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.

翻译：本文旨在表征非均匀流行度与文件大小下去中心化缓存的内存-速率权衡。我们考虑一种近期提出的去中心化改进编码缓存方案（D-MCCS），并构建缓存放置优化问题以最小化该方案的平均速率。为求解这一非凸优化难题，我们首先提出一种逐次几何规划（GP）逼近算法，该算法保证收敛至驻点但计算复杂度较高。随后开发了一种低复杂度的基于文件分组的方法，其中提出一种流行度优先且感知大小（PF-SA）的缓存放置策略，通过考虑文件流行度与大小的非均匀性将文件划分为两组。两种算法均无需预先知晓活跃用户数量即可完成缓存放置。数值结果表明两者性能高度接近。我们进一步将非均匀流行度与文件大小下缓存放置的非凸优化问题转化为下界，并采用类似逐次GP逼近算法求解。研究表明，当每次同时请求文件的活跃用户数不超过两个时，具有优化缓存放置的D-MCCS可达到该下界。对于文件大小均匀但流行度非均匀且最优缓存放置关于文件对称的情形同样成立。在这些情况下，优化后的D-MCCS精确刻画了去中心化缓存的内存-速率权衡。对于一般情形，数值结果表明优化D-MCCS所得平均速率与该下界非常接近。