  Empirical risk minimization is an important class of optimization problems
with many popular machine learning applications, and stochastic variance
reduction methods are popular choices for solving them. Among these methods,
SVRG and Katyusha X (a Nesterov accelerated SVRG) achieve fast convergence
without substantial memory requirement. In this paper, we propose to accelerate
these two algorithms by \textit{inexact preconditioning}, the proposed methods
employ \textit{fixed} preconditioners, although the subproblem in each epoch
becomes harder, it suffices to apply \textit{fixed} number of simple
subroutines to solve it inexactly, without losing the overall convergence. As a
result, this inexact preconditioning strategy gives provably better iteration
complexity and gradient complexity over SVRG and Katyusha X. We also allow each
function in the finite sum to be nonconvex while the sum is strongly convex. In
our numerical experiments, we observe an on average $8\times$ speedup on the
number of iterations and $7\times$ speedup on runtime.
