Modeling with Hadoop kdd2011

Session 2: Modeling with
Hadoop
Algorithms in MapReduce

Vijay K Narayanan
Principal Scientist, Yahoo! Labs, Yahoo!

Outline
• Why learn models in MapReduce framework?
• Types of learning in MapReduce
• Statistical Query Model (SQM)
• SQM Algorithms in MapReduce
• Sequential learning methods and MapReduce
• Challenges and Enhancements
• Apache Mahout

Why learn models in MapReduce?
• High data throughput
– Stream about 100 Tb per hour using 500 mappers
• Framework provides fault tolerance
– Monitors mappers and reducers and re-starts tasks on
other machines should one of the machines fail
• Excels in counting patterns over data records
• Built on relatively cheap, commodity hardware
– No special purpose computing hardware
• Large volumes of data are being increasingly
stored on Grid clusters running MapReduce
– Especially in the internet domain

Why learn models in MapReduce?
• Learning can become limited by computation
time and not data volume
– With large enough data and number of machines
– Reduces the need to down-sample data
– More accurate parameter estimates compared to
learning on a single machine for the same amount of
time

Learning models in MapReduce
• A primer for learning models in MapReduce (MR)
– Illustrate techniques for distributing the learning algorithm
in a MapReduce framework
– Focus on the mapper and reducer computations
• Data parallel algorithms are most appropriate for
MapReduce implementations
• Not necessarily the most optimal implementation for a
specific algorithm
– Other specialized non-MapReduce implementations exist
for some algorithms, which may be better
• MR may not be the appropriate framework for exact
solutions of non data parallel/sequential algorithms
– Approximate solutions using MapReduce may be good
enough

Types of learning in MapReduce
• Three common types of learning models using
MapReduce framework
1. Parallel training of multiple models Use the Grid as a
– Train either in mappers or reducers large cluster
of independent
2. Ensemble training methods machines
– Train multiple models and combine them (with fault
tolerance)
3. Distributed learning algorithms
– Learn using both mappers and reducers

Parallel training of multiple models

• Train multiple models simultaneously using a
learning algorithm that can be learnt in memory
• Useful when individual models are trained using
a subset, filtered or modification of raw data
• Can train 1000’s of models simultaneously
• Train 1 model in each reducer
– Map:
• Input: All data
• Filters subset of data relevant for each model training
• Output: <model_index, subset of data for training this model>
– Reduce
• Train model on data corresponding to that model_index

• Train 1 model in each reducer

Data subgroup 1 "model_1", Data model_1

Train Model_1
Data subgroup 2


Train Model_2

Data subgroup N "model_2", Data model_2

Mapper Reducer

• Train 1 model in each mapper

Map_1 Model (c1 ) • All data is sent to each
c1 mapper (as a cache
archive)
• Mapper partition file
Map_2 Model (c2 ) determines the training
Training c2 configuration and labeling
Data strategy
{x, (ci , c j ...ck )} – e.g., Training one vs. rest
models in multi-class
ci {c1 , c2 ...cM } classification
– Can train 1000s of
classes in parallel
Map_M Model (cM )
cM

Ensemble methods
• Train 1 base model in each mapper on a data partition
• Combine the base models using ensemble methods
(primarily, bagging) in the reducer
• Strictly, bagging requires the data to be sampled with
replacement
– However, if the data set is very large, sampling without
replacement may be ok
• Base models are typically decision trees, SVMs etc.

Ensemble Methods: Random
Subspace Bagging (RSBag)
• Assume that the training data is partitioned randomly into
blocks
– Class distributions are roughly the same across all blocks
• Algorithm (Yan et al. 2007)
– Learn 1 base model per data sub-group
Base-model  hc ( x)
yc {1, 1}
– Optionally, use a random subset of features to train each model
– Combine the multiple base models into a composite classifier as
the final output

Fci ( x)  Fci 1 ( x)  hci ( x)

RSBag in MapReduce
1
Map_1 hc ( x)

Features

2 Combine
Map_2 h ( x)
c base
D models
A into
T final
A hc3 ( x) classifier
Map_3

Map_4 hc4 ( x)

RSBag in MapReduce
• Provides coarse level parallelism at the level of base models
– Base models can be decision trees, SVMs etc.
• Speed-up with SVM base models
Nrd2 rf , rd , rf  data, feature sampling ratios
N  5, rd  0.2, rf  0.5  Speedup  10

• Can achieve similar performance as a single classifier with
theoretical guarantee in less learning time


E * ( Fc )   1  sc2  sc2 Upper bound on
generalization error
  E ,    x  h( x, )h( x, )  
 
Correlation between
classifiers
sc  2 Ex , yc P  h( x,  )  yc   1

Strength of classifier

Robust Subspace Bagging
(RB-SBag)
• Sometimes the base models may over-fit the
training data
– Correlation between base models may be high
• Add a Forward selection step for models
– Iteratively add base models based on their
performance on a validation data (Yan et al. 2009)
• Adds another MapReduce job
– Select the base models using forward selection based
on performance metrics on a validation dataset Vc

RB-SBag in MapReduce
Map_1 " c ",{hc , Pr ediction c (V )}

Validation
Data Map_2

1
hc ( x ),
hc2 ( x ),
....
hcN ( x ) 1. Forward selection of base models
2. Combine base models into composite
Map_N classifier

Mapper Reducer

COMET: Cloud of Massive
Ensemble Trees
• Similar to RSBag, but uses Importance-Sampled Voting
(IVoting) in each base model
• Samples are weighted with non-uniform probability
• Each mapper creates a set of data to train on
• Ensemble after k iterations = E(k)
– Add new sample to training set:
• Always if E(k) incorrectly classifies new sample
• With a lower probability if E(k) correctly classifies new sample
e(k ) / (1  e(k )); e(k )  error on training dataset
• Variant of Random Forests, in which IVoting generates
the training samples instead of bagging
• Use lazy evaluation during prediction
J.D Basilico, M.A. Munson, T.G. Kolda, K.R. Dixon, W.P.Kegelmeyer, COMET: A Recipe for Learning and Using
Large Ensembles on massive data, 2011, http://arxiv.org/PS_cache/arxiv/pdf/1103/1103.2068v1.pdf

Distributed learning algorithms
• Use multiple mappers and reducers to learn 1 model
• Suitable for learning algorithms that
– Have heavy computing per data record
– One or few iterations for learning
– Do not transfer much data between iterations
• Typical algorithms
– Fit the Statistical query model (SQM)
• One/few iterations
– Linear regression, Naïve Bayes, k-means clustering, pair-wise similarity etc.
• More iterations have high overheads, e.g.,
– SVM, Logistic regression etc.
– Divide and conquer
• Frequent item-set mining, Approximate matrix factorization etc.

Statistical Query Model (SQM)
• Learning algorithm can access the
learning problem only through a statistical
query oracle (Kearns 1998)

• Given a function f(x,y) over data instances,
the statistical query oracle returns an
estimate of the expectation of f(x,y)
(averaged over the data distribution).

