  In this paper, we consider a class of finite-sum convex optimization problems
defined over a distributed multiagent network with $m$ agents connected to a
central server. In particular, the objective function consists of the average
of $m$ ($\ge 1$) smooth components associated with each network agent together
with a strongly convex term. Our major contribution is to develop a new
randomized incremental gradient algorithm, namely random gradient extrapolation
method (RGEM), which does not require any exact gradient evaluation even for
the initial point, but can achieve the optimal ${\cal O}(\log(1/\epsilon))$
complexity bound in terms of the total number of gradient evaluations of
component functions to solve the finite-sum problems. Furthermore, we
demonstrate that for stochastic finite-sum optimization problems, RGEM
maintains the optimal ${\cal O}(1/\epsilon)$ complexity (up to a certain
logarithmic factor) in terms of the number of stochastic gradient computations,
but attains an ${\cal O}(\log(1/\epsilon))$ complexity in terms of
communication rounds (each round involves only one agent). It is worth noting
that the former bound is independent of the number of agents $m$, while the
latter one only linearly depends on $m$ or even $\sqrt m$ for ill-conditioned
problems. To the best of our knowledge, this is the first time that these
complexity bounds have been obtained for distributed and stochastic
optimization problems. Moreover, our algorithms were developed based on a novel
dual perspective of Nesterov's accelerated gradient method.
