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Adding transmitters dramatically boosts coded-caching gains for finite file sizes

Eleftherios Lampiris, Petros Elia

arXiv:1802.03389v1cs.IT

TL;DR

Coded caching’s theoretical gains are limited in practice by the subpacketization required for finite files. The paper uses multiple transmitting antennas to reduce that requirement while preserving caching gains, achieving multiplicative effective-DoF improvements and constant-subpacketization regimes when antenna count scales with caching gain.

  • Problem

    Finite file and packet-size constraints create a subpacketization bottleneck that can hard-bound effective coded-caching gains despite theoretically unbounded gains.

  • Method

    The paper exploits transmitter-side dimensionality in multi-antenna, multi-server, and cache-aided interference settings to reduce required subpacketization without sacrificing caching gain.

  • Results

    The scheme reduces subpacketization approximately to its Lth root and can achieve the full cumulative DoF with constant subpacketization when L scales with theoretical caching gain.

  • Takeaways & Limitations

    Multiple transmitters provide a multiplicative mitigation of the subpacketization problem, making transmitter dimensionality more consequential than multiplexing alone over a large range of L.

Abstract

from arXiv · show

In the context of coded caching in the $K$-user BC, our work reveals the surprising fact that having multiple ($L$) transmitting antennas, dramatically ameliorates the long-standing subpacketization bottleneck of coded caching by reducing the required subpacketization to approximately its $L$th root, thus boosting the actual DoF by a multiplicative factor of up to $L$. In asymptotic terms, this reveals that as long as $L$ scales with the theoretical caching gain, then the full cumulative (multiplexing + full caching) gains are achieved with constant subpacketization. This is the first time, in any known setting, that unbounded caching gains appear under finite file-size constraints. The achieved caching gains here are up to $L$ times higher than any caching gains previously experienced in any single- or multi-antenna fully-connected setting, thus offering a multiplicative mitigation to a subpacketization problem that was previously known to hard-bound caching gains to small constants. The proposed scheme is practical and it works for all values of $K,L$ and all cache sizes. The scheme's gains show in practice: e.g. for $K=100$, when $L=1$ the theoretical caching gain of $G=10$, under the original coded caching algorithm, would have needed subpacketization $S_1 = \binom{K}{G}= \binom{100}{10} > 10^{13}$, while if extra transmitting antennas were added, the subpacketization was previously known to match or exceed $S_1$. Now for $L=5$, our scheme offers the theoretical (unconstrained) cumulative DoF $d_L = L+G = 5+10=15$, with subpacketization $S_L=\binom{K/L}{G/L} =\binom{100/5}{10/5} = 190$. The work extends to the multi-server and cache-aided IC settings, while the scheme's performance, given subpacketization $S_L=\binom{K/L}{G/L}$, is within a factor of 2 from the optimal linear sum-DoF.

I. INTRODUCTION

Coded caching creates multicasting opportunities by combining cached content across users, but finite file sizes and packet constraints impose a subpacketization bottleneck that can hard-bound practical gains. The paper motivates reduced-subpacketization algorithms and shows that transmitter dimensionality can preserve caching gains while reducing this burden.

  • Coded caching framework: Coded caching combines placement and delivery phases to encode across different users’ requests, creating multicasting opportunities despite distinct requested files.Each receiver can use cached content from other requested files to cancel interference.
  • Coded caching gains: T = K(1 − γ)/(1 + Kγ) gives the normalized delivery time for any simultaneous requests, approaching 1/M as K increases.Here γ = M/N is the normalized cache size.
  • Subpacketization bottleneck: Finite file sizes and minimum packet sizes force file segmentation limits, so the original algorithm can encode over only a bounded number of users and lose its theoretical caching gain.The resulting effective gain remains hard-bounded by small constants under realistic γ and Smax.
  • Prior approaches: Prior reduced-subpacketization constructions trade theoretical caching gain against packetization cost, but their effective gains remain constrained under realistic operating parameters.The cited constructions include placement-delivery arrays and linear-code designs.

B. Coded caching with multiple transmitters

Prior multi-transmitter and multi-antenna schemes could add multiplexing and theoretical caching gains, but finite subpacketization generally reduced their effective caching gains. The section positions this gap as an unresolved limitation in fully connected settings.