Statistical Query Model (SQM)

Raw Data Raw Data
Learning Statistics
Samples Samples
Algorithm Oracle
(X,Y) (X,Y)
f ( x, y)

• Learning algorithms that calculate sufficient statistics of data,
gradients of a function, etc. fit this model

• These calculations can be expressed in a “summation form”
over subgroups of data (Chu et al. 2006)


subgroup
f ( x, y )

SQM in MapReduce
• Distribute the summation calculations over each
data sub-group
• Map:
– Calculate function estimates over sub-groups of data
• Reduce
– Aggregate the function estimates from various sub-
groups
• Learning algorithm should be able to work with
these summaries alone

SQM in MapReduce
• Assume algorithm depends on 2 functions f(x,y) and g(x,y)

 
Data subgroup 1
 " f ",
subgroup
f ( x, y ) " g ",
subgroup
g ( x, y )

Data subgroup 2


 
N subgroup
f ( x, y),  
N subgroup
g ( x, y)

Data subgroup N

Mapper Reducer

Algorithms in MapReduce
• Many common algorithms can be formulated in
the SQM framework (Chu et al. 2006)
– Classification and Regression
• Linear Regression, Naïve Bayes, Logistic regression,
Support Vector Machine, Decision Trees
– Clustering
• K-means, Canopy clustering, Co-clustering
– Back-propagation neural network
– Expectation Maximization
– PCA
• Recommendations and Frequent Itemset mining
• Graph Algorithms

Classification and Regression
algorithms in MapReduce
• Linear Regression
• Naïve Bayes
• Logistic Regression
• Support Vector Machine
• Decision Trees

Linear regression
• Data vector: xi  ( xi1 , xi 2 ,...xin )T
• Real valued target : yi
• Weight of data point: wi
 x, y, w
m
• Data set of points:

y T x
 *  A1b
m
A   wi ( xi xiT )
i 1
m Summation form
b   wi ( xi yi )
i 1

Linear Regression in MapReduce
• Map:
– Input data Index,  x, y, w from a subgroup of data
– Output
• 2 types of keys
– K1 – for matrix A
» Value1 = N x N matrix
– K2 – for vector b
» Value2 = N x 1 vector

• Reducer:
– Aggregate the individual mapper outputs for each key
– Estimate   A b
* 1

Linear Regression in MapReduce
• A: N x N matrix, b: N x 1 vector

 x, y, w A, b " A ", 
subgroup
wi xi xiT " b ", 
subgroup
wi xi yi
1

 x, y, w
2
A, b  A,  b
 *  A1b

 x, y, w
k
A, b
Mapper Reducer

Naïve Bayes
• Input Data: x  ( x1 , x2 ,...xn ); x j {a1j , a2j ....aPj }
j

Domain of x j
• Categorical target: y  c1 , c2 ...cL 

• Class prediction: Class prior Conditional probability table (CPT)

y*  arg max P( y  ck ) P( x j  a pj | y  ck )
j

y j

• Two types of sufficient statistics

P( x j  a pj | y  ck )
j
Sum counts
P( y  ck ) over sub-groups

Naïve Bayes in MapReduce
• Map
– Input data {x, y} from a subgroup of data
– Output: 3 types of keys

key  ( x j  a pj , y  ck ), value 
j

subgroup
1( x j  a pj | y  ck )
j CPT

key  ( y  ck ), value  
subgroup
1( y  ck ) Class prior

key  " samples ", value  
subgroup
1 Normalization

• Reduce
– Sum all the values of each key
– Compute the class prior and the conditional probabilities

Logistic Regression
• Features: x  ( x1 , x2 ,...xn )
;
• Categorical target: y [0,1]
Data:  x, y 
m
•
• Conditional probability:
1
P ( y | x,  ) 
1  exp( T x)
• Equivalently
 p 
log   T x
 1 p 
– Log odds is a linear function of the features

Logistic Regression
• Estimate the parameters by maximizing the log
conditional likelihood of observed data
LCL  log p   log 1  p 
i: yi 1
i
i: yi 0
i

• Optimize using Newton-Raphson to update 
    H 1 LCL  
Gradient   LCL   j    y i  p i  x ij
i
Summation form
Hessian  H jk  H jk   p  p  1 x x i i i i
j k
i

i  [1, m]; j , k  [1, n]
Data Features

Logistic Regression in MapReduce
• A control program sets up the MapReduce iterations
• Map
– Input:  x, y 
   
– Output:  
key  g , value   j ,  y i  p i x ij  
 isubgroup
 

 
i i 
 
key  h, value   j , k ,  p p  1 x j xk  
i i


 isubgroup 

• Reduce
– Aggregate the values of  LCL   j , H jk from all mappers
– Compute H  LCL  
1

– Update
    H  LCL  
1

• Stop when updates become small 1 update per iteration

Support Vector Machine
• Features: x  Rn ;
• Binary target: y [1, 1]
• Objective function in primal form
min w  C  i p , i  ( wT xi  yi )
2

w ,b
i ,i  0

s.t i
 
y i wT xi  b  (1  i )
p=1 (hinge loss), p=2 (quadratic loss)
• For quadratic loss, batch gradient descent to estimate w
Gw  2w  2C   w.xi  yi  xi
i

Summation form

Support Vector Machine in
MapReduce
• Map
– Input: ( x, y}
– Output:
key  GGW , value  2w  2C   w.x  y  x
subgroup
i i i

• Reduce
– Aggregate the values of gradient from all mappers
– Update
w  w  * Gw
• Driver program sets up the iterations and checks
for convergence

Decision Trees
• Features: x  ( x1 , x2 ,...xn )
• Targets: y [0,1] or yR
Data: D   x, y 
m
•
• Construct Tree
– Each node splits the data by feature value
– Start from root
• Select best feature, value to split the node
– Based on reduction in data impurity between the child and
parent nodes
– Select the next child node
– Repeat the process till some stopping criterion
• Pure node, or data is below some threshold etc.

Decision Trees

Expensive step
for
Large datasets

B. Panda, J. S. Herbach, S. Basu, R. J. Bayardo, PLANET: Massively Parallel
Learning of Tree Ensembles with MapReduce, 2009, Proceedings of The Vldb
Endowment - PVLDB, vol. 2, no. 2, pp. 1426-1437

PLANET for Decision Trees
• Parallel Learner for Assembling Numerous Ensemble
Trees (PLANET- Panda et al. 2009)
– Main idea is to use MapReduce to determine the best feature
value splits for nodes from large datasets
• Each intermediate node has a sub-set of all data falling
into it
• If this sub-set is small enough to fit in memory,
– Grow remaining sub-tree in memory
• Else,
– Launch a MapReduce job to find candidate feature value splits
– Select the best feature split from among the candidates

• 5 main components
1. Controller M
• Monitors and controls the growth of tree a
p
2. Initialization Task R
• Identifies all feature values to be considered for splits e
3. FindBestSplit Task d
u
• Finds best split when there is too much data to fit in memory
c
4. InMemoryGrow Task e
• Grow an entire sub-tree once the data fits in memory
T
5. Model File a
• File describing the state of the model s
k
s

• Maintain 2 queues
– MapReduceQueue (MRQ)
• Contains nodes for which data is too large to fit in memory
– InMemoryQueue (InMemQ)
• Contains nodes for which data fits in memory
• 2 main MapReduce jobs
– MR_ExpandNodes
• Process nodes from the MRQ to find best split
• Output for each node:
– Candidate split positions for node along with
» Quality of split (using summary statistics)
» Predictions in left and right branches
» Size of data going into left and right branches
– MR_InMemory
• Process nodes from the InMemQ.
• For a given set of nodes N, complete tree induction at nodes in N using the
InMemoryGrow algorithm.

