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A Terascale Learning Algorithm
Alekh Agarwal, Olivier Chapelle, Miroslav Dudik, and JohnLangford
... And the Vowpal Wabbit project.
Applying for a fellowship in 1997
Interviewer: So, what do you want to do?John: I’d like to solve AI.I: How?J: I want to use parallel learning algorithms to create fantasticlearning machines!I: You fool! The only thing parallel machines are good for iscomputational windtunnels!The worst part: he had a point. At that time, smarter learningalgorithms always won. To win, we must master the bestsingle-machine learning algorithms, then clearly beat them with aparallel approach.
Applying for a fellowship in 1997
Interviewer: So, what do you want to do?
John: I’d like to solve AI.I: How?J: I want to use parallel learning algorithms to create fantasticlearning machines!I: You fool! The only thing parallel machines are good for iscomputational windtunnels!The worst part: he had a point. At that time, smarter learningalgorithms always won. To win, we must master the bestsingle-machine learning algorithms, then clearly beat them with aparallel approach.
Applying for a fellowship in 1997
Interviewer: So, what do you want to do?John: I’d like to solve AI.
I: How?J: I want to use parallel learning algorithms to create fantasticlearning machines!I: You fool! The only thing parallel machines are good for iscomputational windtunnels!The worst part: he had a point. At that time, smarter learningalgorithms always won. To win, we must master the bestsingle-machine learning algorithms, then clearly beat them with aparallel approach.
Applying for a fellowship in 1997
Interviewer: So, what do you want to do?John: I’d like to solve AI.I: How?
J: I want to use parallel learning algorithms to create fantasticlearning machines!I: You fool! The only thing parallel machines are good for iscomputational windtunnels!The worst part: he had a point. At that time, smarter learningalgorithms always won. To win, we must master the bestsingle-machine learning algorithms, then clearly beat them with aparallel approach.
Applying for a fellowship in 1997
Interviewer: So, what do you want to do?John: I’d like to solve AI.I: How?J: I want to use parallel learning algorithms to create fantasticlearning machines!
I: You fool! The only thing parallel machines are good for iscomputational windtunnels!The worst part: he had a point. At that time, smarter learningalgorithms always won. To win, we must master the bestsingle-machine learning algorithms, then clearly beat them with aparallel approach.
Applying for a fellowship in 1997
Interviewer: So, what do you want to do?John: I’d like to solve AI.I: How?J: I want to use parallel learning algorithms to create fantasticlearning machines!I: You fool! The only thing parallel machines are good for iscomputational windtunnels!
The worst part: he had a point. At that time, smarter learningalgorithms always won. To win, we must master the bestsingle-machine learning algorithms, then clearly beat them with aparallel approach.
Applying for a fellowship in 1997
Interviewer: So, what do you want to do?John: I’d like to solve AI.I: How?J: I want to use parallel learning algorithms to create fantasticlearning machines!I: You fool! The only thing parallel machines are good for iscomputational windtunnels!The worst part: he had a point. At that time, smarter learningalgorithms always won. To win, we must master the bestsingle-machine learning algorithms, then clearly beat them with aparallel approach.
Demonstration
Terascale Linear Learning ACDL11
Given 2.1 Terafeatures of data, how can you learn a good linearpredictor fw (x) =
∑i wixi?
2.1T sparse features17B Examples16M parameters1K nodes
70 minutes = 500M features/second: faster than the IObandwidth of a single machine⇒ we beat all possible singlemachine linear learning algorithms.
(Actually, we can do even better now.)
Terascale Linear Learning ACDL11
Given 2.1 Terafeatures of data, how can you learn a good linearpredictor fw (x) =
∑i wixi?
2.1T sparse features17B Examples16M parameters1K nodes
70 minutes = 500M features/second: faster than the IObandwidth of a single machine⇒ we beat all possible singlemachine linear learning algorithms.
(Actually, we can do even better now.)
Terascale Linear Learning ACDL11
Given 2.1 Terafeatures of data, how can you learn a good linearpredictor fw (x) =
∑i wixi?
2.1T sparse features17B Examples16M parameters1K nodes
70 minutes = 500M features/second: faster than the IObandwidth of a single machine⇒ we beat all possible singlemachine linear learning algorithms.
