  This paper studies a distributed multi-agent convex optimization problem. The
system comprises multiple agents in this problem, each with a set of local data
points and an associated local cost function. The agents are connected to a
server, and there is no inter-agent communication. The agents' goal is to learn
a parameter vector that optimizes the aggregate of their local costs without
revealing their local data points. In principle, the agents can solve this
problem by collaborating with the server using the traditional distributed
gradient-descent method. However, when the aggregate cost is ill-conditioned,
the gradient-descent method (i) requires a large number of iterations to
converge, and (ii) is highly unstable against process noise. We propose an
iterative pre-conditioning technique to mitigate the deleterious effects of the
cost function's conditioning on the convergence rate of distributed
gradient-descent. Unlike the conventional pre-conditioning techniques, the
pre-conditioner matrix in our proposed technique updates iteratively to
facilitate implementation on the distributed network. In the distributed
setting, we provably show that the proposed algorithm converges linearly with
an improved rate of convergence than the traditional and adaptive
gradient-descent methods. Additionally, for the special case when the minimizer
of the aggregate cost is unique, our algorithm converges superlinearly. We
demonstrate our algorithm's superior performance compared to prominent
distributed algorithms for solving real logistic regression problems and
emulating neural network training via a noisy quadratic model, thereby
signifying the proposed algorithm's efficiency for distributively solving
non-convex optimization. Moreover, we empirically show that the proposed
algorithm results in faster training without compromising the generalization
performance.