• Map function in MR_ExpandNodes
– Load the current model file and set of nodes N from MRQ
– For each record
• Determine if record is relevant to any of the nodes in N
• Add record to the summary statistics (SS) for node
• For each feature-value in record
– Add record to the summary statistics for node for split points “s” less than the
value in record “v”
– Output
Split ID
key  (n  N , x  Ordered  feature, s ); value  Tn , x  s  SS of

key  (n  N , x  Categorical  feature); value   v, Tn , x v 
candidate
splits
key  (n  N ); value  SS SS of parent node
 
Tn , x  s   SS    y,  y ,  1 2

 subgroup subgroup subgroup  SS for variance impurity

• Reduce function in MR_ExpandNodes
– For each node
• Aggregate the summary statistics for that node
– For each split (which is node specific)
• Aggregate the summary statistics for that Split ID from all map
outputs of summary statistics
• Compute impurity of data going into left and right branches
• Total impurity = Impurity in left branch + Impurity in right branch
• If Total impurity < Best split impurity so far
– Best split = Current split
– Output the best split found

Clustering algorithms in
MapReduce
• k-means clustering
• Canopy clustering
• Co-clustering

k-means clustering
• Choose k samples as initial cluster centroids
• Iterate till convergence
– Assign membership of each point to closest cluster
MR
– Re-compute new cluster centroids using assigned
members
• Control program to
– Initialize the centroids
• random, initial clustering on sample etc.
– Run the MapReduce iterations
– Determine stopping criterion

k-means clustering in MapReduce
• Map
– Input data points: x1 , x2 ...xN
– Input cluster centroids: C  (c1 , c2 ,...cK )
– Assign each data point to closest cluster
– Output
 
key  ci , value    x j | x j  ci ,  1| x j  ci 
• Reduce  subgroup subgroup 
– Compute new centroids for each cluster ci

 
key  ci subgroup
x j | x j  ci
key  ci , value 
 
key  ci subgroup
1| x j  ci

Complexity of k-means clustering
• Each point is compared with each cluster centroid
• Complexity = N * K * O(d ) where O(d ) is the complexity
of the distance metric
• Typical Euclidean distance is not a cheap operation
• Can reduce complexity using an initial canopy clustering
to partition data cheaply
– Preliminary step to help reduce expensive distance calculations
– Group data into (possibly overlapping) canopies using a cheap
distance metric (McCallum et al. 2000)
– Compute the distance metric between a point and a cluster
centroid only if they share a canopy.

Canopy clustering
• Every point in the dataset is in a canopy
• A point can belong to multiple canopies
• Canopy size = T1
• Algorithm
– Keep a list of canopies, initially an empty list
– Scan each data point:
• If it is within T2 < T1 distance of existing canopies, discard it.
Otherwise, add this point into the list of canopies
– Use a cheap distance metric to construct the
canopies
• e.g. Manhattan distance, L
– Assign points to the closest canopy

A. McCallum, K. Nigam, L. Ungar. Efficient Clustering of High Dimensional Data Sets with Application
to Reference Matching, SIGKDD 2000

Canopy clustering

Image from: http://horicky.blogspot.com/2011/04/k-means-clustering-in-map-reduce.html

Canopy clustering in MapReduce
• Map
– Input data points: x1 , x2 ...xN
– If data point is not within distance
T2 of an existing
candidate canopy, add it as a candidate canopy point
– Output
key  1, value  xi | xi  candidate  canopy
• Reduce
– Keep a list of final canopy points, initially an empty list
– If the canopy point is not within distance
0.5*T2
of an
existing final canopy point, add it as a final canopy
point
– Output
key  1, value  xi | xi  final  canopy

Canopy + k-means clustering
• Final step in canopy clustering assigns all points
to the closest final canopy point
– Map only operation
• Speeding up k-means using canopy clustering
– Initial run of canopy clustering on the data (or on a
sample of data)
• Pick canopy centers
• Assign points to canopies
– Pick initial k-means cluster centroids
• Run k-means iterations
– Compute distance between point and centroid only if
they are in the same canopy

Co-clustering
• Cluster pair-wise relationships in dyadic data
• Simultaneously cluster both rows and clusters,
based on certain criteria
• Identify sub-matrices of rows and columns that
are inter-related
• Commonly used in text mining, recommendation
systems and graph mining

Co-clustering
• Given an m x n matrix
– Find group assignments of rows and columns such that the resulting
sub-matrices are smooth (Papadimitriou & Sun, 2008)
– Assign rows and columns to clusters

r {1, 2....k}m , c {1, 2....l}n , k  m, l  n

01011 r   2 1 2 1
T 11000
10100 11000
01011 c   2 1 2 1 1 00111
T

10100 00111

Co-clustering
• Iteratively re-arrange rows and columns till an
error function keeps reducing
• Algorithm: Input Am x n , k , l
– Initialize r and c
– Compute a group statistics/cost matrix Gk x l
– While cost decreases
• For each row i  1 m do
– For each row group label p  1 k do
» r (i) p if cost decreases
• Update G, r
• Do the same for columns
– Return r and c

S. Papadimitriou, J. Sun, DisCo: Distributed Co-clustering with Map-Reduce, 2008,
ICDM '08. Eighth IEEE International Conference on Data Mining, pp 512-521

Co-clustering in MapReduce
• Assumptions
– Error can be computed using r , c, G only (sufficient statistics)
– Row assignments can be based on r , c, G, ai: (greedy search)
• Map:
– Cost matrix and column cluster assignments are in all mappers
– Input:
• Key = row index i
• Value = adjacency list for row i  ai:
– Compute:
• Row statistics for current column cluster assignment gi (ai: , c)
• Assign row to row cluster r (i) {1 k} that has the lowest cost
– Output:
key  r (i ) Row cluster label for row

value  ( gi ,{i}) Cost of cluster assignment, row

Co-clustering in MapReduce
• Reduce
– For each row cluster label, merge the rows and total cost
p  r (i) gp  
j:r ( j )  r ( i )
gi Ip  Ip i

Row cluster label Total cost Rows in this row cluster
– Output
p,  g p , I p 

• Collect the results for each row cluster
– For each reduce output
g p:  g p
r (i )  p, i  I p

