  Distributed control systems require high reliability and availability
guarantees despite often being deployed at the edge of network infrastructure.
Edge computing resources are less secure and less reliable than centralized
resources in data centers. Replication and consensus protocols improve
robustness to network faults and crashed or corrupted nodes, but these volatile
environments can cause non-faulty nodes to temporarily diverge, increasing the
time needed for replicas to converge on a consensus value, and give Byzantine
attackers too much influence over the convergence process.
  This paper proposes proximal Byzantine consensus, a new approximate consensus
protocol where clients use statistical models of streaming computations to
decide a consensus value. In addition, it provides an interval around the
decision value and the probability that the true (non-faulty, noise-free) value
falls within this interval. Proximal consensus (PC) tolerates unreliable
network conditions, Byzantine behavior, and other sources of noise that cause
honest replica states to diverge. We evaluate our approach for scalar values,
and compare PC simulations against a vector consensus (VC) protocol simulation.
Our simulations demonstrate that consensus values selected by PC have lower
error and are more robust against Byzantine attacks. We formally characterize
the security guarantees against Byzantine attacks and demonstrate attacker
influence is bound with high probability. Additionally, an informal complexity
analysis suggests PC scales better to higher dimensions than convex hull-based
protocols such as VC.
