5GCS (Fifth-Generation Local Training with Client Sampling)

An accelerated fifth-generation local training (LT) federated optimization framework for minimizing mu-strongly convex and L-smooth distributed objectives f(x) = (1/n) sum_{i=1}^n f_i(x) + r(x) under partial client participation (sampling a cohort S^t subseteq {1, ..., n} of size s = |S^t| uniformly at random in each round t). Grounded in the primal-dual lifted reformulation (Equations 11-12) over server primal-dual variables (x^t, z^t, bar{u}^t) and client dual control variates u_i^t in R^d, 5GCS (Algorithm 1) constructs the momentum-interpolated primal query tilde{x}^t = (eta x^t + z^t)/(1 + eta), queries each sampled client i in S^t to compute an Approximate Proximal Point (APP(delta)) solution hat{x}_i^t satisfying ||hat{x}_i^t - x_i^*(tilde{x}^t, u_i^t)||^2 <= delta ||tilde{x}^t - x_i^*(tilde{x}^t, u_i^t)||^2 for the shifted local subproblem x_i^*(tilde{x}^t, u_i^t) = prox_{(1/tau) f_i}(tilde{x}^t + (1/tau) u_i^t), updates active client duals via u_i^{t+1} = u_i^t + alpha(tilde{x}^t - hat{x}_i^t), and updates the server iterates (x^{t+1}, z^{t+1}, bar{u}^{t+1}) to attain accelerated O((n/s) sqrt(L/mu) log(1/epsilon)) communication complexity.