  • Multiplexing and caching gains can be combined additively in theory, but high subpacketization generally reduces the actual caching gain.This limitation appears in both multi-server and cache-aided settings described in the section.
  • A prior multi-server construction achieved T = KTγT + Kγ, within a factor of 2 of optimal one-shot linear sum-DoF.
  • Transmitter-side cache redundancy KTγT can provide cooperative multiplexing gains that add to the receiver-side theoretical caching gain G = Kγ.
  • Earlier multi-antenna approaches maintained theoretical caching gains and added multiplexing gains, while retaining high subpacketization levels.
  • Known fully connected single- and multi-antenna settings had no method supporting more than small effective caching gains under the stated subpacketization constraints.

C. Preview of results and paper outline

The proposed scheme uses transmitter-side dimensionality to reduce subpacketization while preserving theoretical DoF. It achieves either an L-fold effective-DoF increase or the unconstrained DoF, and extends to related caching models.

  • Multiple antennas reduce rather than increase subpacketization, producing accelerated reductions even for small L.
  • The scheme achieves dL = L + G = L + Kγ = KTγT + Kγ with subpacketization approximately the Lth root of the single-antenna requirement.When L matches Kγ, the theoretical DoF can be achieved with SL = K/L.
  • The resulting effective DoF is either L times the single-antenna effective DoF or the theoretical value dL = L + Kγ.
  • The approach extends to multiple underlying coded-caching algorithms, multi-server settings, and cache-aided interference scenarios.
  • The main construction initially assumes L divides K and Kγ, with a modified scheme removing these integer constraints and only a very small performance loss.

D. Notation

This section defines the caching, DoF, and subpacketization quantities and specifies the K-user L-antenna broadcast-channel model. The system uses cache placement followed by request-dependent delivery under high-SNR assumptions.

  • The theoretical caching gain is G = Kγ, while the effective caching gain is the subpacketization-constrained DoF beyond the multiplexing gain L.
  • The effective total DoF is d̄L(γ) = L + ḠL, representing the actual number of users served simultaneously under the subpacketization constraint.
  • The L-antenna broadcast channel assumes high SNR, perfect channel state information at active nodes, and statistically symmetric fading across users.
  • The model has N files, K single-antenna receivers with cache size Mf, and a transmitter that serves arbitrary user requests after placement.
  • The analysis excludes the trivial regime L ≥ K(1 − γ), where interference-free delivery achieves T = 1 − γ and sum-DoF K.

III. DESCRIPTION OF THE SCHEME

The scheme groups users into L-sized groups, applies coded caching across groups, and uses precoding within each group. This yields the theoretical sum-DoF with reduced subpacketization, including a concrete K = 50 example.

  • Users are partitioned into K′ = K/L groups of L users, and the coded-caching algorithm treats each group as a single user.
  • The delivery process serves K′γ + 1 groups per coded-caching transmission and avoids repeating subfiles across group cliques.
  • Within each group, zero-forcing precoders null interference at the other L − 1 receivers, while caches remove undesired out-of-group messages.
  • Kγ + L users are served at a time, achieving DoF Kγ + L with the proposed grouped transmission scheme.
  • For K = 50, L = 5, and γ = 3/10, the scheme achieves sum-DoF 20 with subpacketization 120.

IV. MAIN RESULTS

The main result gives a cache-aided MISO-BC scheme achieving sum-DoF L + Kγ with reduced subpacketization. Extra antennas can multiply effective DoF while subpacketization remains limiting, and the scheme extends to practical parameter settings.

  • Theorem 1: dL(γ) = L + Kγ is achievable with the proposed L-antenna MISO-BC scheme and its stated reduced subpacketization.The result is first established for L dividing K and Kγ, with general cases handled by memory sharing.
  • A. Effective gains and multiplicative boost of effective DoF: L antennas can encode over L times as many users as one antenna under the same subpacketization constraint, up to the available K users.This substitution effectively replaces K by K/L in the subpacketization expression.
  • A. Effective gains and multiplicative boost of effective DoF: The effective DoF is either increased by a factor of L or reaches the unconstrained DoF dL = L + Kγ.The multiplicative boost persists while subpacketization remains an active constraint.
  • Practical implication - Making small caches relevant: For a fixed target caching gain, extra antennas exponentially expand the range of cache sizes that can achieve it.The paper connects this reduction in the minimum applicable γ to substituting K by K/L in the subpacketization requirement.

