Dual-Entangled Polynomial Code: Three-Dimensional Coding for - - PowerPoint PPT Presentation

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Dual-Entangled Polynomial Code: Three-Dimensional Coding for - - PowerPoint PPT Presentation

Jan 20, 2023 427 likes •674 views

Dual-Entangled Polynomial Code: Three-Dimensional Coding for Distributed Matrix Multiplication Pedro Soto, Jun Li, Xiaodi Fan Florida International University Matrix Multiplication Matrix multiplication is a fundamental building block in

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SLIDE 1

Dual-Entangled Polynomial Code: Three-Dimensional Coding for Distributed Matrix Multiplication

Pedro Soto, Jun Li, Xiaodi Fan Florida International University

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SLIDE 2

Matrix Multiplication

  • Matrix multiplication is a fundamental building block in

various machine learning algorithms.

  • When the matrix comes from a large dataset, the

multiplication will be split into smaller multiplications of submatrices on different nodes.

2

slide-3
SLIDE 3

Matrix Multiplication

  • Matrix multiplication is a fundamental building block in

various machine learning algorithms.

  • When the matrix comes from a large dataset, the

multiplication will be split into smaller multiplications of submatrices on different nodes.

2

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slide-4
SLIDE 4

Matrix Multiplication

  • Matrix multiplication is a fundamental building block in

various machine learning algorithms.

  • When the matrix comes from a large dataset, the

multiplication will be split into smaller multiplications of submatrices on different nodes.

2

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slide-5
SLIDE 5

Matrix Multiplication

  • Matrix multiplication is a fundamental building block in

various machine learning algorithms.

  • When the matrix comes from a large dataset, the

multiplication will be split into smaller multiplications of submatrices on different nodes.

2

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worker worker

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slide-6
SLIDE 6

Matrix Multiplication

  • Matrix multiplication is a fundamental building block in

various machine learning algorithms.

  • When the matrix comes from a large dataset, the

multiplication will be split into smaller multiplications of submatrices on different nodes.

2

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worker worker master

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slide-7
SLIDE 7

Straggler

  • Stragglers are common in distributed systems, due to

load imbalance, resource contention, etc.

3

slide-8
SLIDE 8

Straggler

  • Stragglers are common in distributed systems, due to

load imbalance, resource contention, etc.

3

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worker worker master

  • <latexit sha1_base64="jvZ1pEC6o51a+h06RGOKnBfQifU=">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</latexit>
<latexit sha1_base64="t2WTPjNM0fu4hYh83muedb+5xhY=">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</latexit>
slide-9
SLIDE 9

Straggler

  • Stragglers are common in distributed systems, due to

load imbalance, resource contention, etc.

3

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worker worker master

  • <latexit sha1_base64="jvZ1pEC6o51a+h06RGOKnBfQifU=">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</latexit>
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worker worker

<latexit sha1_base64="t2WTPjNM0fu4hYh83muedb+5xhY=">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</latexit>
slide-10
SLIDE 10

Straggler

  • Stragglers are common in distributed systems, due to

load imbalance, resource contention, etc.

3

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worker worker master

  • <latexit sha1_base64="jvZ1pEC6o51a+h06RGOKnBfQifU=">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</latexit>
<latexit sha1_base64="t2WTPjNM0fu4hYh83muedb+5xhY=">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</latexit>

worker

<latexit sha1_base64="lTYhtUX6dlVFUJOfti1wRXAH0dc=">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</latexit>
slide-11
SLIDE 11

Straggler

  • Stragglers are common in distributed systems, due to

load imbalance, resource contention, etc.

3

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worker worker master

  • <latexit sha1_base64="jvZ1pEC6o51a+h06RGOKnBfQifU=">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</latexit>
<latexit sha1_base64="t2WTPjNM0fu4hYh83muedb+5xhY=">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</latexit>

worker

  • Coded matrix multiplication can tolerate stragglers with

fewer tasks.

