Caching in Networks without Regret

We consider the online $\textsf{Bipartite Caching}$ problem where $n$ users are connected to $m$ caches in the form of a bipartite network. Each of the $m$ caches has a file storage capacity of $C$. There is a library consisting of $N >C$ distinct files. Each user can request any one of the files from the library at each time slot. We allow the file request sequences to be chosen in an adversarial fashion. A user's request at a time slot is satisfied if the requested file is already hosted on at least one of the caches connected to the user at that time slot. Our objective is to design an efficient online caching policy with minimal regret. In this paper, we propose $\textsf{LeadCache,}$ an online caching policy based on the $\textsf{Follow the Perturbed Leader}$ (FTPL) paradigm. We show that $\textsf{LeadCache}$ is regret optimal up to a multiplicative factor of $\tilde{O}(n^{0.375}).$ As a byproduct of our analysis, we design a new linear-time deterministic Pipage rounding procedure for the LP relaxation of a well-known NP-hard combinatorial optimization problem in this area. Our new rounding algorithm substantially improves upon the currently best-known complexity for this problem. Moreover, we show the surprising result that under mild Strong-Law-type assumptions on the file request sequence, the rate of file fetches to the caches approaches to zero under the $\textsf{LeadCache}$ policy. Finally, we derive a tight universal regret lower bound for the $\textsf{Bipartite Caching}$ problem, which critically makes use of results from graph coloring theory and certifies the announced approximation ratio.