(Actually, we can do even better now.)
Terascale Linear Learning ACDL11
Given 2.1 Terafeatures of data, how can you learn a good linearpredictor fw (x) =
∑i wixi?
2.1T sparse features17B Examples16M parameters1K nodes
70 minutes = 500M features/second: faster than the IObandwidth of a single machine⇒ we beat all possible singlemachine linear learning algorithms.
(Actually, we can do even better now.)
Compare: Other Supervised Algorithms in Parallel Learningbook
100 1000
10000 100000 1e+06 1e+07 1e+08 1e+09
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eads
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oop+
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Fea
ture
s/s
Speed per method
parallelsingle
The tricks we use
First Vowpal Wabbit Newer Algorithmics Parallel Stuff
Feature Caching Adaptive Learning Parameter Averaging
Feature Hashing Importance Updates Smart Averaging
Online Learning Dimensional Correction Gradient Summing
L-BFGS Hadoop AllReduce
Hybrid LearningWe’ll discuss Hashing, AllReduce, then how to learn.
RA
M
Conventional VW
Weights
Str
ing −
> I
ndex
dic
tionar
y
Weights
Most algorithms use a hashmap to change a word into an index fora weight.VW uses a hash function which takes almost no RAM, is x10faster, and is easily parallelized.Empirically, radical state compression tricks possible with this.
MPI-style AllReduce
2 3 4
6
7
Allreduce initial state
5
1
AllReduce = Reduce+Broadcast
Properties:
1 Easily pipelined so no latency concerns.
2 Bandwidth ≤ 6n.
3 No need to rewrite code!
MPI-style AllReduce
2 3 4
7
8
1
13
Reducing, step 1
AllReduce = Reduce+Broadcast
Properties:
1 Easily pipelined so no latency concerns.
2 Bandwidth ≤ 6n.
3 No need to rewrite code!
MPI-style AllReduce
2 3 4
8
1
13
Reducing, step 2
28
AllReduce = Reduce+Broadcast
Properties:
1 Easily pipelined so no latency concerns.
2 Bandwidth ≤ 6n.
3 No need to rewrite code!
MPI-style AllReduce
2 3 41
28
Broadcast, step 1
28 28
AllReduce = Reduce+Broadcast
Properties:
1 Easily pipelined so no latency concerns.
2 Bandwidth ≤ 6n.
3 No need to rewrite code!
MPI-style AllReduce
28
28 28
Allreduce final state
28 28 28 28
AllReduce = Reduce+Broadcast
Properties:
1 Easily pipelined so no latency concerns.
2 Bandwidth ≤ 6n.
3 No need to rewrite code!
MPI-style AllReduce
28
28 28
Allreduce final state
28 28 28 28
AllReduce = Reduce+BroadcastProperties:
1 Easily pipelined so no latency concerns.
2 Bandwidth ≤ 6n.
3 No need to rewrite code!
An Example Algorithm: Weight averaging
n = AllReduce(1)While (pass number < max)
1 While (examples left)1 Do online update.
2 AllReduce(weights)
3 For each weight w ← w/n
Other algorithms implemented:
1 Nonuniform averaging for online learning
2 Conjugate Gradient
3 LBFGS
An Example Algorithm: Weight averaging
n = AllReduce(1)While (pass number < max)
1 While (examples left)1 Do online update.
2 AllReduce(weights)
3 For each weight w ← w/n
Other algorithms implemented:
1 Nonuniform averaging for online learning
2 Conjugate Gradient
3 LBFGS
What is Hadoop-Compatible AllReduce?
1
DataProgram
“Map” job moves program to data.
2 Delayed initialization: Most failures are disk failures. Firstread (and cache) all data, before initializing allreduce. Failuresautorestart on different node with identical data.
3 Speculative execution: In a busy cluster, one node is oftenslow. Hadoop can speculatively start additional mappers. Weuse the first to finish reading all data once.
The net effect: Reliable execution out to perhaps 10K node-hours.
What is Hadoop-Compatible AllReduce?
1
DataProgram
“Map” job moves program to data.
