Private and Robust Distributed Nonconvex Optimization via Polynomial Approximation

There has been work that exploits polynomial approximation to solve distributed nonconvex optimization problems involving univariate objectives. This idea facilitates arbitrarily precise global optimization without requiring local evaluations of gradients at every iteration. Nonetheless, there remains a gap between existing guarantees and practical requirements, e.g., privacy preservation and robustness to network imperfections. To fill this gap and keep the above strengths, we propose a Private and Robust Chebyshev-Proxy-based distributed Optimization Algorithm (PR-CPOA). Specifically, to ensure both the accuracy of solutions and the privacy of local objectives, we design a new privacy-preserving mechanism. This mechanism leverages the randomness in blockwise insertions of perturbed vector states and hence provides a stronger privacy guarantee in the scope of ($\alpha,\beta$)-data-privacy. Furthermore, to gain robustness against network imperfections, we use the push-sum consensus protocol as a backbone and discuss its specific enhancements. Thanks to the purely consensus-type iterations, we avoid the privacy-accuracy trade-off and the bother of selecting proper step sizes in different settings. We rigorously analyze the accuracy, privacy, and complexity of the proposed algorithm. We show that the advantages brought by introducing polynomial approximation are maintained when all the above requirements exist.