FedFV (Federated Fair Averaging via Loss-Ordered Internal and Historical External Gradient Projection)

A fair federated optimization algorithm (Wang, Fan, Qi, Wen, Wang, & Yu 2021, Section 3, Definitions 1-3, Algorithms 1-3, Theorems 1-4) that mitigates performance unfairness caused by conflicting client gradients (g_i . g_j < 0) with large magnitude disparities by sequentially eliminating internal conflicts among selected clients S_t and external conflicts against unselected clients h not in S_t before server model updates. In round t, after receiving pseudo-gradients g_k^t = theta^t - theta_k^t and local losses l_k^t from k in S_t (|S_t| = m), the server updates its global gradient history GH = [g_1^{t_1}, ..., g_K^{t_K}] and sorts S_t in ascending order of training loss l_{i_1}^t <= ... <= l_{i_m}^t to form the projection target order list PO_t = [g_{i_1}^t, ..., g_{i_m}^t]. In MitigateInternalConflict (Algorithm 2), the top (1 - alpha)m lowest-loss clients k in S_t^alpha iteratively project their gradient g_k^{PC} onto the normal plane of each conflicting target g_{i_j}^t in PO_t (when g_k^{PC} . g_{i_j}^t < 0 and k != i_j) via g_k^{PC} <- g_k^{PC} - ((g_{i_j}^t . g_k^{PC}) / ||g_{i_j}^t||^2) g_{i_j}^t, while the alpha m highest-loss clients retain their original gradients, producing g^t = (1/m) sum_{k=1}^m g_k^{PC}. For t >= tau, MitigateExternalConflict (Algorithm 3) iterates over past rounds t - i (for i = tau, ..., 1), sums historical gradients g_con = sum_{k : t_k = t-i, g^t . g_k^{t_k} < 0} g_k^{t_k}, and when g^t . g_con < 0 projects g^t <- g^t - ((g^t . g_con) / ||g_con||^2) g_con, before rescaling g^t <- (g^t / ||g^t||) ||(1/m) sum_{k in S_t} g_k^t|| and updating theta^{t+1} = theta^t - g^t.