  Structured problems arise in many applications. To solve these problems, it
is important to leverage the structure information. This paper focuses on
convex problems with a finite-sum compositional structure. Finite-sum problems
appear as the sample average approximation of a stochastic optimization problem
and also arise in machine learning with a huge amount of training data. One
popularly used numerical approach for finite-sum problems is the stochastic
gradient method (SGM). However, the additional compositional structure
prohibits easy access to unbiased stochastic approximation of the gradient, so
directly applying the SGM to a finite-sum compositional optimization problem
(COP) is often inefficient.
  We design new algorithms for solving strongly-convex and also convex
two-level finite-sum COPs. Our design incorporates the Katyusha acceleration
technique and adopts the mini-batch sampling from both outer-level and
inner-level finite-sum. We first analyze the algorithm for strongly-convex
finite-sum COPs. Similar to a few existing works, we obtain linear convergence
rate in terms of the expected objective error, and from the convergence rate
result, we then establish complexity results of the algorithm to produce an
$\varepsilon$-solution. Our complexity results have the same dependence on the
number of component functions as existing works. However, due to the use of
Katyusha acceleration, our results have better dependence on the condition
number $\kappa$ and improve to $\kappa^{2.5}$ from the best-known $\kappa^3$.
Finally, we analyze the algorithm for convex finite-sum COPs, which uses as a
subroutine the algorithm for strongly-convex finite-sum COPs. Again, we obtain
better complexity results than existing works in terms of the dependence on
$\varepsilon$, improving to $\varepsilon^{-2.5}$ from the best-known
$\varepsilon^{-3}$.
