Shapley Value Analysis

The Shapley Value Executor (src/applications/shapley/) extends standard centralized federated learning with a fairness and contribution measurement framework based on cooperative game theory.

What Are Shapley Values in FL?

In a federation, not all clients contribute equally to model quality. Some may have rich, high-quality datasets; others may have noisy, redundant, or even adversarially crafted data. Shapley values provide a principled, game-theoretic way to measure each client’s marginal contribution to the global model — without accessing their raw data.

Formally, the Shapley value for client $i$ is:

\[\phi_i = \sum_{S \subseteq N \setminus \{i\}} \frac{|S|!\,(|N|-|S|-1)!}{|N|!} \left[ v(S \cup \{i\}) - v(S) \right]\]

Where:

$N$ is the full set of clients.
$S$ is any coalition (subset) of clients excluding $i$.
$v(S)$ is the model accuracy achieved by coalition $S$.
The sum averages the marginal contribution of client $i$ across all possible orderings in which it could join the coalition.

Clients with consistently high Shapley values are genuinely useful to the federation. Clients with near-zero or negative values may be free-riders or adversaries.

Enabling Shapley Analysis

federated_learning_schema: "TraditionalFederatedLearning"
shapley: true
shapley_type: "value"          # Type of Shapley computation

When shapley: true, app_factory.py routes to shapley_app_executor instead of star_app_executor. The only structural difference is that the server node is replaced with a ShapleyServerNode.

The ShapleyServerNode

src/applications/shapley/shapley_server_node.py extends the standard star server to:

After each round, retrieve a versioned snapshot of each client’s model contribution via federation.pull_version().
Evaluate model quality for every possible coalition subset (or an approximation thereof for large $N$).
Compute the marginal contribution of each client’s update using the Shapley formula.
Log the per-client Shapley values to the standard metrics pipeline (CSV + OpenTelemetry).

The computation is intensive for large federations (exact computation is $O(2^N)$). For large $N$, sampling-based approximations (e.g., Monte Carlo permutation sampling) are used.

Use Cases

Use Case	How Shapley Values Help
Free-rider detection	Identify clients contributing low-quality or recycled data — they will have Shapley values near zero
Incentive mechanisms	Reward clients proportional to their Shapley value — higher contribution earns more global model influence or compensation
Data quality auditing	Rank datasets by their actual impact on model quality without accessing raw data
Adversarial client detection	Clients with consistently negative Shapley values may be poisoning the model

Computational Cost

The exact Shapley computation requires evaluating $v(S)$ for all $2^N$ subsets. For large federations, this is impractical. FedPilot supports the shapley_type config key to select the approximation strategy:

`shapley_type`	Approach	Complexity
`"value"`	Exact marginal contribution evaluation	$O(2^N)$ — use for $N \leq 20$

For production-scale federations ($N > 20$), plan to extend the ShapleyServerNode with a Monte Carlo or kernel SHAP approximation plugged into the standard aggregation hook.