Trajectory Shapley Value in Federated Learning

• Trajectory Shapley Value (TSV) is a metric that attributes client contributions based on the entire optimization trajectory rather than just final performance.
• It computes roundwise Shapley values using a validation-based utility, accumulating contributions across training rounds to adjust client weights in FedTSV.
• Empirical studies on MNIST and CIFAR-10 demonstrate that FedTSV enhances stability and accuracy by differentiating benign, non-IID, and malicious client updates.

Searching arXiv for Trajectory Shapley Value and related Shapley-on-trajectories work. Trajectory Shapley Value (TSV) is a Shapley-style contribution metric introduced for iterative federated learning to evaluate how each client influences the optimization trajectory of the global model, using a validation-based, temporally consistent utility rather than a static endpoint criterion (Kuznetsov et al., 28 May 2026). In this formulation, the attributed object is not merely final validation performance, but the sequence of optimization steps taken across communication rounds. The term also admits a broader conceptual use: earlier work in ride-sharing, routing games, and cooperative-network path integrals did not use the TSV name, but studied Shapley allocations whose characteristic function or attribution rule depends explicitly on ordered routes, induced trajectories, or stochastic coalition paths (Levinger et al., 2019).

1. Definition and conceptual scope

In the federated-learning formulation, TSV modifies classical Shapley valuation by replacing a single global utility with a sequence of per-round coalition games. At round $t$, the active players are the participating clients $\mathcal S_t \subseteq \mathcal N$, and the coalition utility is defined by how closely the coalition’s aggregated update matches a validation-derived reference update. The cumulative contribution of client $i$ is then obtained by summing its per-round Shapley values over training rounds (Kuznetsov et al., 28 May 2026).

The paper’s operational definition is captured by the cumulative update rule

$$\varphi_i^{t+1} = \varphi_i^t + \phi_i(v^t), \qquad \varphi_i^0 = 0,$$

with the effective trajectory-based contribution

$\mathrm{TSV}_i = \sum_{t=0}^{T-1} \phi_i(v^t).$

This preserves the cooperative-game-theoretic Shapley mechanism while changing both the utility and the temporal scope: the utility is trajectory alignment rather than final performance, and the allocation is computed online across rounds rather than once at the end (Kuznetsov et al., 28 May 2026).

The defining conceptual shift is from the question “Did this client help the final model?” to the roundwise question “Did this client’s update move the global model in a direction consistent with the descent direction suggested by server-side validation data?” In the paper’s terminology, learning is interpreted as a discrete approximation of gradient flow,

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

so “trajectory” refers to the optimization path traced by the global model across rounds (Kuznetsov et al., 28 May 2026).

A broader reading of TSV is suggested by adjacent literature. In prioritized ride-sharing and routing games, coalition cost is determined by an ordered subtrajectory induced by a global route order; in cooperative-network theory, the Shapley value is reinterpreted as an expectation of a stochastic path integral over coalition-formation trajectories. These works do not define TSV by name, but they formalize trajectory-dependent Shapley allocation in mathematically explicit ways (Levinger et al., 2019).

2. Formal federated-learning formulation

The federated setting uses clients $\mathcal N = \{1,\dots,n\}$, local datasets $\mathcal D_i$, and a global 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 $\alpha_i \ge 0$ are aggregation weights (Kuznetsov et al., 28 May 2026). The stated motivation is that conventional fixed weights, such as uniform or data-size-based aggregation, do not reflect unequal and time-varying client contributions under data heterogeneity, partial participation, and adversarial behavior.

At round $\mathcal S_t \subseteq \mathcal N$0, for any coalition $\mathcal S_t \subseteq \mathcal N$1, the coalition update is

$\mathcal S_t \subseteq \mathcal N$2

The server also computes a validation reference update using held-out validation data $\mathcal S_t \subseteq \mathcal N$3,

$\mathcal S_t \subseteq \mathcal N$4

where $\mathcal S_t \subseteq \mathcal N$5 is obtained by running the same number of SGD steps on the validation loss $\mathcal S_t \subseteq \mathcal N$6, initialized at $\mathcal S_t \subseteq \mathcal N$7 (Kuznetsov et al., 28 May 2026).

The per-round utility is then

$\mathcal S_t \subseteq \mathcal N$8

with $\mathcal S_t \subseteq \mathcal N$9. In the reported experiments,

$i$0

and the normalization is

$i$1

for small $i$2 (Kuznetsov et al., 28 May 2026). The utility is therefore high when the coalition update is close to the validation reference; if $i$3, then $i$4.

