# Multi-Cell Mobile Edge Coded Computing: Trading Communication and Computing for Distributed Matrix Multiplication

ISIT, pp. 215-220, 2020.

EI

Weibo:

Abstract:

A multi-cell mobile edge computing network is studied, in which each user wishes to compute the product of a user-generated data matrix with a network-stored matrix through data uploading, distributed edge computing, and output downloading. Assuming randomly straggling edge servers, this paper investigates the interplay among upload, comp...More

Code:

Data:

Introduction

- Mobile edge computing (MEC) is an emerging network architecture that enables cloudcomputing capabilities at the edge nodes (ENs) of mobile networks [1].
- For a given NULT τ u, the compute-download latency region is defined as the union of the set for all NCT-NDLT pairs (τ c, τ d), i.e., T ∗(τ u) (τ c, τ d) : (τ u(r), τ c(r, q), τ d(r, q)) is achievable for some (r, q) and τ u ≤ τ u(r), τ c ≥ τ c(r, q), and τ d ≥ τ d(r, q) .

Highlights

- Mobile edge computing (MEC) is an emerging network architecture that enables cloudcomputing capabilities at the edge nodes (ENs) of mobile networks [1]
- This study focuses on the baseline problem of computing the product between user-generated data vectors u’s and a network-stored matrix A
- In Mobile edge computing networks, the servers are embedded in distinct edge nodes, and distributed computation at the edge requires input data uploading via the uplink, computation at the edge nodes, and output result downloading via the downlink
- This paper tackles the question: Given an upload latency, what is the optimal trade-off between computing and download latencies? We focus on the high signal-to-noise ratio (SNR) regime in order to highlight the role of interference management as enabled by spatial redundancy
- For a repetitionrecovery order pair (r, q), as shown in Fig. 2, during task assignment, matrix A is encoded by a cascade of an Maximum Distance Separable code of rate 1/ρ1 and a repetition code of rate 1/ρ2
- As in [4]–[6], Maximum Distance Separable codes can alleviate the impact of stragglers on the computation latency by decreasing the admissible values for the number q of non-straggling edge nodes

Results

- (Convexity of compute-download latency region.) For an input data assignment policy {Ui,K′ }i∈M,K′⊆K with a repetition order r, fix an input uploading strategy achieving NULT τ u.
- 3) Edge Computing: Following Sec. III-A, the cascaded MDS-repetition code rate pair satisfies ρ1∈{1, Kμ/(Kμ−1), Kμ/(Kμ−2), · · · , Kμ} and ρ2=Kμ/ρ1 under the constraint of the total storage size kμ.
- 4) Output Downloading: Following Sec. III-A, for each user, the number of input vectors computed by p1 non-straggling ENs equals
- Similar to the calculation of NULT in Sec. IV-2, by Definition 2, the NDLT for each user to download the outputs replicated at p2 non-stragglers is given by τd p1,p2
- 5) Inner Bound of Compute-Download Latency Region: For an NULT τ u = τau(r) given in (7) for some r ∈ R, where R is given by (10), the feasible values of recovery order q’s should satisfy
- In the rest of this appendix, the authors first derive the lower bounds on the NULT, NCT, and NDLT for a particular task assignment policy Ui,K′ with repetition-recovery order (r, q) ∈ R.
- The minimum NULT over all feasible task assignment is given as τ u∗(r) = min τ u∗(r, γ), γ i.e., it can be lower bounded by the optimal solution of the optimization problem
- 1) Lower bound: For a particular task assignment policy Ui,K′ satisfying (19) and (20), and a particular subset of q ENs denoted as Kq ⊆ K whose outputs are available, each EN k ∈ Kq is assigned ri,kN input vectors from each user i ∈ M and can store μm rows of A.

