APFL (Adaptive Personalized Federated Learning via Convex Global-Local Model Interpolation)

A personalized federated optimization algorithm and generalization framework (Deng, Kamani, & Mahdavi 2020, Sections 2-4, Equations 2-10, Theorem 1, Algorithm 1) that learns for each client i a personalized predictor h_{alpha_i}(w_hat_i) parameterized by a convex combination w_hat_i = alpha_i v_i + (1 - alpha_i) w^* of a local model v_i in R^d and a shared global model w^* = argmin_w (1/n) sum_{j=1}^n f_j(w), maintaining three local sequences during communication-efficient local descent: the global iterate w_i^{(t)} = w_i^{(t-1)} - eta_t nabla f_i(w_i^{(t-1)}; xi_i^t) (synchronized every tau steps to w^{(t)} = (1/|S_t|) sum_{i in S_t} w_i^{(t)}), the local auxiliary iterate v_i^{(t)} = v_i^{(t-1)} - eta_t nabla_{v} f_i(v_hat_i^{(t-1)}; xi_i^t) evaluated at the interpolated point v_hat_i^{(t-1)} = alpha_i v_i^{(t-1)} + (1 - alpha_i) w_i^{(t-1)} (Equations 6-9), and the client mixing weight alpha_i^{(t)} = Proj_{[0,1]}(alpha_i^{(t-1)} - eta_t <v_i^{(t-1)} - w_i^{(t-1)}, nabla f_i(v_hat_i^{(t-1)}; xi_i^t)>) (Equation 10).