2 Delayed initialization: Most failures are disk failures. Firstread (and cache) all data, before initializing allreduce. Failuresautorestart on different node with identical data.
3 Speculative execution: In a busy cluster, one node is oftenslow. Hadoop can speculatively start additional mappers. Weuse the first to finish reading all data once.
The net effect: Reliable execution out to perhaps 10K node-hours.
What is Hadoop-Compatible AllReduce?
1
DataProgram
“Map” job moves program to data.
2 Delayed initialization: Most failures are disk failures. Firstread (and cache) all data, before initializing allreduce. Failuresautorestart on different node with identical data.
3 Speculative execution: In a busy cluster, one node is oftenslow. Hadoop can speculatively start additional mappers. Weuse the first to finish reading all data once.
The net effect: Reliable execution out to perhaps 10K node-hours.
What is Hadoop-Compatible AllReduce?
1
DataProgram
“Map” job moves program to data.
2 Delayed initialization: Most failures are disk failures. Firstread (and cache) all data, before initializing allreduce. Failuresautorestart on different node with identical data.
3 Speculative execution: In a busy cluster, one node is oftenslow. Hadoop can speculatively start additional mappers. Weuse the first to finish reading all data once.
The net effect: Reliable execution out to perhaps 10K node-hours.
Algorithms: Preliminaries
Optimize so few data passes required ⇒ Smart algorithms.Basic problem with gradient descent = confused units.fw (x) =
∑i wixi
⇒ ∂(fw (x)−y)2
∂wi= 2(fw (x)− y)xi which has units of i .
But wi naturally has units of 1/i since doubling xi implies halvingwi to get the same prediction.Crude fixes:
1 Newton: Multiply inverse Hessian: ∂2
∂wi∂wj
−1by gradient to
get update direction.
..but computational complexity kills you.
2 Normalize update so total step size is controlled.
..but this justworks globally rather than per dimension.
Algorithms: Preliminaries
Optimize so few data passes required ⇒ Smart algorithms.Basic problem with gradient descent = confused units.fw (x) =
∑i wixi
⇒ ∂(fw (x)−y)2
∂wi= 2(fw (x)− y)xi which has units of i .
But wi naturally has units of 1/i since doubling xi implies halvingwi to get the same prediction.Crude fixes:
1 Newton: Multiply inverse Hessian: ∂2
∂wi∂wj
−1by gradient to
get update direction...but computational complexity kills you.
2 Normalize update so total step size is controlled...but this justworks globally rather than per dimension.
Approach Used
1 Optimize hard so few data passes required.1 L-BFGS = batch algorithm that builds up approximate inverse
hessian according to:∆w ∆T
w
∆Tw ∆g
where ∆w is a change in weights
w and ∆g is a change in the loss gradient g .
2 Dimensionally correct, adaptive, online, gradient descent forsmall-multiple passes.
1 Online = update weights after seeing each example.2 Adaptive = learning rate of feature i according to 1√P
g2i
where gi = previous gradients.3 Dimensionally correct = still works if you double all feature
values.
3 Use (2) to warmstart (1).
2 Use map-only Hadoop for process control and error recovery.3 Use custom AllReduce code to sync state.4 Always save input examples in a cachefile to speed later
passes.5 Use hashing trick to reduce input complexity.
Open source in Vowpal Wabbit 6.0. Search for it.
Approach Used
1 Optimize hard so few data passes required.1 L-BFGS = batch algorithm that builds up approximate inverse
hessian according to:∆w ∆T
w
∆Tw ∆g
where ∆w is a change in weights
w and ∆g is a change in the loss gradient g .2 Dimensionally correct, adaptive, online, gradient descent for
small-multiple passes.1 Online = update weights after seeing each example.2 Adaptive = learning rate of feature i according to 1√P
g2i
where gi = previous gradients.3 Dimensionally correct = still works if you double all feature
values.
3 Use (2) to warmstart (1).
2 Use map-only Hadoop for process control and error recovery.3 Use custom AllReduce code to sync state.4 Always save input examples in a cachefile to speed later
passes.5 Use hashing trick to reduce input complexity.
Open source in Vowpal Wabbit 6.0. Search for it.