<latexit sha1_base64="lTYhtUX6dlVFUJOfti1wRXAH0dc=">AD2niclVNLbxMxEHazPEp4pXDkYhEhFdFEmxYJDlTq49IDoICaNlJ2FTne2cSq17s2cBqlQs3xJUjV/gB/Bv+Dd7tHkhSDoxkaWa+eXn8eZJKYdH3f280vGvXb9zcvNW8fefuvfutrQdnVmeGw4Brqc1wixIoWCAiUMUwMsmUg4n1wcl/j5HIwVWp1inkKYsKkSseAMnWvcam0fjv1nh+Pe04BHGunRuNX2u34ldF3p1Uqb1NIfbzV+BZHmWQIKuWTWjnp+ijtUMhVZzlzLghkUXMKiGWQWUsYv2BRGbC91oAmLKegE0OTLsFMVS8CGRXJBX3iPBGNtXFHIa28f2cULE2TyYuMmE4s6tY6bwKG2UYvwLodIMQfHLRnEmKWpaboxGwgBHmTuFcSPcVSifMcM4ur02g/fwIXMR/bqaMwDopzRpsAXawHlAB2LuYT9UxiGRaYE1xF0qvmazUDBxzI5ZomQeTDRUV6ao7A4ATkH153Rt5ABfS2mM1ysxuNMqP+JnwGLhJpenXKsVQSqXPuRlE5XAQxyRWBYBhZsAWb1iauhL7CJ867pQ9LGDC6kFW4R1asbK0sXrEf0269IDlV3BV7BJamosv7ScphYyx0i31BXI5JGdSeVKO5b3Vjm9rpztdnt73d13z9sHr2q+b5JH5DHZJj3yghyQE9InA8LJnHwnP8hPL/A+e1+8r5ehjY065yFZEu/bHyuOS5E=</latexit>
slide-12
SLIDE 12

coding matrix partition

recovery threshold

1D

[Lee et al.,

  • Trans. IT, 2018]

x 2D

[Yu et al., NIPS 2017]

xy 3D

[Yu et al., ISIT 2018]

xyz+z-1

Related Works

4

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  • <latexit sha1_base64="6VAgY3mgnbHF0XhL3bUmnvjqD4=">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</latexit>
  • <latexit sha1_base64="ZLfHv2tX0jGIXh8m5NyN+NL3a94=">AEOXiclVNb9QwEPV2WyjLVwtHLhYViEO32m2R4FKptJceAC2o21baRCvHmc1adexgT0qjKH+K7+DA0duiCtHLthphNi0HLAU6Xne+OxZxJlUlgcDL52lrLKzdurt7q3b5z979tfUHx1bnhsOYa6nNacQsSKFgjAIlnGYGWBpJOInODjx/cg7GCq2OsMgTFmixExwhi40XdNBIlQZQyNOKiejUtB1UQ0OA81mgdcIGL/rAKQMV/RAF3JO21rPveSp/SmrUOuEDRtk7XNgZbg3rRq2DYgA3SrNF0felTEGuep6CQS2btZDjIcJNKpmLmbtRyQwKLqHqBbmFjPEzlsCE7WSONGZgE4BTbFIO6hYCjYs6zes6BMXielMG/cpHX0b0fJUmuLNHJKd5W5bXM+eB03yXH2MiyFynIExS8PmuWSoqa+ITQWBjKwgHGjXBXoXzODOPo2tYL3sOH3ClGTa3AQxLX6PNgFdXBL6AvsVCwu4RnIZlrgTXMfTr+nq9QMFHb56xVMgiHRc+O0kLA9BnoM7ndG3kAN9LZI5Vm09zoX6H/0cWCxUcr3lQKsYlH/2fS1jX1wM5ZLrBMAw9yALd+wLHMpdhEu+u7zZ1jAlDWFtOlNWg+932PdxH9VutBA/6e5LHZhSJrRWOy0TDILuZtI96gtyhSxnUvlUrspH7Zn+io43t4a7mxtv3u+sbfXzPsqeUQek2dkSF6QPXJIRmRMOPlCfnWOyvdz91v3e/dH5fSpU7jeUgWVvfnb+EWcow=</latexit>
slide-13
SLIDE 13

Entangled Polynomial Code

  • Entangled Polynomial (EP) code [Yu et al., ISIT 2018] is the

state-of-the-art three-dimensional coding.

