Multinomial Logistic Regression with Apache Spark

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Multinomial Logistic Regression with Apache Spark
DB TsaiMachine Learning Engineer
May 1st, 2014
Machine Learning with Big Data
● Hadoop MapReduce solutions
● MapReduce is scaling well for batch processing● However, lots of machine learning algorithms
are iterative by nature.● There are lots of tricks people do, like training
with subsamples of data, and then average themodels. Why big data with approximation?
+ =
Lightningfast cluster computing
● Empower users to iterate through the data byutilizing the inmemory cache.
● Logistic regression runs up to 100x faster thanHadoop M/R in memory.
● We're able to train exact model without doing any approximation.
Binary Logistic Regressiond=
ax1+bx 2+cx0
√a 2+b2where x0=1
P ( y=1∣⃗x , w⃗ )= exp (d )1+exp (d )
= exp( x⃗ w⃗ )1+exp( x⃗ w⃗ )
P ( y=0∣⃗x , w⃗)= 11+exp( x⃗ w⃗)
logP ( y=1∣⃗x , w⃗ )P( y=0∣⃗x , w⃗)
= x⃗ w⃗
w0=c
√a2+b2
w1=a
√a2+b2
w2=b
√a2+b2
wherew0 is called as intercept
Training Binary Logistic Regression● Maximum Likelihood estimation
From a training data and labels
● We want to find that maximizes thelikelihood of data defined by
● We can take log of the equation, and minimize
it instead. The LogLikelihood becomes theloss function.
X=( x⃗1 , x⃗ 2 , x⃗3 , ...)Y=( y1, y2, y3, ...)
w⃗
L( w⃗ , x⃗1, ... , x⃗N)=P ( y1∣x⃗1 , w⃗)P ( y2∣x⃗2 , w⃗ )... P ( yN∣x⃗N , w⃗ )
l ( w⃗ , x⃗ )=log P ( y1∣x⃗1 , w⃗ )+log P ( y2∣x⃗ 2 , w⃗ )...+log P ( yN∣x⃗N , w⃗ )
Optimization● First Order Minimizer
Require loss, gradient of loss function
– Gradient Decent is step size
– Limitedmemory BFGS (LBFGS)
– OrthantWise Limitedmemory QuasiNewton(OWLQN)
– Coordinate Descent (CD)
– Trust Region Newton Method (TRON)● Second Order Minimizer
Require loss, gradient and hessian of loss function
– NewtonRaphson, quadratic convergence. Fast!
● Ref: Journal of Machine Learning Research 11 (2010) 31833234, ChihJen Lin et al.
w⃗n+1=w⃗ n−γG⃗ , γ
w⃗n+1=w⃗ n−H−1G⃗
Problem of Second Order Minimizer
● Scale horizontally (the numbers of trainingdata) by leveraging Spark to parallelize thisiterative optimization process.
● Don't scale vertically (the numbers of trainingfeatures). Dimension of Hessian is
● Recent applications from documentclassification and computational linguistics areof this type.
dim (H )=[(k−1)(n+1)]2 where k is numof class , nis numof features
LBFGS● It's a quasiNewton method.● Hessian matrix of second derivatives doesn't
need to be evaluated directly. ● Hessian matrix is approximated using gradient
evaluations.● It converges a way faster than the default
optimizer in Spark, Gradient Decent.● We love open source! Alpine Data Labs
contributed our LBFGS to Spark, and it'salready merged in Spark1157.
Training Binary Logistic Regressionl ( w⃗ , x⃗ ) = ∑
k=1
N
log P ( yk∣x⃗k , w⃗ )
= ∑k=1
N
yk logP ( yk=1∣x⃗k , w⃗)+(1− yk ) logP ( yk=0∣x⃗k , w⃗)
= ∑k=1
N
yk logexp ( x⃗k w⃗)
1+exp( x⃗k w⃗)+(1− yk ) log 1
1+exp( x⃗k w⃗ )
= ∑k=1
N
yk x⃗ k w⃗−log(1+exp( x⃗k w⃗))
Gradient : Gi (w⃗ , x⃗)=∂ l ( w⃗ , x⃗ )∂wi
=∑k=1
N
y k xki−exp ( x⃗k w⃗)
1+exp ( x⃗k w⃗ )xki
Hessian : H ij (w⃗ , x⃗)=∂∂ l (w⃗ , x⃗ )∂wi ∂w j
=−∑k=1
N exp( x⃗k w⃗)
(1+exp( x⃗k w⃗ ))2xki x kj
Overfitting
P ( y=1∣⃗x , w⃗)=exp(zd )
1+exp(zd )=
exp( x⃗ w⃗)1+exp( x⃗ w⃗)
Regularization● The loss function becomes
● The loss function of regularizer doesn'tdepend on data. Common regularizers are
– L2 Regularization:
– L1 Regularization:
● L1 norm is not differentiable at zero!
