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Federated learning is a distributed learning framework that allows a set ofclients to collaboratively train a model under the orchestration of a centralserver, without sharing raw data samples. Although in many practical scenariosthe derivatives of the objective function are not available, only few workshave considered the federated zeroth-order setting, in which functions can onlybe accessed through a budgeted number of point evaluations. In this work wefocus on convex optimization and design the first federated zeroth-orderalgorithm to estimate the curvature of the global objective, with the purposeof achieving superlinear convergence. We take an incremental Hessian estimatorwhose error norm converges linearly, and we adapt it to the federatedzeroth-order setting, sampling the random search directions from the Stiefelmanifold for improved performance. In particular, both the gradient and Hessianestimators are built at the central server in a communication-efficient andprivacy-preserving way by leveraging synchronized pseudo-random numbergenerators. We provide a theoretical analysis of our algorithm, named FedZeN,proving local quadratic convergence with high probability and global linearconvergence up to zeroth-order precision. Numerical simulations confirm thesuperlinear convergence rate and show that our algorithm outperforms thefederated zeroth-order methods available in the literature.

FedZeN: Towards superlinear zeroth-order federated learning via incremental Hessian estimation