Co-clustering in MapReduce-
Example
• Assume a row and column partitioning for the matrix
k 2 l2
01011 r  (1,1,1, 2) r  (1, 2,1, 2)
10100 c  (1,1,1, 2, 2)
01011 Cost function = Number of non-zeros per group
10100  4 4  2 4
G=   G=  
 2 0  4 0
Reduce:
Map:
Input: (2,<(2, 0),{2}) )
Input:(2,  1,3 )
Output: g 2   2, 0 
Output:  r (2)  2, ( g 2  (2, 0),{2}) 
I2  I2 {2}
S. Papadimitriou, J. Sun, DisCo: Distributed Co-clustering with Map-Reduce, 2008,
ICDM '08. Eighth IEEE International Conference on Data Mining, pp 512-521

Recommendations and Frequent
Itemset mining
• Item-based collaborative filtering
• Pair-wise similarity
• Low-rank matrix factorization
• Frequent Itemset mining

Item-based collaborative filtering
• Given a user-item ratings matrix, fill in the ratings of the missing
items for each user
ITEM RATING

U 514
S
E ?25
R
432

• Infer missing ratings from available item ratings for user weighted by
similarity between items

R ( u , j )!?
sim(i, j ) * R(u, j )
R(u, i ) 

R ( u , j )!?
sim(i, j )

• Estimate similarity between items as Pearson
correlation of rankings from users who have
rated both items.

  R(u, i)  R (i)  R(u, j )  R ( j ) 
U ij
sim(i, j ) 
  R(u, i)  R (i)    R(u, j )  R ( j ) 
2 2

U ij U ij

U ij  {u | R(i )!  ?, R( j )!  ?}

using MapReduce
• Map • Reduce
– Input: – Input:
key  u, key  (i, j )
Value  {(i, R(i ) | R(i )!  ?)} Value  [( R(i ), R( j )]
– Output: Ratings for – Output:
item pairs
key  (i, j )
key  (i, j ) Value  sim(i, j )
Value  ( R(i ), R( j ))

Pair-wise Similarity
• Compute similarity between pairs of
documents in a corpus
S (di , d j )   wt ,di * wt ,d j   wt ,di * wt ,d j
tV tdi dj

• Generate a postings list for each t V
P(t )  {(di , wt ,di ) | wt ,di  0}
– This is an easy map-reduce job

Pair-wise Similarity in MapReduce
• Generating a postings list of inverted index
– Map
Input di
For each t  di
Emit {t , ( di , wt ,di )}

– Reduce
Emit {t ,[(di , wt ,di )]}

• Map
– Input term postings list t , P(t )
– Take the Cartesian product of the postings list with
itself
• For each pair of (di , d j )  P(t )

Emit <(i, j ), sim(i, j )  wt ,di * wt ,d j 
• Reduce
– For each
key  (i, j ),
Sim(di , d j )   sim(i, j )

• Cartesian product of postings list with itself may produce a large
set of intermediate keys
• Modify the above algorithm as follows
– Split the corpus into blocks of documents and query against postings list
– Map
• Input term postings list t , P(t )
• Load blocks of documents in memory
• For each document d i in block
– If t  di compute partial score for each element
– Reduce
• For each document, aggregate the partial scores from mappers for all other
documents
• Can reduce intermediate keys by implementing term limits when
documents are loaded into memory

Low-rank matrix factorizations
• Useful for analyzing patterns in dyadic data
Vm x n  Wm x d H d x n , d min(m, n)

• Given an application dependent loss function, find
arg min L(V ,W , H )
W ,H

• Most loss functions are sums of local losses
L 
( i , j )Z
l (Vij ,Wij , H ij )

• Use stochastic gradient descent (SGD) for this
factorization

SGD for matrix factorization
Training set Z  Vij | Vij !  ?  , initial values W0 , H 0
While not converged, do
Select a training point (i, j )  Z uniformly at random
Lij (W , H )

Wi*  Wi*   n N
'
l (Vij ,Wi* , H * j ) For local losses,
WWi ' k
i*
depend only on
Lij (W , H )

H* j  H* j   n N l (Vij ,Wi* , H * j ) l (Vij , Wi* , H * j )
HjH kj '
*

Wi*  Wi*
'

end while

R. Gemulla, P.J. Haas, E. Nijkamp, Y. Sismanis, IBM Tech Report , 2011

SGD for matrix factorization in
MapReduce
• Main ideas
– Local loss depends only on Vij ,Wi* , H* j
– If sub-matrices do not share rows and columns, they can be
factored independently and factors combined.
Z b  W bH b
H 1
H 2
H d

W 1 
 W  Z 0 ... 0 
1 11
 2
 2   W 
 W  0 Z 22 ...  W   
H  H 1 , H 2 ...H d 
  0   
   W d 
 W d  0 dd 
 
  0 Z 

– Stratify the input matrix such that each stratum can be processed
in a distributed manner

MapReduce
• Stratify the input matrix (dropping missing values) into subsets
Z s , Z s2 ,
1
Z sd such that
i  i ', j  j ' (i, j )  Z sb1 , (i ', j ')  Z sb 2 , (b1  b2)
• Stratification
– Randomly permute the rows and columns of the input matrix

Z 11 n/d

 V11 V12 V1n  For a permutation j1 , j2 .... jd
  of 1...d
m/d  V21 V22 V2 n 
Z s  Z 1 j1 Z 2 j2 Z 3 j3 ... Z djd
 

V V 
 m1 m 2 Vmn 


MapReduce
Training set Z , initial values W0 , H 0 , cluster size d
W  W0 , H  H 0
Block Z / W / H into d x d / d x 1/1 x d blocks
While not converged, do Epochs
Pick step size 
For s  1 d do Sub-epoch


Pick d blocks Z 1 j1 , Z 2 j2 ,....Z djd  to form a stratum Z s

For b  1 d do Machines
Run SGD on points in Z bjb with step size 
end for
end for
end while


Frequent Itemset Mining
• Set of items I  {a1 , a2 ...aM } and D  {T1 , T2 ...TN }
where Ti  subsets of I
• Pattern A  I is frequent if
support( A)  
• Problem
– Find all complete frequent item-sets of D
• Divide and conquer approach
– Patterns containing A can be found using only transactions
containing A.
– Filter transactions with A – conditional database (CDB) of A
– Find patterns containing A in CDB(A)

• Construct a Frequent Pattern (FP) Tree
– Keep only items with frequency above the minimum support
– Sort each transaction in descending order of frequent items
– Add each sorted transaction to an item prefix tree
– Each node in the FP tree is an item
• Node has count of transactions with that item in that path
• Nodes of same items in different paths are linked together

• FPGrowth algorithm
– Start from CDB of single frequent item
– Build FP Tree of CDB
– Mine frequent patterns from CDBs using recursion
• Recursion terminates when CDB has a single path
• Frequent pattern = Union of all nodes in this tree with support = min. support
of nodes in this tree

Mining frequent patterns without candidate generation, J. Han, J. Pei,Y. Yin. 2000, In SIGMOD, 2000.