l total (w⃗ , x⃗)=lmodel (w⃗ , x⃗)+l reg( w⃗)
lreg (w⃗ )=λ∑i=1
N
wi2
lreg (w⃗ )=λ∑i=1
N
∣wi∣
G⃗ (w⃗ , x⃗)total=G⃗ (w⃗ , x⃗)model+G⃗ ( w⃗)reg
H̄ ( w⃗ , x⃗)total=H̄ (w⃗ , x⃗)model+H̄ (w⃗ )reg
Mini School of Spark APIs
● map(func) : Return a new distributed datasetformed by passing each element of the sourcethrough a function func.
● reduce(func) : Aggregate the elements of thedataset using a function func (which takes twoarguments and returns one). The functionshould be commutative and associative sothat it can be computed correctly in parallel. No “key” thing here compared with HadoopM/R.
Example – No More “Word Count”! Let's compute the mean of numbers.
3.0
2.0
1.0
5.0
Executor 1
Executor 2
map
map
(1.0, 1)
(5.0, 1)
(3.0, 1)
(2.0, 1)
reduce
(5.0, 2)
reduce (11.0, 4)
Shuffle
● The previous example using map(func) willcreate new tuple object, which may causeGarbage Collection issue.
● Like the combiner in Hadoop Map Reduce, for each tuple emitted from map(func), there isno guarantee that they will be combinedlocally in the same executor by reduce(func). It may increase the traffic in shuffle phase.
● In Hadoop, we address this by inmappercombiner or aggregating the result in globalvariables which have scope to entire partition.The same approach can be used in Spark.
Mini School of Spark APIs
● mapPartitions(func) : Similar to map, but runsseparately on each partition (block) of theRDD, so func must be of type Iterator[T] =>Iterator[U] when running on an RDD of type T.
● This API allows us to have global variables onentire partition to aggregate the result locallyand efficiently.
Better Mean of Numbers
Implementation
3.0
2.0
Executor 1
reduce (11.0, 4)
Shuffle
var counts = 0var sum = 0.0
loop
var counts = 2var sum = 5.0
3.0
2.0
(5.0, 2)
1.0
5.0
Executor 2
var counts = 0var sum = 0.0
loop
var counts = 2var sum = 6.0
1.0
5.0
(6.0, 2)
mapPartitions
More Idiomatic Scala Implementation● aggregate(zeroValue: U)(seqOp: (U, T) => U,
combOp: (U, U) => U): U zeroValue is a neutral "zero value"with type U for initialization. This isanalogous to
in previous example.
var counts = 0var sum = 0.0
seqOp is function taking(U, T) => U, where U is aggregatorinitialized as zeroValue. T is eachline of values in RDD. This isanalogous to mapPartition inprevious example.
combOp is function taking(U, U) => U. This is essentiallycombing the results betweendifferent executors. Thefunctionality is the same asreduce(func) in previous example.
Approach of Parallelization in SparkSpark Driver JVM
Tim
e
Spark Executor 1 JVM
Spark Executor 2 JVM
1) Find available resources in cluster, and launch executor JVMs. Also initialize the weights.
2) Ask executors to load the data
into executors' JVMs
2) Trying to load the data into memory. If the data is bigger than memory, it will be partial cached. (The data locality from source will be taken care by Spark)
3) Ask executors to compute loss, and gradient of each training sample (each row) given the current weights. Get the aggregated results after the Reduce Phase in executors.
5) If the regularization is enabled, compute the loss and gradient of regularizer in driver since it doesn't depend on training data but only depends on weights. Add them into the results from executors.
3) Map Phase: Compute the loss and gradient of each row of training data locally given the weights obtained from the driver. Can either emit each result or sum them up in local aggregators.
4) Reduce Phase: Sum up the losses and gradients emitted from the Map Phase
Taking a rest!
6) Plug the loss and gradient from model and regularizer into optimizer to get the new weights. If the differences of weights and losses are larger than criteria, GO BACK TO 3)
Taking a rest!
7) Finish the model training! Taking a rest!
Step 3) and 4) ● This is the implementation of step 3) and 4) in
MLlib before Spark 1.0
● gradient can have implementations ofLogistic Regression, Linear Regression, SVM,or any customized cost function.
● Each training data will create new “grad” object after gradient.compute.
Step 3) and 4) with mapPartitions
Step 3) and 4) with aggregate
● This is the implementation of step 3) and 4) inMLlib in coming Spark 1.0
● No unnecessary object creation! It's helpfulwhen we're dealing with large features trainingdata. GC will not kick in the executor JVMs.
