Guaranteed Privacy of Distributed Nonconvex Optimization via Mixed-Monotone Functional Perturbations

In this paper, we introduce a new notion of guaranteed privacy that requires that the change of the range of the corresponding inclusion function to the true function is small. In particular, leveraging mixed-monotone inclusion functions, we propose a privacy-preserving mechanism for nonconvex distributed optimization, which is based on deterministic, but unknown, affine perturbation of the local objective functions, which is stronger than probabilistic differential privacy. The design requires a robust optimization method to characterize the best accuracy that can be achieved by an optimal perturbation. Subsequently, this is used to guide the refinement of a guaranteed-private perturbation mechanism that can achieve a quantifiable accuracy via a theoretical upper bound that is shown to be independent of the chosen optimization algorithm.