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Online Coded Caching

Ramtin Pedarsani, Mohammad Ali Maddah-Ali, Urs Niesen

arXiv:1311.3646v1cs.ITcs.NI

TL;DR

The paper asks how to minimize shared-link traffic when users request files from a Markov-evolving popular set and cache operations cannot use future requests. It develops online coded caching, characterizes its optimal long-term rate, and evaluates coded LRS; online performance is approximately that of offline caching, while coded LRS outperforms LRU on Netflix-derived demands.

  • Problem

    The problem is to minimize shared-link bits for users with finite caches and sequential demands from a Markov-evolving popular-file set under online, no-future-knowledge constraints.

  • Method

    The paper characterizes optimal online caching, introduces coded LRS, and evaluates it on a Netflix Prize-derived demand time series.

  • Results

    The optimal online scheme performs approximately the same as the optimal offline scheme, while coded LRS significantly outperforms LRU on the Netflix-derived time series.

  • Takeaways & Limitations

    Online caching restrictions have only a small effect relative to offline caching, and coded delivery can substantially improve performance over LRU in the evaluated time series.

Abstract

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We consider a basic content distribution scenario consisting of a single origin server connected through a shared bottleneck link to a number of users each equipped with a cache of finite memory. The users issue a sequence of content requests from a set of popular files, and the goal is to operate the caches as well as the server such that these requests are satisfied with the minimum number of bits sent over the shared link. Assuming a basic Markov model for renewing the set of popular files, we characterize approximately the optimal long-term average rate of the shared link. We further prove that the optimal online scheme has approximately the same performance as the optimal offline scheme, in which the cache contents can be updated based on the entire set of popular files before each new request. To support these theoretical results, we propose an online coded caching scheme termed coded least-recently sent (LRS) and simulate it for a demand time series derived from the dataset made available by Netflix for the Netflix Prize. For this time series, we show that the proposed coded LRS algorithm significantly outperforms the popular least-recently used (LRU) caching algorithm.

I. INTRODUCTION

The paper studies online caching for shared-link content distribution, where caches and server must respond without future-request knowledge. It develops theory for evolving popular files and evaluates coded LRS against LRU on Netflix-derived demands.

  • Online caching must operate cache contents and server transmissions on the fly without knowing future requests.
  • For multiple caches, cache-miss rate is not proportional to network load, motivating coded caching based on multicasting opportunities.
  • The model uses a server, shared bottleneck link, finite user caches, and uniformly selected demands from a Markov-evolving popular-file set.
  • The paper approximately characterizes the optimal long-term average shared-link rate and finds online performance approximately matches offline performance.
  • Coded LRS is simulated on Netflix Prize-derived demands and significantly outperforms baseline LRU.

II. BACKGROUND ON CODED CACHING

Coded caching reduces shared-link load by combining requested content across users, extending offline caching ideas to exploit coded multicasting. Its gains include global multicasting benefits and local availability of requested-file parts.

  • Coded caching creates and exploits coded multicasting opportunities between users with different demands to reduce network load.
  • The offline scheme serves K users requesting files from a fixed set of N files after offline cache placement minimizes peak shared-link load.
  • Each user caches a random subset of MF/N bits from every file, then receives linear combinations enabling recovery of requested files.
  • In the two-user example, sending A∅, B∅, and A2 ⊕ B1 lets both users recover missing file parts using cached side information.
  • The global coded-multicasting gain increases with cache size M, while a second local gain comes from requested content already stored locally.

III. PROBLEM SETTING

The problem setting defines online shared-link caching with time-varying popular files, sequential requests, finite memories, and causal cache and transmission rules. The objective is to minimize normalized long-term average load.

  • At each time t, K users request distinct files uniformly from the time-varying popular set N_t, and each cache has size MF bits.
  • The server observes current requests and transmits R_tF bits so each user reconstructs its requested file from the transmission and cache contents.
  • The objective is to minimize the long-term average shared-link rate while satisfying memory and reconstruction constraints.
  • Online caches depend only on prior cache state, the immediately preceding transmission, and past requests; current messages depend on current demands and caches.
  • LRU stores M whole files, serves hits locally, and transmits a complete requested file on misses before evicting the least-recently used file.
  • The popular-file set remains unchanged with probability 1 − p or replaces one uniformly chosen file with a previously unseen arrival with probability p.