Extension to Multinomial Logistic Regression
● In Binary Logistic Regression
● For K classes multinomial problem where labelsranged from [0, K1], we can generalize it via
● The model, weights becomes(K1)(N+1) matrix, where N is number of features.
logP ( y=1∣⃗x , w̄)P ( y=0∣⃗x , w̄)
= x⃗ w⃗1
logP ( y=2∣⃗x , w̄)P ( y=0∣⃗x , w̄)
= x⃗ w⃗2
...
logP ( y=K−1∣⃗x , w̄)P ( y=0∣⃗x , w̄)
= x⃗ w⃗K−1
logP ( y=1∣⃗x , w⃗)P ( y=0∣⃗x , w⃗)
= x⃗ w⃗
w̄=(w⃗1, w⃗2, ... , w⃗K−1)T
P ( y=0∣⃗x , w̄ )= 1
1+∑i=1
K−1
exp( x⃗ w⃗ i )
P ( y=1∣⃗x , w̄)=exp( x⃗ w⃗2)
1+∑i=1
K−1
exp( x⃗ w⃗ i)
...
P ( y=K−1∣⃗x , w̄ )=exp( x⃗ w⃗K−1)
1+∑i=1
K−1
exp( x⃗ w⃗ i )
Training Multinomial Logistic Regression
l ( w̄ , x⃗ ) = ∑k=1
N
log P ( yk∣x⃗k , w̄ )
= ∑k=1
N
α( y k) log P ( y=0∣x⃗ k , w̄ )+(1−α( yk )) logP ( yk∣x⃗k , w̄)
= ∑k=1
N
α( y k) log 1
1+∑i=1
K−1
exp( x⃗ w⃗i)+(1−α( yk )) log
exp( x⃗ w⃗ yk)
1+∑i=1
K−1
exp( x⃗ w⃗i)
= ∑k=1
N
(1−α( yk )) x⃗ w⃗ y k−log (1+∑
i=1
K−1
exp( x⃗ w⃗i))
Gradient : Gij (w̄ , x⃗)=∂ l ( w̄ , x⃗)∂wij
=∑k=1
N
(1−α( yk )) xkj δi , y k−
exp ( x⃗k w⃗)1+exp( x⃗k w⃗ )
xkj
α( yk )=1 if y k=0α( yk )=0 if yk≠0
Define:
Note that the first index “i” is for classes, and the second index “j” is for features.
Hessian: H ij , lm( w̄ , x⃗ )=∂∂ l (w̄ , x⃗)∂wij ∂wlm
0 5 10 15 20 25 30 350.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0.7
Logistic Regression with a9a Dataset (11M rows, 123 features, 11% nonzero elements)16 executors in INTEL Xeon E31230v3 32GB Memory * 5 nodes Hadoop 2.0.5 alpha cluster
LBFGS Dense Features
LBFGS Sparse Features
GD Sparse Features
GD Dense Features
Seconds
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esa9a Dataset Benchmark
a9a Dataset Benchmark
1 1 3 5 7 9 11 13 150.3
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0.6
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0.7
Logistic Regression with a9a Dataset (11M rows, 123 features, 11% nonzero elements)16 executors in INTEL Xeon E31230v3 32GB Memory * 5 nodes Hadoop 2.0.5 alpha cluster
LBFGS
GD
Iterations
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0 5 10 15 20 25 30
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Logistic Regression with rcv1 Dataset (6.8M rows, 677,399 features, 0.15% nonzero elements)16 executors in INTEL Xeon E31230v3 32GB Memory * 5 nodes Hadoop 2.0.5 alpha cluster
LBFGS Sparse Vector
GD Sparse Vector
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esrcv1 Dataset Benchmark
news20 Dataset Benchmark
0 10 20 30 40 50 60 70 800
0.2
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0.8
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Logistic Regression with news20 Dataset (0.14M rows, 1,355,191 features, 0.034% nonzero elements)16 executors in INTEL Xeon E31230v3 32GB Memory * 5 nodes Hadoop 2.0.5 alpha cluster
LBFGS Sparse Vector
GD Sparse Vector
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Alpine Demo
Alpine Demo
Alpine Demo
Conclusion● Alpine supports state of the art Cloudera CDH5
● Spark runs programs 100x faster than Hadoop
● Spark turns iterative big data machine learning problemsinto single machine problems
● MLlib provides lots of state of art machine learningimplementations, e.g. Kmeans, linear regression,logistic regression, ALS, collaborative filtering, andNaive Bayes, etc.
● Spark provides ease of use APIs in Java, Scala, orPython, and interactive Scala and Python shell
● Spark is Apache project, and it's the most active big dataopen source project nowadays.