adasum: adaptive summation of gradients for deep learning

Adasum: Adaptive Summation of Gradients for Deep Learning

Saeed Maleki, Madan Musuvathi, Todd Mytkowicz, Olli Saarikivi

Microsoft Research

Emad Barsoum, Sergii Dymchenko, Jaliya Ekanayake, Vadim Eksarevskiy, Maxim Lukiyanov, Tianju Xu

Microsoft

TH I S I S AN E L E PHANT R EA L L Y !DATA TH I S I S AN E L E PHANT R EA L L Y !

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Motivation: Neural Network Training is Inherently Sequential• Stochastic Gradient Descent (SGD)

• Workhorse for training a neural network

• Inherently sequential

TH I S I D AN E L A PHANT T EA L L Y A

do not achieve the same“per-epoch accuracy”

Motivation: More parallelism = less accuracy

Adasum addresses this

Background: SGD and Gradients

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=+

𝐷1

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𝐿𝑜𝑠𝑠

𝑤1

𝐷2𝐷1

𝑤𝑥

𝐿𝑜𝑠𝑠

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𝐿𝑖⋯

𝑤0 𝑤1

𝑤2

𝐷2𝐷1

𝑤0

𝑤3

𝑤0+Δ𝑤

𝑤1

𝑤0

𝑤2

𝑤3

𝑆𝐺𝐷 𝑤0 + Δ𝑤 = 𝑆𝐺𝐷 𝑤0 + +𝑂( Δ𝑤 2)𝑆𝐺𝐷′ 𝑤0 ⋅ Δ𝑤

𝜟𝒘

local model 𝑤3

+ 𝑓(Δ𝑤)+𝑀 ⋅ Δ𝑤

• 𝑀 = 𝐼 − 𝛼𝐻 where 𝐻 is the Hessian matrix and 𝛼 is the learning rate

• Very costly to calculate 𝐻

a matrix 𝑀

Adasum: Symbolic SGD

+ 𝐼 − 𝛼𝐻 ⋅ Δ𝑤+ Δ𝑤 − 𝛼𝐻 ⋅ Δ𝑤

𝑤1

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𝐿𝑜𝑠𝑠

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𝐿𝑖⋯

𝑤0 𝑤1

𝑤2

𝐷2𝐷1

𝑤0

𝑤3

𝑤0+Δ𝑤

𝑤1

𝑤0

𝑤2

𝑤3𝜟𝒘

Adasum: Symbolic SGD

+ Δ𝑤 − 𝛼𝐻 ⋅ Δ𝑤

𝜟𝒘 𝑤0

𝑤3

usually ignored

𝑤0

𝑤3𝒘𝟑 + 𝜟𝒘

ok to ignore

𝒘𝟑 + 𝜟𝒘not ok to ignore

𝜟𝒘

𝜟𝒘′

𝒘𝟑 + 𝜟𝒘′

now ok to ignore

orthogonal space

𝐷2

gradient gradient

𝐷2𝐷1Δ𝑤′ is a “good” direction

𝑤1

𝑤0

𝑤2

𝑤3𝜟𝒘

𝑤0

𝑤1gradient𝜟𝒘

𝜟𝒘′

• Is Δ𝑤′ a “good” direction to move along?

• Δ𝑤 is the gradient w.r.t. the green bowl

• Any direction that has a positiveinner product with the gradientsdecays the loss• Δ𝑤′ is a “good” direction

• Adasum operator sums Δ𝑤 with projection

orthogonal space tothe orange gradient

Adasum: Adaptive Sum

• Adasum combines Δ𝑤 from any number of processors

• Adasum combines Δ𝑤 from different processors by projection and summation. Effectively:• They are added when they are orthogonal• Only one is taken when they are parallel

• Traditionally, Δ𝑤 from different processors are:• Either summed: can be too aggressive• Or Averaged: can be too conservative

Orthogonality of Gradients

• We use Pythagorean theorem to define orthogonality

• For P gradients it ranges between:• 1 for all orthogonal

• 1/P for all parallel

• Gradients start out all parallel

• Later in the training they becomemore orthogonal

• Convergence starts out slowbut speeds up later

BERT with 64 GPUs

Adasum is in Horovod

• Horovod is an open-source distributed training framework by Uber that supports both PyTorch and TensorFlow• Adasum is integrated in Horovod

• Adasum is easy to use:• horovod.allreduce(gradients, op=hvd.adasum)

• No hyperparameter

• Adasum allows scaling SGD• Minimizes convergence slowdown in scale

Result – Adasum vs. Allreduce/Averaging Latency on 64 GPUs• Adasum has some

computation overhead• But no communication overhead

• Almost negligible overhead• Communication latency

dominates the computationlatency

64 GPUs

Results – Adasum Convergence on MNIST

• Standard 2-layer CNN gets 99.3% in 2 epochs sequentially with batch size 32

• Tuned LR for averaging and Adasum

• Default Horovod fails at 16 GPUs, while Adasum still works with 32

• With tuned learning rates Adasumhas much better convergence for 16 and 32 GPUs

97.8%

97.9%

98.0%

98.1%

98.2%

98.3%

98.4%

98.5%

98.6%

98.7%

98.8%

98.9%

99.0%

99.1%

99.2%

99.3%

4 GPUs 8 GPUs 16 GPUs 32 GPUs

Test Accuracy at 2 epochs

Adasum (tuned LR) Adasum

Horovod (tuned LR) Default Horovod

Results – Adasum Convergence on word2vec

• word2vec model: embedding for words• London to England::Paris to France

• Runs on 32 nodes

• Tuned LR for averaging

• Adasum matches sequential accuracy

AS

Results – Adasum Convergence on Resnet

• The MLPerf Resnet50 on ImageNet • 16K batch size• 64 GPUs

• Adasum reaches MLPerftarget accuracy 74.9% in 69 epochs

• Default Horovod never reaches target accuracy in 90 epochs

Epochs

Validation Accuracy — Adasum — Default Horovod

Results – Adasum Convergence on BERT

• BERT is a common modeltrained nowadays

• Bigger batch size = more parallelism• 64k and 128k

• LAMB is the optimizerused at large scales

• Adasum beats LAMBwith 128k batchsize

Please use Adasum!

• And let us know what you think!

• https://github.com/horovod/