Linear Convergence of Pre-Conditioned PI Consensus Algorithm under
  Restricted Strong Convexity
Kushal Chakrabarti Mayank Baranwal
This paper considers solving distributed convex optimization problems in
peer-to-peer multi-agent networks. The network is assumed to be synchronous and
connected. By using the proportional-integral (PI) control strategy, various
algorithms with fixed stepsize have been developed. The earliest among them is
the PI consensus algorithm. Using Lyapunov theory, we guarantee exponential
convergence of the PI consensus algorithm for restricted strongly convex
functions with rate-matching discretization, without requiring convexity of
individual local cost functions, for the first time. In order to accelerate the
PI consensus algorithm, we incorporate local pre-conditioning in the form of
constant positive definite matrices and numerically validate its efficiency
compared to the prominent distributed convex optimization algorithms. Unlike
classical pre-conditioning, where only the gradients are multiplied by a
pre-conditioner, the proposed pre-conditioning modifies both the gradients and
the consensus terms, thereby controlling the effect of the communication graph
between the agents on the PI consensus algorithm.