GRADES: Gradient Descent for Similarity Caching

Abstract

A similarity cache can reply to a query for an object with similar objects stored locally. In some applications of similarity caches, queries and objects are naturally represented as points in a continuous space. Examples include 360 • videos where user's head orientation-expressed in spherical coordinatesdetermines what part of the video needs to be retrieved, and recommendation systems where the objects are embedded in a finite-dimensional space with a distance metric to capture content dissimilarity. Existing similarity caching policies are simple modifications of classic policies like LRU, LFU, and qLRU and ignore the continuous nature of the space where objects are embedded. In this paper, we propose GRADES, a new similarity caching policy that uses gradient descent to navigate the continuous space and find the optimal objects to store in the cache. We provide theoretical convergence guarantees and show GRADES increases the similarity of the objects served by the cache in both applications mentioned above.

Content Placement in Networks of Similarity Caches

“…Note that for β = 0 we recover the cost resulting from (11) in the case of a tandem network. Computing the derivative of the above cost with respect to w i,1 we get:…”

“…where notice that we have separated the contribution of cache 1, and taken as decision variables vectors w j , with j > 1, since w 1 = 1 − N j=2 w j . Moreover, notice that the constraints in (11) are sufficient to guarantee that also the following obvious constraints hold:…”

“…Previous result allows us to prove the following interesting property about the structure of the optimal solution: Proposition 4.2. In the case of a chain topology, with requests arriving only at the first cache, the best solution of the continuous-domain, finite-M problem (11) is characterized by a set of popularity thresholds λ…”

“…By doing so the total cost in the system of parallel chains is surely not larger than the total cost achievable in the original tree, and, in general, it might be smaller (this because we can independently place objects in every chain so as to minimize the cost induced by the requests arriving at the corresponding leaf). On the other hand, the optimal allocation on each chain is the same, since the objective function in (11) is linear with respect to parameter β . Therefore, by adopting such equal allocation on each cache of the same level in the original tree, we obtain exactly the same total cost achieved in the system of parallel chains, hence this allocation is optimal.…”

“…An adversarial setting was studied in [10] by competitive analysis. The authors of [11] have proposed a similarity caching policy tailored for the case when cached objects may be embedded in R d with a distance that captures dissimilarity costs. The work most closely related to this paper is [12], where we have analyzed a single similarity cache in the offline, adversarial, and stochastic settings, proposing also some dynamic online policies to manage the cache.…”