition: clients are rewarded to the extent that their updates help the global model track a validation-supported optimization path over time (Kuznetsov et al., 28 May 2026).