  We consider the distributed optimization problem where $n$ agents each
possessing a local cost function, collaboratively minimize the average of the
$n$ cost functions over a connected network. Assuming stochastic gradient
information is available, we study a distributed stochastic gradient algorithm,
called exact diffusion with adaptive stepsizes (EDAS) adapted from the Exact
Diffusion method and NIDS and perform a non-asymptotic convergence analysis. We
not only show that EDAS asymptotically achieves the same network independent
convergence rate as centralized stochastic gradient descent (SGD) for
minimizing strongly convex and smooth objective functions, but also
characterize the transient time needed for the algorithm to approach the
asymptotic convergence rate, which behaves as
$K_T=\mathcal{O}\left(\frac{n}{1-\lambda_2}\right)$, where $1-\lambda_2$ stands
for the spectral gap of the mixing matrix. To the best of our knowledge, EDAS
achieves the shortest transient time when the average of the $n$ cost functions
is strongly convex and each cost function is smooth. Numerical simulations
further corroborate and strengthen the obtained theoretical results.