Approach Used
1 Optimize hard so few data passes required.1 L-BFGS = batch algorithm that builds up approximate inverse
hessian according to:∆w ∆T
w
∆Tw ∆g
where ∆w is a change in weights
w and ∆g is a change in the loss gradient g .2 Dimensionally correct, adaptive, online, gradient descent for
small-multiple passes.1 Online = update weights after seeing each example.2 Adaptive = learning rate of feature i according to 1√P
g2i
where gi = previous gradients.3 Dimensionally correct = still works if you double all feature
values.
3 Use (2) to warmstart (1).
2 Use map-only Hadoop for process control and error recovery.
3 Use custom AllReduce code to sync state.4 Always save input examples in a cachefile to speed later
passes.5 Use hashing trick to reduce input complexity.
Open source in Vowpal Wabbit 6.0. Search for it.
Approach Used
1 Optimize hard so few data passes required.1 L-BFGS = batch algorithm that builds up approximate inverse
hessian according to:∆w ∆T
w
∆Tw ∆g
where ∆w is a change in weights
w and ∆g is a change in the loss gradient g .2 Dimensionally correct, adaptive, online, gradient descent for
small-multiple passes.1 Online = update weights after seeing each example.2 Adaptive = learning rate of feature i according to 1√P
g2i
where gi = previous gradients.3 Dimensionally correct = still works if you double all feature
values.
3 Use (2) to warmstart (1).
2 Use map-only Hadoop for process control and error recovery.3 Use custom AllReduce code to sync state.
4 Always save input examples in a cachefile to speed laterpasses.
5 Use hashing trick to reduce input complexity.
Open source in Vowpal Wabbit 6.0. Search for it.
Approach Used
1 Optimize hard so few data passes required.1 L-BFGS = batch algorithm that builds up approximate inverse
hessian according to:∆w ∆T
w
∆Tw ∆g
where ∆w is a change in weights
w and ∆g is a change in the loss gradient g .2 Dimensionally correct, adaptive, online, gradient descent for
small-multiple passes.1 Online = update weights after seeing each example.2 Adaptive = learning rate of feature i according to 1√P
g2i
where gi = previous gradients.3 Dimensionally correct = still works if you double all feature
values.
3 Use (2) to warmstart (1).
2 Use map-only Hadoop for process control and error recovery.3 Use custom AllReduce code to sync state.4 Always save input examples in a cachefile to speed later
passes.
5 Use hashing trick to reduce input complexity.
Open source in Vowpal Wabbit 6.0. Search for it.
Approach Used
1 Optimize hard so few data passes required.1 L-BFGS = batch algorithm that builds up approximate inverse
hessian according to:∆w ∆T
w
∆Tw ∆g
where ∆w is a change in weights
w and ∆g is a change in the loss gradient g .2 Dimensionally correct, adaptive, online, gradient descent for
small-multiple passes.1 Online = update weights after seeing each example.2 Adaptive = learning rate of feature i according to 1√P
g2i
where gi = previous gradients.3 Dimensionally correct = still works if you double all feature
values.
3 Use (2) to warmstart (1).
2 Use map-only Hadoop for process control and error recovery.3 Use custom AllReduce code to sync state.4 Always save input examples in a cachefile to speed later
passes.5 Use hashing trick to reduce input complexity.
Open source in Vowpal Wabbit 6.0. Search for it.
Approach Used
1 Optimize hard so few data passes required.1 L-BFGS = batch algorithm that builds up approximate inverse
hessian according to:∆w ∆T
w
∆Tw ∆g
where ∆w is a change in weights
w and ∆g is a change in the loss gradient g .2 Dimensionally correct, adaptive, online, gradient descent for
small-multiple passes.1 Online = update weights after seeing each example.2 Adaptive = learning rate of feature i according to 1√P
g2i
where gi = previous gradients.3 Dimensionally correct = still works if you double all feature
values.
3 Use (2) to warmstart (1).
2 Use map-only Hadoop for process control and error recovery.3 Use custom AllReduce code to sync state.4 Always save input examples in a cachefile to speed later
passes.5 Use hashing trick to reduce input complexity.
Open source in Vowpal Wabbit 6.0. Search for it.