Within each round, TSV uses the standard Shapley construction restricted to active clients: $i$5 The marginal contribution $i$6 measures how much adding client $i$7 improves the coalition’s closeness to the validation-induced descent step (Kuznetsov et al., 28 May 2026).

This construction is deliberately temporally additive. The paper invokes the Shapley linearity idea

$i$8

and applies the same decomposition to the roundwise utilities $i$9. This yields a discrete, cumulative trajectory valuation rather than a continuous-time path integral (Kuznetsov et al., 28 May 2026).

3. Computation and use in FedTSV

TSV is the contribution metric; FedTSV is the adaptive aggregation method that converts cumulative TSV scores into dynamic client weights (Kuznetsov et al., 28 May 2026). Each round proceeds by collecting local client models, computing the validation reference update, estimating per-round Shapley values for the active set, accumulating these values, truncating negative totals at zero, and then aggregating with the resulting weights.

The cumulative score update is

$$\varphi_i^{t+1} = \varphi_i^t + \phi_i(v^t), \qquad \varphi_i^0 = 0,$$0

and the aggregation weight is

$$\varphi_i^{t+1} = \varphi_i^t + \phi_i(v^t), \qquad \varphi_i^0 = 0,$$1

The global model is then updated by weighted averaging: $$\varphi_i^{t+1} = \varphi_i^t + \phi_i(v^t), \qquad \varphi_i^0 = 0,$$2 If all participating clients have zero adaptive weight, the method falls back to uniform aggregation for that round (Kuznetsov et al., 28 May 2026).

The server-side requirements are explicit. The server must receive the standard client updates $$\varphi_i^{t+1} = \varphi_i^t + \phi_i(v^t), \qquad \varphi_i^0 = 0,$$3, maintain a held-out validation set $$\varphi_i^{t+1} = \varphi_i^t + \phi_i(v^t), \qquad \varphi_i^0 = 0,$$4, and use the same number $$\varphi_i^{t+1} = \varphi_i^t + \phi_i(v^t), \qquad \varphi_i^0 = 0,$$5 of SGD steps and learning rate $$\varphi_i^{t+1} = \varphi_i^t + \phi_i(v^t), \qquad \varphi_i^0 = 0,$$6 for the validation reference update. No extra raw client data are required beyond normal FL updates; the additional requirement is the server-side validation data (Kuznetsov et al., 28 May 2026).

Exact Shapley evaluation remains intractable, so the paper uses Monte Carlo estimation “following the efficient estimation strategy in [Zhang et al., 2023].” The stated complexity reduction for approximate computation is from

$$\varphi_i^{t+1} = \varphi_i^t + \phi_i(v^t), \qquad \varphi_i^0 = 0,$$7

and the practical savings in FedTSV come from computing TSV only over the participating subset $$\varphi_i^{t+1} = \varphi_i^t + \phi_i(v^t), \qquad \varphi_i^0 = 0,$$8, using a bounded utility, and relying on Monte Carlo sampling (Kuznetsov et al., 28 May 2026).

The paper is explicit about its theoretical status. It does not present full theorems proving convergence, fairness, or unbiasedness of FedTSV, and 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 strongest claims for TSV in this setting are empirical and methodological rather than theorem-backed.

4. Empirical behavior and interpreted meaning

The experiments used MNIST and CIFAR-10, with 100 clients total: 70 benign IID clients, 10 benign non-IID clients, and 20 malicious clients; non-IID data were generated via Dirichlet concentration $$\varphi_i^{t+1} = \varphi_i^t + \phi_i(v^t), \qquad \varphi_i^0 = 0,$$9, malicious clients performed fixed label shuffling, each round sampled 5 clients, local training used one local epoch, MNIST ran for 400 rounds with a simple 1-hidden-layer NN, and CIFAR-10 ran for 1000 rounds with ResNet-20 (Kuznetsov et al., 28 May 2026).

Against FedAvg, CGSV, and LOO, the paper reports that FedTSV consistently outperforms the baselines in accuracy/loss trends and is more stable. The text states that FedAvg saturates around $\mathrm{TSV}_i = \sum_{t=0}^{T-1} \phi_i(v^t).$0 on MNIST and $\mathrm{TSV}_i = \sum_{t=0}^{T-1} \phi_i(v^t).$1 on CIFAR-10, while LOO and FedTSV improve over FedAvg by mitigating corrupted updates; FedTSV is described as delivering the most stable and consistent performance across settings, and the conclusion and abstract claim higher accuracy and faster convergence than baselines (Kuznetsov et al., 28 May 2026).

