A causal filter of gradient information for enhanced robustness and resilience in distributed convex optimization

Exchange or communication of gradient samples is crucial in many distributed convex optimization set-ups, featured in modern Cyber-Physical Systems applications (i.e. smart grid, coordination of mobile robots/vehicles, predictive maintenance, just to cite a few). It is, however, prone to noise injection of various nature, including malicious attacks. In this paper we propose a causal filter to process a (possibly corrupted) sequence of sub gradient samples of an unknown convex function, so as to restore, by minimally perturbing the sequence, compatibility with the underlying convexity prior of the original function. The algorithm is recursive in nature (to reduce its computational complexity) and computes an optimal filtered gradient sequence that, in real-time, minimizes the square of the perturbation applied to its latest sample under the assumption that prior (filtered) samples are accurate. The filter is initially tested on simple convex functions to illustrate its performance. Then, a consensus-based distributed optimization scheme is considered to emphasize the robustness benefits to convergence of the protocol achieved through the filter, in the presence of corrupt data.