IV. MAIN RESULTS

The main-results section introduces the paper’s online coded caching algorithm and its theoretical and empirical comparisons with optimal online performance and LRU.

  • The paper introduces an online coded caching algorithm, analyzes a simplified variant, and compares it with LRU using Netflix Prize-derived demand data.

A. An Online, Coded Caching Algorithm

The paper proposes coded least-recently sent (LRS), an online caching algorithm combining coded delivery, partial caching, and least-recently sent eviction. Its delivery procedure sends coded combinations across user subsets, while cache updates replace least-recently sent files with sampled bits from newly requested files.

  • Algorithm proposal: Coded LRS is an online version of the offline coded caching algorithm from [18].The algorithm is organized into delivery and cache update procedures.
  • Delivery procedure: The delivery procedure sends one linear file combination for each subset of users.Requested-file bits are grouped by the users that cache them exclusively and combined element-wise with XOR.
  • Delivery procedure: When a requested file is absent from all caches, coded LRS sends the entire file uncoded over the shared link.In that case, the empty-set subfile is the entire file.
  • Cache update procedure: Cache updates evict the least-recently sent file and replace it with MF/N′ randomly chosen bits from a newly requested file.This update occurs when the requested file is not already partially cached.
  • Example: In the illustrative configuration, α = 3/2 makes each user cache one-third of N′ = 3 files.The example uses N = 2 popular files, K = 2 users, and M = 1.
  • Example: When a new file D is not cached, the example yields rate 15/9 after transmitting its uncoded content and coded terms.The cache then evicts A, the least-recently sent file, and partially caches D.
  • Comparison with LRU: Coded LRS differs from LRU through coded delivery, partial caching of many files, and eviction based on requests from all users.LRU instead uses uncoded delivery and caches fewer whole files.

B. Theoretical Results

The paper characterizes the optimal online shared-link rate and establishes that online caching is approximately as effective as offline caching. Simulations compare LRU, uncoded LRS, and coded LRS across cache sizes, highlighting the gains from coded delivery and partial caching.

  • Theorem: The main theorem characterizes the long-term average rate of the optimal online caching scheme.The theorem provides the paper’s central performance result, with its displayed bound stated in the theorem itself.
  • Theorem: The optimal online caching rate is approximately the same as the optimal offline caching rate.This remains true despite online caches observing only request sequences and paying shared-link costs for cache updates.
  • Numerical comparison: For N = 1000, K = 30, and p = 0.1, Fig. 2 plots long-term average shared-link rate against cache memory for LRU, uncoded LRS, and coded LRS.The proposed coded LRS curve is compared with dashed green LRU and dashed black uncoded LRS.
  • Numerical comparison: For 0 ≤ M ≤750, uncoded LRS and LRU perform quite similarly; for 750 < M ≤1000, uncoded LRS significantly improves over LRU.The stated explanation is that LRS learns newly arriving popular files with fewer cache misses when M is close to N.
  • Numerical comparison: Coded LRS significantly outperforms uncoded LRS except when M is very close to N.The paper attributes the main gain away from large M to partial caching of many files and coded delivery.

C. Empirical Results

The empirical evaluation constructs and validates a Netflix-derived demand time series, then compares coded LRS with baseline LRU. Coded LRS significantly outperforms LRU, achieving a rate of 8.2 versus 36.2 for LRU at M = 100.

  • Demand time series: The evaluation uses Netflix Prize ratings from 2005 for movies released in 2004 or 2005, yielding about 10^7 ratings across 1948 unique movies.Rating time is used as a proxy for viewing time; filtering recent releases addresses delays for older movies.
  • Demand time series: Ratings for “Troy” and “National Treasure” rise around their DVD release weeks, remain high for several weeks, and then decline.This pattern supports rating time as a viewing-time proxy for recently released movies.
  • Demand time series: The observed rating dynamics also support the model of time-varying popular files used in the theoretical analysis.The validation concerns the viewing behavior represented by the filtered Netflix time series.
  • Caching comparison: Fig. 4 compares long-term average shared-link rate against cache memory size for LRU and coded LRS with K = 100 caches.LRU is shown as dashed green and coded LRS as solid blue.
  • Caching comparison: 8.2 versus 36.2: at M = 100, coded LRS achieves a shared-link rate of 8.2 compared with 36.2 for LRU.The comparison is reported for the Netflix demand time series.