The contribution visualizations on CIFAR-10 show a separation pattern: malicious clients receive low scores under FedTSV, IID benign clients are concentrated in the positive region, and non-IID benign clients lie in between. The paper emphasizes that this separation is more coherent under FedTSV than under LOO or CGSV (Kuznetsov et al., 28 May 2026). This supports the intended interpretation that a high TSV corresponds to updates that repeatedly help coalition averages align with the validation-induced descent direction, whereas a low or negative TSV indicates little benefit or persistent misalignment.

The reported advantages are stated in five forms: trajectory-aware rather than endpoint-only evaluation, validation-based reference direction, coalition-aware interaction modeling through Shapley value, temporally cumulative scoring across rounds, and robustness to scale changes near stationary points because the utility is normalized and bounded (Kuznetsov et al., 28 May 2026). The stated caveats are equally specific: TSV requires server-side validation data, adds computational overhead through the validation SGD pass and repeated utility evaluation, depends on Monte Carlo approximation quality, lacks a full convergence theory, and uses update-space proximity as a proxy rather than direct loss reduction.

A plausible implication is that TSV is best viewed not as a generic fairness axiom, but as a validation-guided online control signal for aggregation. That implication follows from the way cumulative Shapley scores are directly transformed into aggregation weights, rather than merely being recorded for ex post attribution (Kuznetsov et al., 28 May 2026).

5. Trajectory-dependent Shapley before the TSV name

The TSV term was introduced in federated learning, but earlier arXiv work already studied Shapley allocations with explicit trajectory dependence. In “Fair Sharing: The Shapley Value for Ride-Sharing and Routing Games” (Levinger et al., 2019), the players are passengers $\mathrm{TSV}_i = \sum_{t=0}^{T-1} \phi_i(v^t).$2, all starting from a common origin $\mathrm{TSV}_i = \sum_{t=0}^{T-1} \phi_i(v^t).$3, and coalition cost depends on a route through destinations. In the prioritized scenario, where the drop-off order is fixed, the coalition cost is

$\mathrm{TSV}_i = \sum_{t=0}^{T-1} \phi_i(v^t).$4

so the characteristic function is determined by the ordered subtrajectory induced by the coalition. In the non-prioritized scenario, the coalition cost is the length of the shortest path that starts at $\mathrm{TSV}_i = \sum_{t=0}^{T-1} \phi_i(v^t).$5 and traverses all destinations in $\mathrm{TSV}_i = \sum_{t=0}^{T-1} \phi_i(v^t).$6, so trajectory dependence remains but the route is endogenous to the coalition (Levinger et al., 2019).

That paper’s most TSV-like identity is the local insertion-cost formula

$\mathrm{TSV}_i = \sum_{t=0}^{T-1} \phi_i(v^t).$7

where $\mathrm{TSV}_i = \sum_{t=0}^{T-1} \phi_i(v^t).$8 and $\mathrm{TSV}_i = \sum_{t=0}^{T-1} \phi_i(v^t).$9 are the nearest predecessor and successor of $\theta'(t) = -\nabla F\big(\theta(t)\big),$0 in the fixed priority order among the predecessors in permutation $\theta'(t) = -\nabla F\big(\theta(t)\big),$1 (Levinger et al., 2019). In that setting, a Shapley marginal contribution is exactly the cost of replacing one bypass segment $\theta'(t) = -\nabla F\big(\theta(t)\big),$2 with two new segments $\theta'(t) = -\nabla F\big(\theta(t)\big),$3 and $\theta'(t) = -\nabla F\big(\theta(t)\big),$4. The paper further gives a closed-form segment decomposition,

$\theta'(t) = -\nabla F\big(\theta(t)\big),$5

with coefficients $\theta'(t) = -\nabla F\big(\theta(t)\big),$6 computable in polynomial time (Levinger et al., 2019). This is a player-level Shapley allocation derived from weighted trajectory-segment terms.