f:4
p: { f c a m / f c a m / c b }
c:4
facdgimp a:3 fcamp
b:3
m: { f c a / f c a / f c a b }
m:3
abcflmo p:3 fcabm
b: { f c a / f c }
o:2
bfhjo d:1 fb
e:1
a: { f c / f c / f c }
g:1
bcksp h:1 cbp
i:1
c: { f / f / f }
k:1
afcelpmn l:1 fcamp
f: {}
n:1

Original Sorted Conditional databases of
Frequent
transactions transactions Frequent items
items

Frequent Itemset Mining in
MapReduce
• Identifying frequent items = 1 MapReduce job
– Find the set of items and the associated frequency
• Prune this frequent items list keeping only items more
frequent than minimum support
• Mine subsequent projected CDBs in MapReduce
iterations (Li et al. 2008)
– Project transactions in CDB by least frequent item in the mapper
– Breadth first search of the FP Tree using a MapReduce iteration
– Once projected CDB fits in memory of reducer
• Run FPGrowth algorithm in reducer
• No more growth of the sub-tree

Frequent Itemset Mining in
MapReduce
p: { f c a m / f c a m / c b }
pc: {}
pc:3
p
D|p c m: { f c a / f c a / f c a b }
D|am D|cam am: { f c / f c / f c }
a
m cm: { f / f / f }
D|m c cam: { f / f / f }
D|cm fcam: {}
b mf:3, mc:3, ma:3, mfc:3,
D D|b mfa:3, mca:3, mfca:3
c b: { f c a / f c }
a
D|a D|ca a: { f c / f c / f c }
ca: { f / f / f }
c fa: {}
D|c af:3, ac:3, afc: 3
c: { f / f / f }
fc: {}
MR Iteration 1 MR Iteration 2 MR Iteration 3 cf:3

Graph Algorithms
• Ubiquitous in web applications
– Web-graph, Social network graph, User-item
graph
• Typical problems
– Popularity (e.g. PageRank)
– Shortest paths
– Clustering, semi-clustering etc.

Graph algorithms in MapReduce
• Vertex centric approach
– Work with the adjacency list of each vertex
– Especially useful for sparse adjacency matrices
• Breadth first search
– Each MR iteration advances the horizon by one
level
• In each iteration
1. Compute on each vertex
2. Pass values to connected vertices for aggregation
in the reducer
3. Pass the adjacency list of each node to the reducer

Breadth first search on Graphs in
MapReduce
3 • Easy (iterative)
2 implementations exist for
some common algorithms
– Single source shortest path
1 2 3
– PageRank

2 3

3

MR Iteration 1 MR Iteration 2

Single source shortest path in
MapReduce
• Find the shortest path from a given node to any reachable node
• Given a start node:
– Distance to adjacent nodes = 1
– Distance to any other node reachable from a set of nodes S
DistanceTo(n) = 1 + min(DistanceTo(m), m  S)

• Map • Reduce
• Node “n” • “p”, “D+1” from all nodes
• D, Adjacency list of “n” pointing to “p”
Pass the
– Output: graph from • “n”, Adjacency list of “n”
• For each node “p” in 1 iteration – Output:
adjacency list To the next • “p”, min(“D+1” from all
– <p, (D+1)> nodes pointing to “p”)
• <n,Adjacency list of “n”> • “n”, Adjacency list of “n”

PageRank
• Given a node A
PR(Ti )
PR( A)  d  (1  d ) * 
{Ti :Ti  A} C (Ti )

d  random jump probability
Ti  node pointing to A
C (Ti )  out-degree of Ti

• Iterate this equation till convergence
• Driver program to check if the page rank for each
node has converged

PageRank in MapReduce
• In each iteration (i)
• Map • Reduce
• Node “n”, PRi-1(n) • <“p”, V from all nodes “n”
• Adjacency list of “n” pointing to “p”>
– Compute • Adjacency list of “n”
• V = PRi-1(n) / |Adjacency – Compute
list of n| • PRi(p) = Sum(V)
– Output: – Output:
• For each node “p” in • <p, PRi(p) >
adjacency list • <n,Adjacency list of “n”>
– <p, V>
• <n,Adjacency list of “n”>

Frameworks for graph algorithms
• MapReduce is not a good fit for graph algorithms
– 1 iteration for each level of the graph has large overheads
• “Bulk synchronous processing model” for graph
processing.
– Components – for either compute or storage
– Router – to deliver point to point messages
– Synchronization at periodic intervals (called supersteps) that are
atomic
• In each superstep, vertex can
– Receive messages sent by other vertices in previous superstep
– Compute using the data in that vertex and the received
messages
– Send messages to other vertices

Frameworks for graph algorithms
• Vertex can vote to go to halt state
• Computation stops when all vertices have voted to halt.
• Vertices can also mutate the graph
– Add/remove edges and other vertices
– Mutations implemented in next superstep
• Framework also supports aggregators
– Can maintain global summaries over the graph
– Values communicated to all vertices before the next
superstep
• Large scale graph processing tools leveraging Grid
– Pregel (in Google)
– Open source implementation Giraph
https://github.com/aching/Giraph

Sequential learning methods
• Some learning algorithms are inherently sequential in nature, e.g.,
– Stochastic Gradient Descent (SGD) minimization
– Conditional Maximum Entropy using SGD
– Perceptron

• Difficult to distribute sequential algorithms over data partitions
– Need frequent communication of intermediate parameter values

• Some sequential algorithms can be trained in a cluster environment.
– Theoretical and empirical analysis show that parameters
converge to the values from sequential training over all data

Sequential learning methods in
MapReduce
• Types of sequential learning in MapReduce
– Single M/R job:
• Learn parameters on each data partition in mappers over
multiple epochs
• Average the model parameters from all mappers in a reducer
– Multiple M/R jobs:
• Learn parameters on each data partition in each mapper for
1 epoch
• Average the model parameters from all mappers in a
reducer
• Start the next iteration for next epoch in the mapper with the
average parameter values from previous iteration
– Communicate between nodes
• Launch MPI on Hadoop cluster

Stochastic Gradient Descent (SGD)
methods
• Many learning algorithms involve optimizing an objective
function (maximizing log likelihood, minimizing root
mean square error etc.) over the training data to
determine the optimal parameters
w*  arg min  L( xi , y i , w)
itraining  data

w  w  * 
itraining  data
 w L(xi , y i , w)

• Stochastic Gradient techniques update the parameter
one example at a time
w  w  * w L( xi , y i , w)
• Parameter updates are inherently sequential

Parallelized SGD
• Partition the training data into multiple partitions, each
with T examples chosen at random
• Perform stochastic gradient updates on each data
partition separately with constant learning rate.
• Average the solutions between different machines.
• For large scale data, (Zinkevich et al. 2010) show that
– Parameter values converge to sequential estimates
– For k partitions, averaging the parameters reduces
variance by O(k 1 2 )
– Bias in parameter estimates decreases as well