V. PROOF OF THEOREM 1

The proof of Theorem 1 establishes both an upper and a lower bound on the optimal long-term average online caching rate.

  • Proof strategy: Theorem 1 is proved by establishing an upper bound and a lower bound on the optimal long-term average rate R̄⋆.The upper-bound argument and lower-bound argument are developed in separate sections.

A. Upper Bound in Theorem 1

The upper-bound proof analyzes a simplified coded LRS scheme through a Markov process tracking correctly cached popular files. It derives the bound R̄ ≤ 2R(M, N, K) + 6.

  • Scheme and rate: Coded random eviction randomly replaces cached files when newly requested files are not partially cached, inserting MF/N′ bits from each new file.This simplified scheme provides the upper-bound analysis for coded LRS.
  • Scheme and rate: The per-slot rate combines uncoded transmission of Y_t newly requested files with coded delivery of the remaining K − Y_t partially cached files.The coded portion requires rate R(M, αN, K − Y_t).
  • State-process analysis: The process X_t counts popular files partially stored in caches, while Y_t counts requested files that are not currently partially cached.Y_t can take values from 0 through K.
  • State-process analysis: {X_t} is a Markov process with a unique stationary distribution, enabling long-term analysis through its steady-state behavior.The recurrent class consists of states from ⌈K/2⌉ − 1 through N.
  • State-process analysis: The update for correctly cached files adds newly requested cached files and subtracts wrongly evicted files and files that cease to be popular.The auxiliary processes W_t and U_t represent these two loss mechanisms.
  • Bound derivation: The expected normalized number of correctly cached files satisfies a quadratic equation whose resulting lower bound does not depend on the variance σ^2.This lower bound is used to control the additional rate term.
  • Bound derivation: 2R(M, N, K) + 6: the analyzed scheme’s long-term average rate is bounded by this expression.The optimal online scheme is no worse than the analyzed coded random-eviction scheme.

B. Lower Bound in Theorem 1

The lower-bound proof uses an offline caching setting that knows the current popular-file set and may update caches before each request. Its rate lower-bounds the optimal online rate.

  • Offline comparison: In the offline setting, caches know the current popular-file set and may access all its files at the start of each slot, but they do not know future requests.Because this setting is more informed than the online setting, its optimal rate is a lower bound on the online optimum.
  • Offline comparison: The offline lower bound is used as the comparison baseline for the optimal online caching rate.The supplied proof passage states that the offline scheme’s rate is a lower bound for the online setting.
  • Offline comparison: The offline problem equals a prefetching problem whose instantaneous and long-term average rate is lower bounded by 1/12R(M, N, K).This establishes the lower-bound ingredient used for Theorem 1.

APPENDIX A PROOF OF (11) Set

The appendix proves bounds by rewriting an expression as a quadratic in the average random-sequence value, selecting the smaller real solution, and combining inequalities across two N/M regimes.

  • APPENDIX A PROOF OF (11): The proof rewrites (10) as a¯x2 + b¯x + c = 0 and analyzes its solutions.Because ¯x is the average of a real random sequence, the quadratic has real solutions.
  • APPENDIX A PROOF OF (11): The smaller quadratic solution is obtained using the negative sign in −b±.
  • APPENDIX A PROOF OF (11): The proof repeatedly uses β ≥0 to establish intermediate inequalities and lower-bound ¯x.
  • APPENDIX B PROOF OF (12): Appendix B splits the proof of (12) into the regimes N/M ≥1/(2 −α) and N/M < 1/(2 −α).The first regime assumes α < 2, while the second handles the complementary case.
  • APPENDIX B PROOF OF (12): Combining the inequalities from the two regimes yields the stated result.
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