  • For example, if and , i.e.,

x=y=z=2, a task will be:

5

  • <latexit sha1_base64="zplapaOcuLfTEOc8i6HhxaMPVo=">AED3iclVPLbtQwFHU7PMrwamHJxmIKYjGtknYBm5H6EFIXgArqY6RJNHKcOxmrjhPsm9IoykfwHezYsENsWbIEfgY7jSpmWhZYsnR8Xz6+9zjKpTDoeT8XFjvXrt+4uXSre/vO3Xv3l1ceHJms0BwOeSYzPYyYASkUHKJACcNcA0sjCcfRya7zH5+CNiJTB1jmEKYsUWIiOENrGi+/XN0eBEkQlVRylCLs3p7XHl9r6ZPaYP8Ogc8i9svrOBi8yVsfLPW/daxa9DPwW9Ei79scri5+DONFCgq5ZMaMfC/HPpVMxYzy7NiGgWXUHeDwkDO+AlLYMQ2c+vUYZVAlgLqctZtoWIpmLBqOlPTJ9YS0m7VZIG+vfGRVLjSnTyEbat0zNvM8Zr/KNCpy8Cuh8gJB8fOLJoWkmFHXZhoLDRxlaQHjWtinUD5lmnG0w+gG7+B9YSP2r2ABhWjqPJgdeXAhyBNYOlhMEBDMOqUIJnMaw1/LrdQMEHlzxhqZBlEGVx6Y6jsNoDeQr2dkbfQAH0lUimWM/H41So/4mfAouFSq5O2c1UDMq1fSeTsSMXw4QVEpsCwLDQYKrXLM9tiQHC2Zrd7g4DmLKWyLy7TxspuzM2Q/wX05kBuv9jq5gZkbTSmJ20THIDhVWkbeqcS5exmUplS1uV+/OavgyONtb9zfWNtxu9rZ1W70vkEXlMnhGfPCdbZI/sk0PCySfyg/wivzsfO186XzvfzkMXF9qch2Rmdb7/AUubX/I=</latexit>
  • <latexit sha1_base64="ALuQFau4pkI7Wg92fMCvZg8VTg=">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</latexit>
<latexit sha1_base64="9Ht+Sz2s4QVEGo8Neig7CP5q4oY=">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</latexit>
slide-14
SLIDE 14

Entangled Polynomial Code

  • Entangled Polynomial (EP) code [Yu et al., ISIT 2018] is the

state-of-the-art three-dimensional coding.

  • For example, if and , i.e.,

x=y=z=2, a task will be:

5

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  • <latexit sha1_base64="ALuQFau4pkI7Wg92fMCvZg8VTg=">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</latexit>
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slide-15
SLIDE 15

Entangled Polynomial Code

  • Entangled Polynomial (EP) code [Yu et al., ISIT 2018] is the

state-of-the-art three-dimensional coding.

  • For example, if and , i.e.,

x=y=z=2, a task will be:

