  In this work, we study the problem of minimizing the sum of strongly convex
functions split over a network of $n$ nodes. We propose the decentralized and
asynchronous algorithm ADFS to tackle the case when local functions are
themselves finite sums with $m$ components. ADFS converges linearly when local
functions are smooth, and matches the rates of the best known finite sum
algorithms when executed on a single machine. On several machines, ADFS enjoys
a $O (\sqrt{n})$ or $O(n)$ speed-up depending on the leading complexity term as
long as the diameter of the network is not too big with respect to $m$. This
also leads to a $\sqrt{m}$ speed-up over state-of-the-art distributed batch
methods, which is the expected speed-up for finite sum algorithms. In terms of
communication times and network parameters, ADFS scales as well as optimal
distributed batch algorithms. As a side contribution, we give a generalized
version of the accelerated proximal coordinate gradient algorithm using
arbitrary sampling that we apply to a well-chosen dual problem to derive ADFS.
Yet, ADFS uses primal proximal updates that only require solving
one-dimensional problems for many standard machine learning applications.
Finally, ADFS can be formulated for non-smooth objectives with equally good
scaling properties. We illustrate the improvement of ADFS over state-of-the-art
approaches with simulations.
