pFedGP (Personalized Federated Learning with Gaussian Processes and Shared Deep Kernel Trees)

A Bayesian personalized federated learning algorithm (Achituve, Shamsian, Navon, Chechik, & Fetaya 2021, Sections 3-4, Equations 1-14, Theorem 1, Algorithm 1) that learns a shared deep neural network feature extractor g_theta across all clients while fitting a separate personalized Gaussian Process (GP) classifier with Pólya-Gamma (PG) data augmentation on each client's latent embeddings z_i = g_theta(x_i), extending binary PG-augmented GP classification to C-class classification via a full binary decision tree of C - 1 internal GP nodes (where node m models p(y_m = 1 | x, omega_m) with conditionally Gaussian posterior p(f_m | X, y_m, omega_m) = N(f_m | mu_m, Sigma_m), Sigma_m = (K_m^{-1} + Omega_m)^{-1}, mu_m = Sigma_m kappa_m, kappa_m = y_m - 1/2, Equations 1-6), and introducing two inducing-point extensions — pFedGP-IP-data (which optimizes local inducing inputs X_bar in input space to regularize zero-shot adaptation) and pFedGP-IP-compute (which applies the Fully Independent Training Conditional (FITC) approximation K_{NN} approx Q_{NN} + diag(K_{NN} - Q_{NN}), Q_{NN} = K_{NM} K_{MM}^{-1} K_{MN} to reduce cubic O(N^3) matrix inversion to O(M^2 N + M^3), Equations 9-12).