Conclusion

- For a particular computation results distribution {Si,k}i∈M,k∈Kq, the authors adopt the arguments proved in [11, Lemma 6] to derive the lower bound of the NDLT, i.e., intuitively, given any subset of t signals received at t ≤ min{q, M}
- Ausse{rSs,i,dke}ni∈oMte,dk∈aQsq−{tY, ia}lil∈M trat,nasmnditttehde stored computation results information of q −t ENs, denoted signals {Xk}k∈K and all the desired outputs {vi,j}i∈M,j∈[N]

Summary

- Mobile edge computing (MEC) is an emerging network architecture that enables cloudcomputing capabilities at the edge nodes (ENs) of mobile networks [1].
- For a given NULT τ u, the compute-download latency region is defined as the union of the set for all NCT-NDLT pairs (τ c, τ d), i.e., T ∗(τ u) (τ c, τ d) : (τ u(r), τ c(r, q), τ d(r, q)) is achievable for some (r, q) and τ u ≤ τ u(r), τ c ≥ τ c(r, q), and τ d ≥ τ d(r, q) .
- (Convexity of compute-download latency region.) For an input data assignment policy {Ui,K′ }i∈M,K′⊆K with a repetition order r, fix an input uploading strategy achieving NULT τ u.
- 3) Edge Computing: Following Sec. III-A, the cascaded MDS-repetition code rate pair satisfies ρ1∈{1, Kμ/(Kμ−1), Kμ/(Kμ−2), · · · , Kμ} and ρ2=Kμ/ρ1 under the constraint of the total storage size kμ.
- 4) Output Downloading: Following Sec. III-A, for each user, the number of input vectors computed by p1 non-straggling ENs equals
- Similar to the calculation of NULT in Sec. IV-2, by Definition 2, the NDLT for each user to download the outputs replicated at p2 non-stragglers is given by τd p1,p2
- 5) Inner Bound of Compute-Download Latency Region: For an NULT τ u = τau(r) given in (7) for some r ∈ R, where R is given by (10), the feasible values of recovery order q’s should satisfy
- In the rest of this appendix, the authors first derive the lower bounds on the NULT, NCT, and NDLT for a particular task assignment policy Ui,K′ with repetition-recovery order (r, q) ∈ R.
- The minimum NULT over all feasible task assignment is given as τ u∗(r) = min τ u∗(r, γ), γ i.e., it can be lower bounded by the optimal solution of the optimization problem
- 1) Lower bound: For a particular task assignment policy Ui,K′ satisfying (19) and (20), and a particular subset of q ENs denoted as Kq ⊆ K whose outputs are available, each EN k ∈ Kq is assigned ri,kN input vectors from each user i ∈ M and can store μm rows of A.
- For a particular computation results distribution {Si,k}i∈M,k∈Kq, the authors adopt the arguments proved in [11, Lemma 6] to derive the lower bound of the NDLT, i.e., intuitively, given any subset of t signals received at t ≤ min{q, M}
- Ausse{rSs,i,dke}ni∈oMte,dk∈aQsq−{tY, ia}lil∈M trat,nasmnditttehde stored computation results information of q −t ENs, denoted signals {Xk}k∈K and all the desired outputs {vi,j}i∈M,j∈[N]

Funding

- Tao is supported by the National Natural Science Foundation of China under Grant 61941106 and the National Key R&D Project of China under Grant 2019YFB1802702
- Simeone is funded by the European Research Council under the European Union’s Horizon 2020 Research and Innovation Programme (Grant Agreement No 725731)