Parallelized SGD in MapReduce

Map: Reduce:
In each mapper i 1...k Aggregate from all mappers:
Machines
wi ,0  0
1 k
v   wi ,t
Average across all
For t  1...T Data
k i 1 machines

wi ,t  wi ,(t 1)   *  w L( x, y , wi ,t 1 )
end for
end for

Parallelized SGD in MapReduce
• Multi-pass parallel SGD (Weimer, Rao, Zinkevich 2010)
– Divide the data randomly among all machines
ctj  t th example sent to j th machine
*
– Initialize weight vector w
– For i {1...T } iterations do Iterations
• For each machine j {1...k} do Machines
wi  w*
Shuffle data uniformly at random p :{1...m}  {1...m}
For each t {1...m} do Data
Initial value
for next w j  w j c p (t ) (w j )
j

end for
iteration
end for
1 k j
w  w
* Average across all machines in
k j 1 each iteration
end for

Conditional MaxEnt models
• Used in both binary and multi-class classification problems
• Commonly used in NLP and computer vision

S  {( x1 , y1 ), ( x2 , y2 )...( xm , ym )
1
pw ( y | x )  exp  w. ( x, y )  ,  ( x, y )  feature
Z ( x)
Z ( x)   exp  w. ( x, y) 
yY

1 m
w  arg min FS ( w)  arg min  w   log pw ( y | x)
2

w w m i 1
y  arg max pw ( y | x)
y

Conditional MaxEnt in MapReduce
• Mixture weighting method (Mann et al. 2009)
– Train a model in each of mappers using standard gradient ascent on a
M
subsample of the data.

k th mapper; wk  0
for t  1...T do
wk  wk   *  wk FS ( wk )
return w
– Average the weights from all the mappers in 1 reducer

M
w   k wk
M

k 1
k  0  k 1
– Mann et al. (2009) show that the mixture weighting1 estimate converges to the
k
sequential estimate

Perceptron algorithm
• Online algorithm used in NLP for structure prediction e.g.,
– Parsing, Named entity recognition, Machine translation etc.

Perceptron( D  {xi , yi })
w(0)  0; k  0
for n  1...N N epochs
for t  1... | D | Data

y '  arg max wk . f ( xt , y ' ) Predict using
y' current weights

if ( y '  yt )
Add weight to features for
w( k 1)  wk  f ( xt , yt )  f ( xt , y ' ) correct output
k  k 1 Remove weights to features for
incorrect output
return wk

Perceptron in MapReduce
• Iterative parameter mixing
– Train using data sub-group for 1 epoch in each mapper
– Average the weights in reducer
– Communicate back to mapper
– Train next epoch in mapper OneEpochPerceptron( D, w)
w(0)  w; k  0
w 0 for t  1... | D |
for n  1...N
y '  arg max wk . f ( xt , y ' )
w(i ,n ) = OneEpochPerceptron( Di , w) y'

w   i , n w (i ,n ) if ( y '  yt )
w( k 1)  wk  f ( xt , yt )  f ( xt , y ' )
i

return w
k  k 1
Average across all machines in return wk
each iteration

Perceptron in MapReduce
• McDonald et al. (2010) show that averaging
parameters after each epoch:
– Has as good or better performance as sequential
training on all data
– Trains better classifiers quicker than training
sequentially on all data
– Performs better than averaging parameters from
training model in each partition for multiple epochs to
convergence

Challenges for ML algorithms on
Hadoop
• Hadoop is optimized for large batch data processing
– Assumes data parallelism
– Ideal for shared nothing computing
• Many learning algorithms are iterative
– Incur significant overheads per iteration
• Multiple scans of the same data
– Typically once per iteration  high I/O overhead reading data
into mappers per iteration
– In some algorithms static data is read into mappers in each
iteration
• e.g. input data in k-means clustering.
• Need a separate controller outside the framework to:
– coordinate the multiple MapReduce jobs for each iteration
– perform some computations between iterations and at the end
– measure and implement stopping criterion

Challenges for ML algorithms on
Hadoop
• Incur multiple task initialization overheads
– Setup and tear down mapper and reducer tasks per iteration
• Transfer/shuffle static data between mapper and reducer
repeatedly
– Intermediate data is transferred through index/data files on local
disks of mappers and pulled by reducers
• Blocking architecture
– Reducers cannot start till all map jobs complete
• Availability of nodes in a shared environment
– Wait for mapper and reducer nodes to become available in each
iteration in a shared computing cluster

Iterative algorithms in MapReduce

Overhead per Iteration:
Data (each pass)

Pass Result
•Job setup
•Data Loading
•Disk I/O

Enhancements to Hadoop
• Many proposals to overcome these challenges
• All try to retain the core strengths of data partitioning and
fault tolerance of Hadoop to various degrees
• Proposed enhancements and alternatives to Hadoop
– Worker/Aggregator framework
– HaLoop
– MapReduce Online
– iMapReduce
– Spark
– Twister
– Hadoop ML
– …..

Worker/Aggregator framework
• Worker
- Load data in memory
- Iterate:
› Iterates over data using user specified functions
› Communicates state
› Waits for input state of next pass
• Aggregator
– Receive state from the workers
– Aggregate state using user specified functions
– Send state to all workers
• Communicate between workers and aggregators using TCP/IP
• Leverage the fault tolerance, and data locality of Hadoop

M. Weimer, S. Rao, M. Zinkevich, 2010, NIPS 2010 Workshop on Learning
on Cores, Clusters and Clouds

Parallelized SGD in
Worker/Aggregator

Advantages:

Final Result
Initial Data

•Schedule once per Job
•Data stays in memory
•P2P communication

102 9/7/2011

HaLoop
• Programming model and architecture for iterations
– New APIs to express iterations in the framework
• Loop-aware task scheduling
– Physically co-locate tasks that use the same data in different
iterations
– Remember association between data and node
– Assign task to node that uses data cached in that node
• Caching for loop invariant data:
– Detect invariants in first iteration, cache on local disk to reduce
I/O and shuffling cost in subsequent iterations
– Cache for Mapper inputs, Reducer Inputs, Reducer outputs
• Caching to support fixpoint evaluation:
– Avoids the need for a dedicated MR step on each iteration

HaLoop: Efficient Iterative Data Processing on Large Clusters by Yingyi Bu, Bill Howe, Magdalena Balazinska,
Michael D. Ernst. In VLDB'10

HaLoop vs. MapReduce
Application
Application

Framework

Framework

• HaLoop framework controls the loop
• First iteration is similar to that on Hadoop.
• Framework identifies data  node mappings, caches and indexes
for fast access, and controls looping
• Subsequent iterations leverage the above optimizations


New, additional
API HaLoop Design

Leverage
data
locality

Caching
for fast
Starts new access
MR jobs
repeatedly


HaLoop Programming API
Name Functionality
Map() & Reduce() Specify a map & reduce function
AddMap() & AddReduce() Specify a step in loop Iteration
SetDistanceMeasure() Specify a distance for results inputs
SetInput() Specify inputs to iterations
AddInvariantTable() Specify loop-invariant data
SetFixedPointThreshold() A loop termination condition Loop
SetMaxNumberOfIterations() Specify the max number of control
iterations
SetReducerInputCache() Enable/disable reducer input cache
SetReducerOutputCache() Enable/disable reducer output Cache
cache control
SetMapperInputCache() Enable/disable mapper input cache


k-means clustering in HaLoop
• k-means in HaLoop
1. Job job = new Job();
2. job.AddMap(Map_Kmeans,1);  Assign data point to closest
cluster
3. job.AddReduce(Reduce_Kmeans,1);  Re-compute centroids
4. job.SetDistanceMeasure(ResultDistance);
– # of changes in cluster membership
5. job.SetFixedPointThreshold(0.01);
6. job.SetMaxNumOfIterations(12);  Stopping criteria
7. job.SetInput(IterationInput);  Same input data to each iteration
8. job.SetMapperInputCache(true);
– Enable mapper input caching for mappers to read data from local
disk node
9. job.Submit();

