Toward Energy-Efficient and Low-Carbon Intrusion Detection in Edge and Cloud Computing Based on GreenShield Cybersecurity Framework

. 2026 Mar 11;26(6):1780. doi: 10.3390/s26061780

Algorithm 2: Hierarchical Federated Learning with Gradient Compression

Initial model $θ^{(0)}$ ,
Edge nodes $K$ ,
Fog groups ${G_{g}}$ ,
Learning rate $η$
Updated global model $θ$
Train local model on $D_{k}$
Compute local gradient $\nabla F_{k}^{(t)}$
Compress gradient: ${\tilde{\nabla}}_{k}^{(t)} \leftarrow Compress (\nabla F_{k}^{(t)})$
Send ${\tilde{\nabla}}_{k}^{(t)}$ to fog node
Aggregate compressed gradients: ${\tilde{\nabla}}_{g}^{(t)} \leftarrow \sum_{k \in G_{g}} \frac{|D_{k}|}{\sum_{j \in G_{g}} |D_{j}|} {\tilde{\nabla}}_{k}^{(t)}$
Update fog model: $θ_{g}^{(t)} \leftarrow θ^{(t)} - η {\tilde{\nabla}}_{g}^{(t)}$
Aggregate fog-level models at cloud: $θ^{(t + 1)} \leftarrow \sum_{g = 1}^{G} \frac{\sum_{k \in G_{g}} |D_{k}|}{|D|} θ_{g}^{(t)}$
Broadcast $θ^{(t + 1)}$ to all nodes
End if
End for
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