  The problem of designing distributed optimization algorithms that are
resilient to Byzantine adversaries has received significant attention. For the
Byzantine-resilient distributed optimization problem, the goal is to
(approximately) minimize the average of the local cost functions held by the
regular (non adversarial) agents in the network. In this paper, we provide a
general algorithmic framework for Byzantine-resilient distributed optimization
which includes some state-of-the-art algorithms as special cases. We analyze
the convergence of algorithms within the framework, and derive a geometric rate
of convergence of all regular agents to a ball around the optimal solution
(whose size we characterize). Furthermore, we show that approximate consensus
can be achieved geometrically fast under some minimal conditions. Our analysis
provides insights into the relationship among the convergence region, distance
between regular agents' values, step-size, and properties of the agents'
functions for Byzantine-resilient distributed optimization.