The same paper establishes a sharp tractability boundary. Exact Shapley computation is polynomial-time in the fixed-order case, but “There is no polynomial time algorithm that computes the Shapley value for a given passenger in the non-prioritized ride-sharing problem unless $\theta'(t) = -\nabla F\big(\theta(t)\big),$7” (Levinger et al., 2019). It also proposes SHAPO, “SHapley Approximation based on a Prioritized Order,” which freezes a chosen route order and uses the fixed-order Shapley formula as a proxy for the non-prioritized problem. In experiments on a Toulouse road network, SHAPO significantly outperformed Depot Distance, Shortcut Distance, and Re-routed Margin across percent error, MAE, RMSE, and maximum error (Levinger et al., 2019).

A closely related transport-cost study, “A Study of Proxies for Shapley Allocations of Transport Costs” (Aziz et al., 2014), formulates the traveling salesperson game (TSG), in which customer locations are the players and $\theta'(t) = -\nabla F\big(\theta(t)\big),$8 is the length of the shortest tour visiting coalition $\theta'(t) = -\nabla F\big(\theta(t)\big),$9. The Shapley allocation is

$\mathcal N = \{1,\dots,n\}$0

or equivalently an average over permutations of incremental route-cost increases (Aziz et al., 2014). The paper proves that “There is no polynomial-time $\mathcal N = \{1,\dots,n\}$1-approximation of the Shapley value of the location in a TSG for constant $\mathcal N = \{1,\dots,n\}$2 unless $\mathcal N = \{1,\dots,n\}$3,” reinforcing the hardness of coalition-reoptimized route valuation (Aziz et al., 2014).

These route-based papers do not define TSV as a formal term. Still, they study exactly the structural distinction that later became central in trajectory-aware attribution: fixed or order-preserving trajectories permit local decompositions and tractable allocation, whereas coalition-specific route reoptimization induces global nonlocality and hardness (Levinger et al., 2019).

6. Path-integral and Hodge-theoretic generalization

A distinct theoretical precursor appears in “Cooperative networks and Hodge-Shapley value” (Lim, 2022), which reinterprets the Shapley value as an expectation of a stochastic path integral over coalition-formation trajectories. The player set is $\mathcal N = \{1,\dots,n\}$4, the coalition graph has node set $\mathcal N = \{1,\dots,n\}$5, and a Markov chain starts at $\mathcal N = \{1,\dots,n\}$6 and performs an unbiased random walk on the coalition hypercube, allowing both player additions and removals (Lim, 2022).

For player $\mathcal N = \{1,\dots,n\}$7, the edgewise partial differential is

$\mathcal N = \{1,\dots,n\}$8

with antisymmetry on reverse edges. The path integral up to target $\mathcal N = \{1,\dots,n\}$9 is

$\mathcal D_i$0

and the expected path integral is

$\mathcal D_i$1

The central theorem states

$\mathcal D_i$2

so the classical Shapley value at the grand coalition is recovered as expected accumulated contribution over stochastic coalition trajectories rather than merely as an average over monotone permutations (Lim, 2022).

The paper then generalizes from game-derived marginals to arbitrary antisymmetric edge flows $\mathcal D_i$3, defining an $\mathcal D_i$4-Shapley value and showing that the expected path attribution $\mathcal D_i$5 solves the Poisson equation

$\mathcal D_i$6

where $\mathcal D_i$7 is the graph Laplacian (Lim, 2022). In this formulation, trajectory attribution becomes a Hodge-theoretic potential associated with an edge flow.

This is not TSV in the federated-learning sense: the trajectories are coalition-state trajectories rather than optimization paths, and the valued objects are cooperative players or agents rather than model updates (Lim, 2022). Even so, the structural parallel is direct. Both constructions define local contribution on transitions, accumulate it along trajectories, and obtain a final value by averaging over paths. This suggests that TSV belongs to a broader family of path-based Shapley constructions in which the classical permutation average is replaced by an explicitly trajectory-indexed accumulation rule.

The combined literature therefore delineates three layers of the concept. The federated-learning paper supplies the explicit term “Trajectory Shapley Value” and a concrete online algorithmic use in adaptive aggregation (Kuznetsov et al., 28 May 2026). The ride-sharing and routing papers provide route-dependent characteristic functions, local insertion-cost decompositions, tractable fixed-trajectory formulas, and hardness results for coalition-reoptimized trajectories (Levinger et al., 2019). The cooperative-network paper supplies a general path-integral interpretation and a Hodge-theoretic computational framework for trajectory-based Shapley allocation on state graphs (Lim, 2022). Taken together, these works define TSV as both a named FL metric and a broader trajectory-dependent mode of Shapley valuation.