Resilient Approximation-Based Distributed Nonconvex Optimization

There has been an approximation-based distributed optimization algorithm that solves univariate nonconvex problems to arbitrary precision. The key idea is to construct approximations of local objectives and address a more structured approximate version of the problem. By representing diverse local objectives with compressed coefficients vectors, such algorithms enjoy gradient-free iterations but face severe security issues when adversaries occur. In this paper, we propose a resilient approximation-based distributed nonconvex optimization algorithm termed R-ADOA to defend attacks from malicious nodes. First, errors caused by adversaries are quantified and unified as the perturbation of coefficient vectors of approximations. Next, we propose a filtering mechanism and resilient stopping mechanism to limit errors arising in consensus-based iterations. Finally, an upper bound of the deviations of the obtained solutions from optimal solutions is given based on the eigenvalue perturbation theory of matrices. Numerical experiments are provided to illustrate the effectiveness of our algorithm. Compared to existing resilient distributed optimization algorithms, R-ADOA addresses nonconvex problems, converges exponentially fast, and contains explicit bounds for the deviations of solutions.