B. Subpacketization cost of complementing the multiplexing gains

The subpacketization cost is governed by the desired ratio between total DoF and multiplexing gain rather than directly by K, L, or the caching gain alone. Matching this ratio yields explicit cache and subpacketization requirements.

  • B. Subpacketization cost of complementing the multiplexing gains: The relevant design parameter is x = dL(γ) / dL(γ=0), the ratio between total DoF and multiplexing gain.The paper expresses the target DoF as x times the cache-free multiplexing gain.
  • B. Subpacketization cost of complementing the multiplexing gains: A subpacketization level specified by the corollary can yield a DoF x times the multiplexing gain.The construction uses γ = λ(x − 1) with λ = L/K.
  • B. Subpacketization cost of complementing the multiplexing gains: With γ = λ and subpacketization SL = 1/λ = K/L, caching doubles the cache-free DoF; with γ = 2λ, it triples it.These examples correspond to serving 2L and 3L users at a time, respectively.

C. Subpacketization scaling and algorithmic simplicity from matching multiplexing gain with caching gain

When transmitter antennas scale with the receiver caching gain, the full cumulative DoF can be achieved with constant subpacketization. The same scaling principles extend to cache-aided interference and multi-server settings.

  • C. Subpacketization scaling and algorithmic simplicity from matching multiplexing gain with caching gain: As L scales with Kγ, the full sum-DoF L + Kγ is achievable with constant subpacketization.In particular, setting L = Kγ achieves this DoF with subpacketization independent of K and L.
  • C. Subpacketization scaling and algorithmic simplicity from matching multiplexing gain with caching gain: In cache-aided interference, transmitter redundancy KTγT can multiply effective DoF or reach the unconstrained DoF KTγT + Kγ.If transmitter redundancy scales with receiver redundancy, the full sum-DoF is achievable with constant subpacketization.
  • C. Subpacketization scaling and algorithmic simplicity from matching multiplexing gain with caching gain: With KT = 2 and LT = 5, or KT = 4 cooperating base stations, the example reduces the subpacketization needed for a caching gain of 20 to 500.The single-base-station case would require subpacketization greater than 10^61.
  • C. Subpacketization scaling and algorithmic simplicity from matching multiplexing gain with caching gain: With γ = 1/50 and either LT = 100 or five base stations with LT = 20, the scheme achieves dL(γ) = 300 while caching adds 200 served users.The corresponding subpacketization is reported as 10000/100.

E. Near-optimality of schemes

The schemes are one-shot linear and therefore admit an outer-bound comparison showing near-optimal linear sum-DoF performance. Removing integer divisibility constraints causes only modest changes.

  • E. Near-optimality of schemes: After removing integer constraints, subpacketization increases by at most a marginal amount while achieved DoF decreases relatively little.The stated target remains dL(γ) = L + Kγ up to the reported approximation.
  • E. Near-optimality of schemes: The schemes achieve performance within a factor of 2 of the theoretical optimal linear-DoF.The comparison applies through the schemes’ one-shot linear property and the cited outer bound.
  • E. Near-optimality of schemes: The multiplicative gap is bounded above by 5/3 when L > Kγ and by 4/3 when L < Kγ, and it converges to 1 as K increases.These bounds quantify the effect of relaxing the integer constraints.

F. Elevating different coded caching algorithms to the L antenna setting

The elevation procedure adapts single-stream coded-caching algorithms to L antennas by grouping users and replacing XOR summands with precoded vectors. This preserves the underlying algorithmic structure while multiplying differences in effective caching gains by up to L.