MapReduce Online
• Pipeline data between operators as it is produced
– Decouple computation and data transfer schedules
– Intra-job:
• between mapper and reducer
– Inter-job:
• schedule multiple dependent jobs simultaneously
• between reducer of one job and mapper of next job
• “Push” data from producers instead of a “pull” by consumers
• Intermediate data is considered tentative till map job completes
– Also stored on disk for fault tolerance/recovery
• Reducer starts as soon as some data is available from mappers
– Can compute approximate answers from partial data
• Mappers and Reducers can also run continuously
– Enables stream processing

Mapreduce online, T. Condie, N. Conway, P. Alvaro, J. M. Hellerstein, K. Elmeleegy, R. Sears, 2010, NSDI'10,
Proceedings of the 7th USENIX conference on Networked systems design and implementation

iMapReduce
• Iterative processing
– Persistent map/reduce tasks
– Each reduce task has a locally connected
corresponding map task
• Maintain static data locally
– On local disk of mapper
• Asynchronous map execution
– Persistent socket between reducemap
– Completion of reduce triggers map
– Mappers do not need to wait

iMapReduce: A Distributed Computing Framework for Iterative Computation, Y. Zhang, Q. Gao, L. Gao,
C. Wang, DataCloud 2011

iMapReduce – Iterative Processing

iMapReduce – Asynchronous map
execution

T
I
M
E

MapReduce iMapReduce

Spark
• Open source cluster computing model:
– Different from MapReduce, but retains some basic character
• Optimized for:
– iterative computations
• Applies to many learning algorithms
– interactive data mining
• Load data once into multiple mappers and run multiple queries
• Programming model using working sets
– applications reuse intermediate results in multiple parallel operations
– preserves the fault tolerance of MapReduce
• Supports
– Parallel loops over distributed datasets
• Loads data into memory for (re)use in multiple iterations
– Access to shared variables accessible from multiple machines
• Implemented in Scala,
• www.spark-project.org

Spark: Cluster Computing with Working Sets. M. Zaharia, M. Chowdhury, M.J. Franklin, S. Shenker,
I. Stoica. 2010, USENIX HotCloud 2010.

Mahout
• Goal
– Create scalable, machine learning algorithms under the Apache license.
• Scalable:
– to large datasets
– business use cases
– community
• Contains both:
– Hadoop implementations of algorithms that scale linearly with data.
– Fast sequential (non MapReduce) algorithms
• Latest release is Mahout 0.5 on 27th May 2011 (circa Aug 4,
2011)
• Wiki:
– https://cwiki.apache.org/confluence/display/MAHOUT/Mahout+Wiki
• Mailing lists
– User, Developer, Commit notification lists
– https://cwiki.apache.org/confluence/display/MAHOUT/Mailing+Lists

Algorithms in Mahout
• Classification:
– Logistic Regression
– Naïve Bayes, Complementary Naïve Bayes
– Random Forests
• Clustering
– K-means, Fuzzy k-means
– Canopy
– Mean-shift clustering
– Dirichlet Process clustering
– Latent Dirichlet allocation
– Spectral clustering
• Parallel FP growth
• Item based recommendations
• Stochastic Gradient Descent (sequential)

Acknowledgment
Numerous wonderful colleagues!

Questions?

Exercise problem
• Problem:
– Predict the age of abalone as a function of physical attributes
– Useful for ecological and commercial fishing purposes
• Dataset:
– Dataset from the Marine Resources Division at the Department of
Primary Industry and Fisheries, Tasmania
– Attributes:
• Gender, Length, Diameter, Height, 4 different weights – 8 attributes
– Target:
• Number of Rings in shell
• Age (in years) = 1.5 + number of rings in shell
– At: http://www.stat.duke.edu/data-sets/rlw/abalone.dat
• Learn a linear relation between the age and the physical
attributes

Exercise dataset
• Original data sample size = 4177
• Generate larger dataset by replicating each record
– Add Gaussian noise for each feature with the sample variance
– Do not add variance for Gender and # of rings
– Replicate by factors of:
• 10x, 1k x, 8k x, 16k x, 32k x
• Datasets of about 40k, 4MM, 32 MM, 64MM and 128 MM records.
• For all attributes, compared to the original dataset, the
larger datasets have:
– same mean
– higher sample variance

Exercise: Model training
• Train a linear regression model
8
Rings   wi xi x0  1
i 0
w*  A1b
8 8
A   ( xi x ) T
i b   ( xi yi )
i 0 i 0

• Split the training data into 10 parts
• Mapper:
– Compute the matrix A and vector b on each partition
• Reducer
– Aggregate the values of A and b from all mappers
– Compute the weights w*  A1b

Exercise: Model Results
• For replication factor of 10x
– w[Sex] = 0.747
– w[Length] = 1.894
– w[Diameter] = 2.844
– w[Height] = 7.213
– w[Whole] = 0.311
– w[Shucked] = -0.558
– w[Viscera] = 0.840
– w[Shell] = 3.288
– w[1] = 5.046

Training Times: Sequential vs
Hadoop
9000

8000
Training Time (seconds)

7000
Hadoop Sequential
6000

5000

4000

3000

2000

1000

0
0 20 40 Data60 (MM records)
size 80 100 120 140

References
1. M. Kearns. Efficient noise-tolerant learning from
statistical queries. Journal of the ACM, Vol. 45, No. 6,
November 1998, pp. 983–1006.
2. C. Chu, S.K.Kim, Y. Lin, Y. Yu, G. Bradski, A.Y. Ng, K.
Olukotun, Map-Reduce for Machine Learning on
Multicore. In Proceedings of NIPS 2006, pp. 281-288.
3. W. Zhao, H. Ma, Q. He. Parallel K-Means Clustering
Based on MapReduce. CloudCom '09 Proceedings of
the 1st International Conference on Cloud Computing
2009, pp. 674-679
4. R. Ho. http://horicky.blogspot.com/2011/04/k-means-
clustering-in-map-reduce.html

References
5. Cluster Computing and MapReduce, Lecture 4.
http://www.youtube.com/watch?v=1ZDybXl212Q
6. A. McCallum, K. Nigam, L. Ungar. Efficient Clustering
of High Dimensional Data Sets with Application to
Reference Matching, Proceedings of the sixth ACM
SIGKDD international conference on Knowledge
discovery and data mining, 2000, pp.169-178
7. C. Elkan, 2011.
http://cseweb.ucsd.edu/~elkan/250B/logreg.pdf
8. B. Panda, J. S. Herbach, S. Basu, R. J. Bayardo,
PLANET: Massively Parallel Learning of Tree
Ensembles with MapReduce, 2009, Proceedings of
The Vldb Endowment - PVLDB, vol. 2, no. 2, pp. 1426-
1437.