Robustness & Speedup
0
1
2
3
4
5
6
7
8
9
10
10 20 30 40 50 60 70 80 90 100
Spe
edup
Nodes
Speed per method
Average_10Min_10
Max_10linear
Webspam optimization
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0 5 10 15 20 25 30 35 40 45 50
test
log
loss
pass
Webspam test results
OnlineWarmstart L-BFGS 1
L-BFGS 1Single machine online
How does it work?
The launch sequence (on Hadoop):
1 mapscript.sh hdfs output hdfs input maps asked
2 mapscript.sh starts spanning tree on the gateway. Each vwconnects to spanning tree to learn which other vw is adjacentin the binary tree.
3 mapscript.sh starts hadoop streaming map-only job: runvw.sh
4 runvw.sh calls vw twice.
5 First vw call does online learning while saving examples in acachefile. Weights are averaged before saving.
6 Second vw uses online solution to warmstart L-BFGS.Gradients are shared via allreduce.
Example: mapscript.sh outdir indir 100Everything also works in a nonHadoop cluster: you just need toscript more to do some of the things that Hadoop does for you.
How does this compare to other cluster learning efforts?
Mahout was founded as the MapReduce ML project, but hasgrown to include VW-style online linear learning. For linearlearning, VW is superior:
1 Vastly faster.
2 Smarter Algorithms.
Other algorithms exist in Mahout, but they are apparently buggy.AllReduce seems a superior paradigm in the 10K node-hoursregime for Machine Learning.
Graphlab(@CMU) doesn’t overlap much. This is mostly aboutgraphical model evaluation and learning.
Ultra LDA(@Y!), overlaps partially. VW has noncluster-parallelLDA which is perhaps x3 more effecient.
Can allreduce be used by others?
It might be incorporated directly into next generation Hadoop.
But for now, it’s quite easy to use the existing code.void all reduce(char* buffer, int n, std::string master location,size t unique id, size t total, size t node);buffer = pointer to some floatsn = number of bytes (4*floats)master location = IP address of gatewayunique id = nonce (unique for different jobs)total = total number of nodesnode = node id numberThe call is stateful: it initializes the topology if necessary.
Further Pointers
search://Vowpal Wabbit
mailing list: vowpal [email protected]
VW tutorial (Preparallel): http://videolectures.net
Machine Learning (Theory) blog: http://hunch.net
Bibliography: Original VW
Caching L. Bottou. Stochastic Gradient Descent Examples on ToyProblems, http://leon.bottou.org/projects/sgd, 2007.
Release Vowpal Wabbit open source project,http://github.com/JohnLangford/vowpal_wabbit/wiki,2007.
Hashing Q. Shi, J. Petterson, G. Dror, J. Langford, A. Smola, andSVN Vishwanathan, Hash Kernels for Structured Data,AISTAT 2009.
Hashing K. Weinberger, A. Dasgupta, J. Langford, A. Smola, and J.Attenberg, Feature Hashing for Large Scale MultitaskLearning, ICML 2009.
Bibliography: Algorithmics
L-BFGS J. Nocedal, Updating Quasi-Newton Matrices with LimitedStorage, Mathematics of Computation 35:773–782, 1980.
Adaptive H. B. McMahan and M. Streeter, Adaptive BoundOptimization for Online Convex Optimization, COLT 2010.
Adaptive J. Duchi, E. Hazan, and Y. Singer, Adaptive SubgradientMethods for Online Learning and Stochastic Optimization,COLT 2010.
Import. N. Karampatziakis, and J. Langford, Online ImportanceWeight Aware Updates, UAI 2011.
Bibliography: Parallel
grad sum C. Teo, Q. Le, A. Smola, V. Vishwanathan, A ScalableModular Convex Solver for Regularized Risk Minimization,KDD 2007.
averaging G. Mann, R. Mcdonald, M. Mohri, N. Silberman, and D.Walker. Efficient large-scale distributed training of conditionalmaximum entropy models, NIPS 2009.
averaging K. Hall, S. Gilpin, and G. Mann, MapReduce/Bigtable forDistributed Optimization, LCCC 2010.
(More forthcoming, of course)