5

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  • <latexit sha1_base64="ALuQFau4pkI7Wg92fMCvZg8VTg=">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</latexit>
  • <latexit sha1_base64="NpM6f5OmzMzyR5mwCKNIXAYic=">AGxHiclZRb9MwFMezFcotw0eYlYQUO7KN7KtpdJWyehPQAaDdpKcVxTltrjhNsZ1tkhUde4bPwbfgovOFkWde03RiWohz7HP/98/GxvYhRqRzn98Rk5c7d6r2p+7UHDx89fjI98/RAhrEgsE9CFojD0tglMO+orBUSQABx6DQ+9kO/MfnoKQNOR7KomgFeAupx1KsDJD7ZnJP64HXcq1wl7MsEg1IStub1MsFZ3FWU+6K207RmMzN9YAp/dur2K7vungva7SksRHim03xoq62dBSdtjUyv8voeZdBR825Cs5VDq4F+Km+Cs5+83kXc5N3Vz8daGBtzOlfRy3XtG5FRP3I8cpkZ9YVvwYzKzGiYeaXE3JdujGdGo2lu3MSMRvLcuJ4ZlfOMrs3zmwH3QJ5X/8mM+pG3Ye4nY/UWzKjMPJLntRJzX3rdMF+Wswvc7xd6rT096yw5ebNHDVQYs1bRds1l+eX6IYkD4IowLOUxciK1YDPMfUmwuVwaC0UJA3OBYgkRJie4C8d4JTJO0dJdCANQIim7jclxALKl852n9ksz4tudUJiPKzsfHZyhcSBlEngmMsCqJ4d92eA43GsOustTXkUK+DkYqFOzGwV2tnbYPtUAFEsMQYmgpqt2KSHBSbKvCA19xN8jU3EbqFmOqBaOmOUEWSPxlBABrAoVcJgYw+OWjrmlIQ+LOZ8tZrL4Syb3MEBZYnrhX6SdY9begfYKZjVsf0BYrDfZcecDserHuX/E98D7FPeHT9lO+Q+8CztzZD5GZwPHRwzlQsAVrEAqd/jKDISG6ZKF7NKNWtIUAEuQIbdC3b+/m70i/o60tIBZo+UZGlIilKo3zSrBtJiE1FmqQOuUTiyx7jRtpUORqu6VHjYHkJrSwtf2zMbjaLep+ynlsvrDkLWvWprVj7Vr7Fql8qXyv/Kj8rL6tsqsxhehkxPFnGdWqVW/QVqfE0J</latexit>
  • <latexit sha1_base64="5pVqGWX6BC/ympYAIw9sv9NhoXc=">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</latexit>
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slide-16
SLIDE 16

Motivation

  • As the input matrices are divided smaller and smaller, the

number of results uploaded and decoded at the master also increase.

  • the master’s incoming traffic becomes congested, or
  • the master is overwhelmed by the decoding complexity
  • Dual entangled polynomial codes allows a tradeoff

between computation and communication/decoding

  • verhead.

6

slide-17
SLIDE 17

Dual Entangled Polynomial Code

  • Dual Entangled Polynomial code (DEP) doubles the

computational complexity with two multiplications, lowering the recovery threshold to . For example, when x=y=z=2, a task will be

7

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slide-18
SLIDE 18

Dual Entangled Polynomial Code

  • Dual Entangled Polynomial code (DEP) doubles the

computational complexity with two multiplications, lowering the recovery threshold to . For example, when x=y=z=2, a task will be

7

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slide-19
SLIDE 19

Dual Entangled Polynomial Code

  • Dual Entangled Polynomial code (DEP) doubles the

computational complexity with two multiplications, lowering the recovery threshold to . For example, when x=y=z=2, a task will be

7

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  • +

noise + + noise +

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slide-20
SLIDE 20

Dual Entangled Polynomial Code

  • Dual Entangled Polynomial code (DEP) doubles the

computational complexity with two multiplications, lowering the recovery threshold to . For example, when x=y=z=2, a task will be

7

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  • +

noise + + noise +

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Compared to EP codes, the recovery threshold is reduced from 9 tasks to 6 tasks.

slide-21
SLIDE 21

Evaluation

  • We implemented DEP codes with Open MPI, and ran the evaluation on

24 workers hosted on Microsoft Azure.

  • EP X2 runs two tasks on each worker.
  • DEP tolerates similar stragglers, while significantly saving job

completion time.

8

(2,2,2) (2,3,2) (2,2,4)

(x,y,z) 10 20 30 (D) # strDgglers tolerDEle (3 (3 X2 D(3

(2,2,2) (2,3,2) (2,2,4)

(x,y,z) 1 2 (E) joE coPpletion tiPe (sec.) (3 (3 X2 D(3

(2,2,2) (2,3,2) (2,2,4)

(x,y,z) 0.00 0.25 0.50 0.75 1.00 (c) decoding tiPe (sec.) (3 (3 X2 D(3

slide-22
SLIDE 22

Conclusion

  • We propose the dual entangled polynomial code, another

three-dimensional coding scheme, for distributed matrix multiplication.

  • The extra computation at each node allows DEP codes to

have a significantly lower recovery threshold than EP codes, leading to a lower communication overhead and decoding complexity.

  • Future work: explore more flexible tradeoff between

computation and communication/decoding overhead.

9

slide-23
SLIDE 23

Thank you.

10