References
9. J.S. Herbach, 2009.
http://fora.tv/2009/08/12/Josh_Herbach_PLANET_MapR
educe_and_Tree_Learning#fullprogram
10. R. Yan, J. Tesic, and J. R. Smith. Model-shared
subspace boosting for multi-label classification, 2007, In
Proceedings of the 13th ACM SIGKDD Intl. Conf. on
Knowledge discovery and data mining, pp 834-843.
11. R. Yan, M. Fleury, M. Merler, A. Natsev, J.R. Smith,
2009, Proceedings of the First ACM workshop on Large-
scale multimedia retrieval and mining, pp 35-42
12. J.D Basilico, M.A. Munson, T.G. Kolda, K.R. Dixon,
W.P.Kegelmeyer, COMET: A Recipe for Learning and
Using Large Ensembles on massive data, 2011,
http://arxiv.org/PS_cache/arxiv/pdf/1103/1103.2068v1.pd
f

References
13. S. Papadimitriou, J. Sun, DisCo: Distributed Co-
clustering with Map-Reduce, 2008,ICDM '08. Eighth
IEEE International Conference on Data Mining, pp
512-521
14. M.A. Zinkevich, M. Weimer, A. Smola, A., L. Li,
Parallelized Stochastic Gradient Descent, 2010, NIPS.
15. T. Elsayed, J. Lin, and D. Oard. Pairwise document
similarity in large collections with MapReduce, 2008, In
ACL, Companion Volume, pp 265-268, 2008
16. J. Lin, Brute Force and Indexed Approaches to
Pairwise Document Similarity Comparisons with
MapReduce., Proceedings of the 32nd Annual
International ACM SIGIR Conference on Research and
Development in Information Retrieval (SIGIR) 2009.

References
17. M. Weimer, S. Rao, M. Zinkevich, 2010, NIPS 2010
Workshop on Learning on Cores, Clusters and Clouds
18. HaLoop: Efficient Iterative Data Processing on Large
Clusters by Yingyi Bu, Bill Howe, Magdalena
Balazinska, Michael D. Ernst. In VLDB'10: The 36th
International Conference on Very Large Data Bases,
Singapore, 24-30 September, 2010.
19. G. Mann, R. McDonald, M. Mohri, N. Silberman, D. D.
Walker, 2009, in Advances in Neural Information
Processing Systems 22 (2009), edited by: Y. Bengio, D.
Schuurmans, J. Lafferty, C. K. I. Williams, A. Culotta
pp. 1231-1239.
20. R. McDonald, K. Hall, G. Mann, Distributed training
strategies for the structured perceptron , 2010, In
Human Language Technologies: The 2010 Annual
Conference of the North American Chapter of the
Association for Computational Linguistics (2010), pp.
456-464.

References
21. H. Li, Y. Wang, D. Zhang, M. Zhang, E.Y. Chang,
2008, In Proceedings of the 2008 ACM conference on
Recommender systems (2008), pp. 107-114.
22. R. Gemulla, P.J. Haas, E. Nijkamp, Y. Sismanis, IBM
Tech Report , 2011
http://www.almaden.ibm.com/cs/people/peterh/dsgdTe
chRep.pdf
23. Pregel: a system for large-scale graph processing, G.
Malewicz, M. H. Austern, A. J.C Bik, J. C. Dehnert, A.H
Horn, N. Leiser, G. Czajkowski, 2010, SIGMOD
'10 Proceedings of the 2010 international conference
on Management of data

References
24. Mapreduce online, T. Condie, N. Conway, P. Alvaro, J.
M. Hellerstein, K. Elmeleegy, R. Sears, 2010, NSDI'10,
Proceedings of the 7th USENIX conference on
Networked systems design and implementation
25. iMapReduce: A Distributed Computing Framework for
Iterative Computation, Y. Zhang, Q. Gao, L. Gao, C.
Wang, 2011, DataCloud 2011
26. Spark: Cluster Computing with Working Sets. M.
Zaharia, M. Chowdhury, M.J. Franklin, S. Shenker, I.
Stoica. 2010, USENIX HotCloud 2010.
27. Mining frequent patterns without candidate generation,
J. Han, J. Pei,Y. Yin. 2000, In SIGMOD, 2000.

• Controller
– Determines the state of the tree and grows it
• Decides if nodes are pure or have small data to become leaves
• Data fits in memory  Launch a MapReduce job to
grow the entire sub-tree in memory
• Data does not fit in memory  Launch a MapReduce job to find
candidate best splits
• Collect results from MR jobs and choose the best split for a node
• Update the Model File
– Periodically checkpoints the system

• Model File
– Contains the state of the tree constructed so far
– Used by the controller to check which nodes to split or grow next

• Maintain 2 queues
– MapReduceQueue (MRQ)
• Contains nodes for which data is too large to fit in memory
– InMemoryQueue (InMemQ)
• Contains nodes for which data fits in memory

• Initialization Task (MapReduce)
– Identifies candidate attribute values for node splits
– Continuous attributes
• Compute an approximate equi-depth histogram
• Boundary points of histogram used for potential splits
– Categorical attributes
• Identify attribute's domain
• Sort values by average values of Y and use this for ordering
– Generate a file with list of attributes to be used by other tasks

• 2 main MapReduce jobs
– MR_ExpandNodes
• Process nodes from the MRQ to find best split
• Output for each node:
– Candidate split positions for node along with
» Quality of split (using summary statistics)
» Predictions in left and right branches
» Size of data going into left and right branches
– MR_InMemory
• Process nodes from the InMemQ.
• For a given set of nodes N, complete tree induction at nodes
in N using the InMemoryGrow algorithm.

• Map function in MR_ExpandNodes
– Load the current model file M and set of nodes N
– For each record
• Determine if record is relevant to any of the nodes in N
• Add record to the summary statistics (SS) for node
• For each feature-value in record
– Add record to the summary statistics for node for split points “s” less than the
value in record “v”
– Output
Split ID
key  (n  N , x  Ordered  feature, s ); value  Tn , x  s  SS of

key  (n  N , x  Categorical  feature); value   v, Tn , x v 
candidate
splits
key  (n  N ); value  SS SS of parent node
 
Tn , x  s   SS    y,  y ,  1 2

 subgroup subgroup subgroup  SS for variance impurity

• InMemoryGrow
– Task to grow the entire subtree once the data for it fits
in memory
– Similar to parallel training
– Map
• Load the current model file
• For each record identify the node that needs to be grown,
• Output <Node_id, Record>
– Reduce
• Initialize the feature value file from Initialization task
• For each <Node_id, List<Record>> run the basic tree
growing algorithm on the records
• Output the best split for each node in the subtree

Modeling with Hadoop kdd2011

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