  This paper aims to characterize the memory-rate tradeoff for decentralized
caching under nonuniform file popularity and size. We consider a recently
proposed decentralized modified coded caching scheme (D-MCCS) and formulate the
cache placement optimization problem to minimize the average rate for the
D-MCCS. To solve this challenging non-convex optimization problem, we first
propose a successive Geometric Programming (GP) approximation algorithm, which
guarantees convergence to a stationary point but has high computational
complexity. Next, we develop a low-complexity file-group-based approach, where
we propose a popularity-first and size-aware (PF-SA) cache placement strategy
to partition files into two groups, taking into account the nonuniformity in
file popularity and size. Both algorithms do not require the knowledge of
active users beforehand for cache placement. Numerical results show that they
perform very closely to each other. We further develop a lower bound for
decentralized caching under nonuniform file popularity and size as a non-convex
optimization problem and solved it using a similar successive GP approximation
algorithm. We show that the D-MCCS with the optimized cache placement attains
this lower bound when no more than two active users request files at a time.
The same is true for files with uniform size but nonuniform popularity and the
optimal cache placement being symmetric among files. In these cases, the
optimized DMCCS characterizes the exact memory-rate tradeoff for decentralized
caching. For general cases, our numerical results show that the average rate
achieved by the optimized D-MCCS is very close to the lower bound.
