A package for computing the average marginal contribution (Shapley Value) for each client in a Federated Learning environment

# ShapleyValueFL

A pip library for computing the marginal contribution (Shapley Value) for each client in a Federated Learning environment.

## Brief

The Shapley Value is a game theory concept that explores how to equitably distribute rewards and costs among members of a coalition. It is extensively used in incentive mechanisms for Federated Learning to fairly distribute rewards to clients based on their contribution to the system.

Let $v(S)$ where $S\subset N$ is defined as the contribution of the model collaboratively trained by the subset $S$. $N$ is a set of all the participants in the system. The $i-th$ participant’s Shapley Value $\phi(i)$ is defined as

$$\phi(i) = \sum_{S\subset N \backslash {i}} \frac{|S|!(N-|S|-1)!}{|N|!}(v(S\cup {i}) - v(S))$$

The marginal contribution of the $i-th$ participant is defined as $(v(S \cup {i}) - v(S))$ when they join this coalition.

Let's see this equation in action, consider a Federated Learning environment with three clients, so $N = {0, 1, 2}$. We list the contribution of each subset within this coalition. Let's consider the contribution to be measured in terms of model accuracy.

$v(\emptyset) = 0$    $v({0}) = 40$    $v({1}) = 60$    $v({2}) = 80$

$v({0,1}) = 70$    $v({0,2}) = 75$    $v({1,2}) = 85$

$v({0,1,2}) = 90$

Subset Client #0 Client #1 Client #2
$0 \leftarrow 1 \leftarrow 2$ 40 30 20
$0 \leftarrow 2 \leftarrow 1$ 40 15 35
$1 \leftarrow 0 \leftarrow 2$ 10 60 20
$1 \leftarrow 2 \leftarrow 0$ 5 60 25
$2 \leftarrow 0 \leftarrow 1$ 0 10 80
$2 \leftarrow 1 \leftarrow 0$ 5 5 80
$Sum$ 100 180 260
$\phi(i)$ 16.67 30 20

The arrow signifies the order in which each client joins the coalition. Consider the first iteration $0 \leftarrow 1 \leftarrow 2$, we calculate the marginal contribution of each client using the above equation.

• Client 0's marginal contribution is given as $v({0}) = 40$.
• Client 1's marginal contribution is given as $v({0, 1}) - v({0}) = 30$.
• Client 2's marginal contribution is given as $v({0, 1, 2}) - v({0, 1}) - v({0}) = 20$.

The marginal contribution is calculated for each permutation likewise, and the Shapley Value is derived by averaging all of these marginal contributions.

## Usage

from svfl.svfl import calculate_sv

models = {
"client-id-1" : ModelUpdate(),
"client-id-2" : ModelUpdate(),
"client-id-3" : ModelUpdate(),
}

def evaluate_model(model):
# function to compute evaluation metric, ex: accuracy, precision
return metric

def fed_avg(models):
# function to merge the model updates into one model for evaluation, ex: FedAvg, FedProx
return model

# returns a key value pair with the client identifier and it's respective Shapley Value
contribution_measure = calculate_sv(models, evaluate_model, fed_avg)

## Future Work

• Built-in support for standard averaging methods like FedAvg, & FedProx.

## Feedback

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