FO-RI-FedAvg

Implements FO-RI-FedAvg, a roughness-informed federated averaging method with a fractional-order coordinate-wise preconditioner.

Each client runs local SGD on a proximal-regularized objective, where the proximal pull toward the global weights \(\theta_t\) is scaled by a roughness index \(\mathcal{I}_k\) of the client's loss landscape. The gradient is then preconditioned element-wise by \(p_t\), a fractional-order memory term built from the magnitude of the last local displacement raised to the power \(1-\alpha\): large recent moves are damped and small ones amplified, with \(\alpha=1\) recovering plain SGD (\(p_t=\mathbf{1}\)). After \(H\) local steps the server forms the standard data-size-weighted average of the client weights.

\[ \begin{aligned} p_t^{k} &= \frac{1}{\Gamma(2-\alpha)}\,\bigl(|\theta_t^{k}-\theta_{t-1}^{k}|+\delta\mathbf{1}\bigr)^{\odot(1-\alpha)} \\ g_t^{k} &= \nabla_\theta \ell(\theta_t^{k}; b_t^{k}) + \lambda_t\, r(\mathcal{I}_k)\,(\theta_t^{k}-\theta_t) \\ \theta_{t+1}^{k} &= \theta_t^{k} - \eta_t\,(g_t^{k}\odot p_t^{k}) \\ \theta_{t+1} &= \sum_{k\in S_t}\frac{n_k}{n_t}\,\theta_{t+1}^{k},\qquad r(\mathcal{I}_k)=\frac{\mathcal{I}_k}{\mathcal{I}_k+\tau_{\mathcal{I}}} \end{aligned} \]

where \(\theta^k\) are client \(k\)'s parameters, \(\theta_t\) the broadcast global weights, \(\eta_t\) the local learning rate, \(\lambda_t\) the base proximal strength, \(\alpha\in(0,1]\) the fractional order, \(\Gamma\) the Gamma function, \(\delta>0\) a numerical stabilizer, \(\odot\) element-wise operations, \(\mathcal{I}_k\) the roughness index with saturation constant \(\tau_{\mathcal{I}}\), \(S_t\) the participating clients, and \(n_k/n_t\) their data-proportional aggregation weights.

Reference: Mohammad Partohaghighi, Roummel Marcia, Bruce J. West, YangQuan Chen, "Fractional Order Federated Learning for Battery Electric Vehicle Energy Consumption Modeling", arXiv preprint 2026. https://arxiv.org/abs/2602.12567

---

Back to the Canon