| # Federated Averaging (FedAvg) Algorithm Specification |
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| ## 1. Problem Solved |
| Federated Averaging (McMahan et al., 2017) resolves the distributed optimization problem of fitting a global fraud classification model across $K$ decentralized banking institutions without transferring or pooling raw transaction data. |
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| Mathematically, it seeks to minimize the global empirical objective: |
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| $$\min_{w \in \mathbb{R}^d} f(w) \quad \text{where} \quad f(w) \triangleq \sum_{k=1}^K \frac{n_k}{N} F_k(w)$$ |
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| where $n_k$ is the number of local transaction samples at Bank $k$, $N = \sum_{k=1}^K n_k$, and $F_k(w) = \frac{1}{n_k} \sum_{i \in \mathcal{D}_k} \ell(w; x_i, y_i)$ is Bank $k$'s local empirical risk. |
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| ## 2. Implementation in CF-Intelligence |
| - **Location**: [`backend/app/application/services/fl_engine.py`](../../backend/app/application/services/fl_engine.py) |
| - **Aggregation Formulation**: |
| At round $t$, the central coordinator broadcasts global weights $w_t$. Each participant bank executes $E$ local epochs of mini-batch SGD: |
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| $$w_k^{t+1} \leftarrow \mathrm{ClientUpdate}(k, w_t)$$ |
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| The coordinator receives model update vectors $\Delta w_k = w_k^{t+1} - w_t$ and computes the weighted aggregate: |
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| $$w_{t+1} = w_t + \sum_{k \in \mathcal{S}_t} \frac{n_k}{\sum_{j \in \mathcal{S}_t} n_j} \Delta w_k$$ |
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| where $\mathcal{S}_t \subseteq \{1, \dots, K\}$ is the active cohort selected for round $t$. |
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| ## 3. Threat Model & Security Assumptions |
| - **Honest-but-Curious Coordinator**: FedAvg assumes the coordinator faithfully aggregates weights but may inspect unmasked gradients. |
| - **Privacy Limitation**: Raw FedAvg without Differential Privacy or Secure Aggregation is vulnerable to gradient inversion attacks (Deep Leakage from Gradients) and membership inference. |
| - **Robustness Limitation**: Assumes all selected clients are honest ($f = 0$ malicious clients). A single adversarial client can poison $w_{t+1}$ by submitting arbitrarily scaled gradients. |
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| ## 4. Non-IID Dirichlet Client Partitioning & Client Drift Dynamics |
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| Cross-bank federated fraud detection operates inherently under heterogeneous data distributions across financial institutions. In **CF-Intelligence**, this institutional heterogeneity is mathematically modeled using a Dirichlet distribution $\mathrm{Dir}(\alpha \cdot \mathbf{1}_K)$ implemented in [`experiments/paysim/partitioner.py`](../../experiments/paysim/partitioner.py). |
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| ### 4.1 Dirichlet Allocation Formulation |
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| For each binary transaction class $c \in \{0, 1\}$ (legitimate transactions and fraudulent transfers), a probability vector $\mathbf{q}_c = (q_{c,1}, \dots, q_{c,K})$ is sampled independently: |
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| $$\mathbf{q}_c \sim \mathrm{Dir}(\alpha \cdot \mathbf{1}_K), \quad \text{where } \sum_{k=1}^K q_{c,k} = 1, \; q_{c,k} \ge 0$$ |
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| where $\alpha > 0$ denotes the Dirichlet concentration parameter and $K$ is the number of participating banking institutions (default $K=3$: Bank A, Bank B, Bank C). |
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| The number of samples of class $c$ assigned to institution $k$ is $n_{c,k} = \lfloor q_{c,k} N_c \rfloor$, with residual samples allocated to maintain exact dataset conservation: |
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| $$\sum_{k=1}^K n_{c,k} = N_c, \quad \mathcal{D}_j \cap \mathcal{D}_k = \emptyset \quad \forall j \neq k$$ |
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| ### 4.2 Impact of Concentration Parameter $\alpha$ on Client Drift |
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| The gradient variance between local client objectives and the global consortium objective is directly bounded by the Dirichlet concentration: |
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| $$\mathbb{E}_{k \sim \mathcal{P}_K} \left[ \left\| \nabla F_k(w) - \nabla f(w) \right\|^2 \right] \le G^2(\alpha)$$ |
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| - **$\alpha = 0.1$ (Extreme Non-IID Heterogeneity)**: Induces severe label distribution skew and high Kullback-Leibler divergence ($D_{\mathrm{KL}}(P_k \parallel P_{\mathrm{global}}) \gg 1.0$). Certain banks encounter negligible fraud while others experience high alert volume. Local SGD directions diverge sharply, resulting in client drift that accelerates the need for proximal regularization (FedProx $\mu > 0$) or control variates (SCAFFOLD). |
| - **$\alpha = 0.5$ (Realistic Consortium Skew)**: Calibrated to empirical banking consortium structures, simulating volume differences between Tier-1 retail banks, regional commercial lenders, and digital challenger banks. |
| - **$\alpha = 1.0$ (Uniform Dirichlet / Moderate Heterogeneity)**: Approaching balanced participation where all institutions observe representative cross-bank fraud signals. |
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| ### 4.3 Statistical Divergence Metrics |
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| The platform quantifies client distribution divergence via Kullback-Leibler (KL) divergence and Total Variation Distance (TVD): |
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| $$D_{\mathrm{KL}}(P_k \parallel P_{\mathrm{global}}) = \sum_{c \in \{0, 1\}} P_k(c) \ln \frac{P_k(c) + \epsilon}{P_{\mathrm{global}}(c) + \epsilon}$$ |
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| $$\mathrm{TVD}(P_k, P_{\mathrm{global}}) = \frac{1}{2} \sum_{c \in \{0, 1\}} \lvert P_k(c) - P_{\mathrm{global}}(c) \rvert$$ |
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| ## 5. Operational Limitations |
| - **Client Drift under Extreme Skew**: When local label distributions diverge significantly ($\alpha \le 0.1$ in Dirichlet skew), vanilla FedAvg weights oscillate between local minima, degrading global test PR-AUC. |
| - **System Stragglers**: In synchronous FedAvg, round completion latency is bounded by the slowest banking node. |
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| ## 6. Test Suite Verification & Empirical Benchmarking |
| - **Unit Tests**: [`backend/tests/unit/test_fl_engine.py`](../../backend/tests/unit/test_fl_engine.py), [`backend/tests/unit/test_dirichlet_partition.py`](../../backend/tests/unit/test_dirichlet_partition.py), [`backend/tests/unit/test_dirichlet_sweep.py`](../../backend/tests/unit/test_dirichlet_sweep.py) |
| - **Partitioner Implementation**: [`experiments/paysim/partitioner.py`](../../experiments/paysim/partitioner.py) |
| - **Multi-Alpha Sweep Runner**: [`experiments/ablations/dirichlet_sweep.py`](../../experiments/ablations/dirichlet_sweep.py) |
| - **Empirical Dossier**: [`experiments/ablations/audit_dossier.md`](../../experiments/ablations/audit_dossier.md) |
| - **Publication Figure**: [`docs/figures/benchmark_fl_convergence.png`](../figures/benchmark_fl_convergence.png) |
| - **Benchmark Runner**: [`benchmarks/runners/run_fl_benchmark.py`](../../benchmarks/runners/run_fl_benchmark.py) |
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