  Most existing caching solutions for wireless networks rest on an unrealistic
assumption that the file popularity distribution is perfectly known. In this
paper, we consider optimal caching designs for perfect, imperfect and unknown
file popularity distributions in large-scale multi-tier wireless networks.
First, in the case of perfect file popularity distribution, we formulate the
optimization problem to maximize the successful transmission probability (STP),
which is a nonconvex problem. We develop an efficient parallel iterative
algorithm to obtain a stationary point using parallel successive convex
approximation (SCA). Then, in the case of imperfect file popularity
distribution, we formulate the robust optimization problem to maximize the
worst-case STP. To solve this challenging maximin problem, we transform it to
an equivalent complementary geometric programming (CGP), and develop an
efficient iterative algorithm which is shown to converge to a stationary point
using SCA. To the best of our knowledge, this is the first work explicitly
considering the estimation error of file popularity distribution in the
optimization of caching design. Next, in the case of unknown file popularity
distribution, we formulate the stochastic optimization problem to maximize the
stochastic STP (i.e., the STP in the stochastic form), which is a challenging
nonconvex stochastic optimization problem. Based on stochastic parallel SCA, we
develop an efficient iterative algorithm to obtain a stationary point, by
exploiting structural properties of the stochastic STP and making full use of
instantaneous file requests. As far as we know, this is the first work
considering stochastic optimization in a large-scale multi-tier wireless
network. Finally, by numerical results, we show that the proposed solutions
achieve significant gains over existing schemes in all three cases.