Reference

- ETSI, “Mobile edge computing: A key technology towards 5G,” White Paper, Sep. 2015. [Online]. Available: http://www.etsi.org/images/files/ETSIWhitePapers/etsi wp11 mec a key technology towards 5g.pdf
- R. Bekkerman, M. Bilenko, and J. Langford, Scaling up machine learning: Parallel and distributed approaches. Cambridge University Press, 2011.
- S. Li, M. A. Maddah-Ali, Q. Yu, and A. S. Avestimehr, “A fundamental tradeoff between computation and communication in distributed computing,” IEEE Trans. Inf. Theory, vol. 64, no. 1, pp. 109–128, 2017.
- K. Lee, M. Lam, R. Pedarsani, D. Papailiopoulos, and K. Ramchandran, “Speeding up distributed machine learning using codes,” IEEE Trans. Inf. Theory, vol. 64, no. 3, pp. 1514–1529, 2017.
- S. Li, M. A. Maddah-Ali, and A. S. Avestimehr, “A unified coding framework for distributed computing with straggling servers,” in Proc. IEEE Globecom Workshops (GC Wkshps), Dec. 2016, pp. 1–6.
- J. Zhang and O. Simeone, “On model coding for distributed inference and transmission in mobile edge computing systems,” IEEE Commun. Letters, vol. 23, no. 6, pp. 1065–1068, Jun. 2019.
- S. Kiani, N. Ferdinand, and S. C. Draper, “Exploitation of stragglers in coded computation,” in Proc. IEEE Int. Symp. Inf. Theory (ISIT), Jun. 2018, pp. 1988–1992.
- S. Dutta, V. Cadambe, and P. Grover, “Coded convolution for parallel and distributed computing within a deadline,” in Proc. IEEE Int. Symp. Inf. Theory (ISIT), Jun. 2017, pp. 2403–2407.
- J. Zhang and O. Simeone, “Improved latency-communication trade-off for map-shuffle-reduce systems with stragglers,” in Proc. IEEE Int. Conf. Acoustics, Speech and Signal Process. (ICASSP), May 2019, pp. 8172–8176.
- N. Naderializadeh, M. A. Maddah-Ali, and A. S. Avestimehr, “Fundamental limits of cache-aided interference management,” IEEE Trans. Inf. Theory, vol. 63, no. 5, pp. 3092–3107, May 2017.
- A. Sengupta, R. Tandon, and O. Simeone, “Fog-aided wireless networks for content delivery: Fundamental latency tradeoffs,” IEEE Trans. Inf. Theory, vol. 63, no. 10, pp. 6650–6678, 2017.
- F. Xu, M. Tao, and K. Liu, “Fundamental tradeoff between storage and latency in cache-aided wireless interference networks,” IEEE Trans. Inf. Theory, vol. 63, no. 11, pp. 7464–7491, Nov. 2017.
- M. Tao, D. Gndz, F. Xu, and J. S. P. Roig, “Content caching and delivery in wireless radio access networks,” IEEE Trans. Commun., vol. 67, no. 7, pp. 4724–4749, Jul. 2019.
- J. Dean and L. A. Barroso, “The tail at scale,” Communications of the ACM, vol. 56, no. 2, pp. 74–80, 2013.
- S. Li, M. A. Maddah-Ali, and A. S. Avestimehr, “Communication-aware computing for edge processing,” in Proc. IEEE Int. Symp. Inf. Theory (ISIT), Jun. 2017, pp. 2885–2889.
- K. Li, M. Tao, and Z. Chen, “Exploiting computation replication for mobile edge computing: A fundamental computationcommunication tradeoff study,” IEEE Trans. Wireless Commun., pp. 1–1, 2020.
- F. Li, J. Chen, and Z. Wang, “Wireless MapReduce distributed computing,” IEEE Trans. Inf. Theory, vol. 65, no. 10, pp. 6101–6114, 2019.
- J. Hachem, U. Niesen, and S. N. Diggavi, “Degrees of freedom of cache-aided wireless interference networks,” IEEE Trans. Inf. Theory, vol. 64, no. 7, pp. 5359–5380, 2018.
- B. C. Arnold, N. Balakrishnan, and H. Nagaraja, A First Course in Order Statistics. SIAM, 2008, vol. 54.
- K. Li, M. Tao, and Z. Chen, “Exploiting computation replication in multi-user multi-server mobile edge computing networks,” in Proc. IEEE Global Commun. Conf. (GLOBECOM), Dec. 2018, pp. 1–7.

Full Text

Tags

Comments