Symmetry, Outer Bounds, and Code Constructions: A Computer-Aided Investigation on the Fundamental Limits of Caching

Abstract

Abstract:We illustrate how computer-aided methods can be used to investigate the fundamental limits of the caching systems, which are significantly different from the conventional analytical approach usually seen in the information theory literature. The linear programming (LP) outer bound of the entropy space serves as the starting point of this approach; however, our effort goes significantly beyond using it to prove information inequalities. We first identify and formalize the symmetry structure in the problem, which enables us to show the existence of optimal symmetric solutions. A symmetry-reduced linear program is then used to identify the boundary of the memory-transmission-rate tradeoff for several small cases, for which we obtain a set of tight outer bounds. General hypotheses on the optimal tradeoff region are formed from these computed data, which are then analytically proven. This leads to a complete characterization of the optimal tradeoff for systems with only two users, and certain partial characterization for systems with only two files. Next, we show that by carefully analyzing the joint entropy structure of the outer bounds for certain cases, a novel code construction can be reverse-engineered, which eventually leads to a general class of codes. Finally, we show that outer bounds can be computed through strategically relaxing the LP in different ways, which can be used to explore the problem computationally. This allows us firstly to deduce generic characteristic of the converse proof, and secondly to compute outer bounds for larger problem cases, despite the seemingly impossible computation scale.

Characterizing the rate-memory tradeoff in cache networks within a factor of 2

“…Based on the proof idea of Theorem 1, we can completely characterize the rate-memory tradeoff for the two-user case, for any possible values of N and M , for both peak rate and average rate. Prior to this work, the peak rate vs. memory tradeoff for the two-user case was characterized in [2] for N ≤ 2, and is characterized in [25] for N ≥ 3 very recently using non-parallel bounding techniques. However the average rate vs. memory tradeoff has never been completely characterized for any non-trivial case.…”

“…In prior works, despite various attempts, this tradeoff has only been exactly characterized in two instances: the single-user case [2] and, more recently, the two-user case [25]. Our second converse bound improves the bounds introduced in those results, and allows us to characterize the rate-memory tradeoff for systems with up to 5 users.…”

“…In this case, the number of subfiles becomes large, but fraction of files (or sub-files) that can be stored at each user is fixed. In this paper, we completely characterize this tradeoff for systems with up to 5 users, for both peak rate and average rate, while in prior works, this tradeoff has only been exactly characterized in two instances: the single-user case [2] and, more recently, the two-user case [25].…”

Demand-Private Coded Caching and the Exact Trade-off for N=K=2

“…For N = 2 and N K = 4, the demands in D RS and their correspondingc(d s ) are given in Table I. A related concept is the "demand type" used in [14]. Definition 2 (Demand Types) In (N, K)-non-private coded caching problem, for a given demand vectord, let t i denote the number of users requesting file i, where i = 0, .…”

“…The proof will construct an (N, K, M, R)-private scheme using an (N, N K, M, R) D RS -non-private scheme as a blackbox using ideas from [3]. We first give an example to illustrate this construction for N = 2, K = 2 using only the restricted demand subset for a (2, 4, 1 3 , 4 3 ) D (2,2) -non-private scheme from [14]. We will see that this allows a better achievable rate ( 1 3 , 4 3 ) for the (N = 2, K = 2) demand-private coded caching problem than what can be achieved for the N = 2, K = 4 non-private caching problem.…”