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Multinomial Logistic Regression with Apache Spark DB Tsai Machine Learning Engineer May 1 st , 2014

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Logistic Regression can not only be used for modeling binary outcomes but also multinomial outcome with some extension. In this talk, DB will talk about basic idea of binary logistic regression step by step, and then extend to multinomial one. He will show how easy it's with Spark to parallelize this iterative algorithm by utilizing the in-memory RDD cache to scale horizontally (the numbers of training data.) However, there is mathematical limitation on scaling vertically (the numbers of training features) while many recent applications from document classification and computational linguistics are of this type. He will talk about how to address this problem by L-BFGS optimizer instead of Newton optimizer. Bio: DB Tsai is a machine learning engineer working at Alpine Data Labs. He is recently working with Spark MLlib team to add support of L-BFGS optimizer and multinomial logistic regression in the upstream. He also led the Apache Spark development at Alpine Data Labs. Before joining Alpine Data labs, he was working on large-scale optimization of optical quantum circuits at Stanford as a PhD student.

### Transcript of Multinomial Logistic Regression with Apache Spark

Multinomial Logistic Regression with Apache Spark

DB TsaiMachine Learning Engineer

May 1st, 2014

Machine Learning with Big Data

● 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?

+ =

Lightning-fast cluster computing

● Empower users to iterate through the data byutilizing the in-memory 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 Log-Likelihood 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

– Limited-memory BFGS (L-BFGS)

– Orthant-Wise Limited-memory Quasi-Newton(OWLQN)

– Coordinate Descent (CD)

– Trust Region Newton Method (TRON)● Second Order Minimizer

Require loss, gradient and hessian of loss function

● Ref: Journal of Machine Learning Research 11 (2010) 3183-3234, Chih-Jen 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

L-BFGS● It's a quasi-Newton 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 L-BFGS to Spark, and it'salready merged in Spark-1157.

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 in-mappercombiner 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.

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.

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, K-1], we can generalize it via

● The model, weights becomes(K-1)(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% non-zero elements)16 executors in INTEL Xeon E3-1230v3 32GB Memory * 5 nodes Hadoop 2.0.5 alpha cluster

L-BFGS Dense Features

L-BFGS 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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Logistic Regression with a9a Dataset (11M rows, 123 features, 11% non-zero elements)16 executors in INTEL Xeon E3-1230v3 32GB Memory * 5 nodes Hadoop 2.0.5 alpha cluster

L-BFGS

GD

Iterations

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0.1

0.2

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Logistic Regression with rcv1 Dataset (6.8M rows, 677,399 features, 0.15% non-zero elements)16 executors in INTEL Xeon E3-1230v3 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

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1.2

Logistic Regression with news20 Dataset (0.14M rows, 1,355,191 features, 0.034% non-zero elements)16 executors in INTEL Xeon E3-1230v3 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. K-means, 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.