  • Elevation procedure: The elevation procedure splits K users into K′ groups of L users and applies a single-stream coded-caching algorithm as if each group were one user.Users within each group share cache contents, and the single-stream algorithm determines cache placement and delivery structure.
  • Elevation procedure: Each XOR summand is replaced by a precoded L-length vector, and the resulting vectors are combined into a composite transmission.A composite vector serves L·d′_1(γ) users at a time before the scheme proceeds to subsequent XORs.
  • Gain comparison: The elevated MN construction treats d′_1(γ)=L+Kγ users at a time, whereas elevated alternatives such as PD and LC treat d′_1,pd=Kγ groups at a time.For the latter schemes, the corresponding total is d_L,pd(γ)=L·d′_1,pd(γ), subject to L≤Kγ.
  • Gain comparison: The elevated PD and LC algorithms remain subject to constraints on γ, while their effective gain is bounded by the unconstrained theoretical caching gain.For PD, the theoretical caching gain is Kγ−L, and the effective gain depends on the allowable subpacketization Smax.
  • Gain comparison: Differences in effective caching gain between single-stream algorithms are magnified by up to L after elevation to the L-antenna setting.The improvement can be small when L=1 but increases as a multiple of L in the multi-antenna setting.
  • Implications: The construction increases the value of developing single-stream coded-caching algorithms with reduced subpacketization rather than eliminating that need.Any improvement from such underlying algorithms can be amplified after elevation.

V. CONCLUSIONS

The scheme uses transmitter-side dimensionality and joint cache-placement/physical-layer design to reduce subpacketization while preserving caching gains. In subpacketization-constrained settings, extra antennas can yield multiplicative effective-DoF improvements, with extensions to multi-server and interference settings.

  • V. CONCLUSIONS: The scheme substantially reduces required subpacketization without sacrificing the theoretical caching gain.It combines zero-forcing and low-dimensional coded caching through transmitter-side dimensionality.
  • V. CONCLUSIONS: Multiplicative effective-DoF gains can exceed the additive multiplexing gain from extra antennas in subpacketization-constrained systems.The conclusions contrast an additive increase of L−1 with a multiplicative increase of L, identifying receiver-side caching enhancement as the main impact over a range of L.
  • V. CONCLUSIONS: The design relies on grouping receivers with identical caches, reducing the number of distinct cache pairings needed for coded delivery.Multi-node precoding enables the construction to use fewer distinct cache configurations while retaining implementable zero-forcing and coded-caching components.
  • V. CONCLUSIONS: The construction works for all values of K, L, γ, K_T, and γ_T, supporting practical use when subpacketization is the major bottleneck.The conclusions also note that multiple antennas and transmitter cooperation are standard wireless ingredients.
  • V. CONCLUSIONS: The scheme maintains substantial, though not complete, robustness when the exact network structure is unknown during cache placement.Universal schemes retain an advantage for some network-structure uncertainty, whereas the proposed non-separated approach can achieve better effective gains when structure is exploited.
  • V. CONCLUSIONS: Joint consideration of cache placement and physical-layer structure can produce unboundedly better effective gains than universal schemes.The comparison attributes this potential to exploiting network structure through a non-separated coded-caching and PHY design.

VI. APPENDIX

The appendix extends the construction to cache-aided interference networks with independent transmitters and removes divisibility constraints through memory sharing. It establishes near-optimal performance, with a multiplicative gap bounded by 2 in the stated regime.

  • VI. APPENDIX: The multi-transmitter setting assumes full connectivity and no information exchange between transmitters.Each receiver is connected to all K_T transmitters, and transmitter-side placement provides each subfile at exactly K_Tγ_T transmitters.
  • VI. APPENDIX: The cache-aided interference construction serves K_Tγ_T+Kγ users at a time using transmitter-side and receiver-side caching gains.Precoding separates L=K_Tγ_T streams within receiver groups, while caching permits simultaneous treatment of additional groups.
  • VI. APPENDIX: With K_T transmitters each having L_T antennas, the scheme achieves sum-DoF K_TL_Tγ_T+Kγ under the stated transmitter-cache setting.The result applies when each subfile is available at K_TL_Tγ_T transmitting antennas.
  • VI. APPENDIX: Memory sharing removes the divisibility constraints on L, K, and Kγ by splitting files and combining placements with different cache redundancies.The construction uses hypothetical users for L∤K and memory sharing when L∤Kγ.
  • VI. APPENDIX: The achieved DoF is within a multiplicative factor of 2 of the relevant bound when L<Kγ, and the gap vanishes as Kγ/L increases.For Kγ in intervals (qL,qL+1), the intermediate gap is q+1 before approaching the stated bound.
  • VI. APPENDIX: Memory sharing adds at most a factor K to subpacketization from splitting files, while the resulting performance changes only marginally after integer constraints are removed.The extra factor is bounded because the sharing proportion is at least 1/K when Kγ is an integer.
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