  This paper presents a lower bound for optimizing a finite sum of $n$
functions, where each function is $L$-smooth and the sum is $\mu$-strongly
convex. We show that no algorithm can reach an error $\epsilon$ in minimizing
all functions from this class in fewer than $\Omega(n +
\sqrt{n(\kappa-1)}\log(1/\epsilon))$ iterations, where $\kappa=L/\mu$ is a
surrogate condition number. We then compare this lower bound to upper bounds
for recently developed methods specializing to this setting. When the functions
involved in this sum are not arbitrary, but based on i.i.d. random data, then
we further contrast these complexity results with those for optimal first-order
methods to directly optimize the sum. The conclusion we draw is that a lot of
caution is necessary for an accurate comparison, and identify machine learning
scenarios where the new methods help computationally.
