### Large-scale Machine Learning in Distributed Environments

Chih-Jen Lin

National Taiwan University eBay Research Labs

Tutorial at ACM ICMR, June 5, 2012

### Outline

1 Why distributed machine learning?

2 Distributed classification algorithms Kernel support vector machines Linear support vector machines Parallel tree learning

3 Distributed clustering algorithms k-means

Spectral clustering Topic models

4 Discussion and conclusions

### Why Distributed Machine Learning

The usual answer is that data are too big to be stored in one computer

Some say that because “Hadoop” and “MapReduce are buzzwords

No, we should never believe buzzwords

I will argue that things are more complicated than we thought

In this talk I will consider only machine learning in data-center environments

That is, clusters using regular PCs

I will not discuss machine learning in other parallel environments:

GPU, multi-core, specialized clusters such as supercomputers

Slides of this talk are available at

http://www.csie.ntu.edu.tw/~cjlin/talks/

icmr2012.pdf

### Let’s Start with An Example

Using a linear classifier LIBLINEAR (Fan et al., 2008) to train the rcv1 document data sets (Lewis et al., 2004).

# instances: 677,399, # features: 47,236 On a typical PC

$time ./train rcv1_test.binary Total time: 50.88 seconds

Loading time: 43.51 seconds For this example

loading time running time

### Loading Time Versus Running Time I

Let’s assume the memory hierarchy contains only disk

Assume # instances is l

Loading time: l × (a big constant)

Running time: l^{q} × (some constant), where q ≥ 1.

Running time often larger because q > 1 (e.g., q = 2 or 3) and

l^{q−1} > a big constant

### Loading Time Versus Running Time II

Traditionally machine learning and data mining papers consider only running time

For example, in this ICML 2008 paper (Hsieh et al., 2008), some training algorithms were compared for rcv1

### Loading Time Versus Running Time III

DCDL1 is what LIBLINEAR used

We see that in 2 seconds, final testing accuracy is achieved

But as we said, this 2-second running time is misleading

So what happened? Didn’t you say that
l^{q−1} > a big constant??

The reason is that when l is large, we usually can afford using only q = 1 (i.e., linear algorithm) Now we see different situations

### Loading Time Versus Running Time IV

- If running time dominates, then we should design algorithms to reduce number of operations

- If loading time dominates, then we should design algorithms to reduce number of data accesses

Distributed environment is another layer of memory hierarchy

So things become even more complicated

### Data in a Distributed Environment

One apparent reason of using distributed clusters is that data are too large for one disk

But in addition to that, what are other reasons of using distributed environments?

On the other hand, now disk is large. If you have several TB data, should we use one or several machines?

We will try to answer this question in the following slides

### Possible Advantages of Distributed Systems

Parallel data loading

Reading several TB data from disk ⇒ a few hours Using 100 machines, each has 1/100 data in its local disk ⇒ a few minutes

Fault tolerance

Some data replicated across machines: if one fails, others are still available

Of course how to efficiently/effectively do this is a challenge

### An Introduction of Distributed Systems I

Distributed file systems

We need it because a file is now managed at different nodes

A file split to chunks and each chunk is replicated

⇒ if some nodes fail, data still available

Example: GFS (Google file system), HDFS (Hadoop file system)

Parallel programming frameworks

A framework is like a language or a specification.

You can then have different implementations

### An Introduction of Distributed Systems II

Example:

MPI (Snir and Otto, 1998): a parallel programming framework

MPICH2 (Gropp et al., 1999): an implementation Sample MPI functions

MPI Bcast: Broadcasts to all processes.

MPI AllGather: Gathers the data contributed by each process on all processes.

MPI Reduce: A global reduction (e.g., sum) to the specified root.

MPI AllReduce: A global reduction and

sending result to all processes.

### An Introduction of Distributed Systems III

They are reasonable functions that we can think about

MapReduce (Dean and Ghemawat, 2008). A

framework now commonly used for large-scale data processing

In MapReduce, every element is a (key, value) pair Mapper: a list of data elements provided. Each element transformed to an output element

Reducer: values with same key presented to a single reducer

### An Introduction of Distributed Systems IV

See the following illustration from Hadoop Tutorial http:

//developer.yahoo.com/hadoop/tutorial

### An Introduction of Distributed Systems V

### An Introduction of Distributed Systems VI

Let’s compare MPI and MapReduce MPI: communication explicitly specified

MapReduce: communication performed implicitly In a sense, MPI is like an assembly language, but MapReduce is high-level

MPI: sends/receives data to/from a node’s memory MapReduce: communication involves expensive disk I/O

MPI: no fault tolerance

MapReduce: support fault tolerance

### An Introduction of Distributed Systems VII

Because of disk I/O, MapReduce can be inefficient for iterative algorithms

To remedy this, some modifications have been proposed

Example: Spark (Zaharia et al., 2010) supports - MapReduce and fault tolerance

- Cache data in memory between iterations MapReduce is a framework; it can have different implementations

### An Introduction of Distributed Systems VIII

For example, shared memory (Talbot et al., 2011) and distributed clusters (Google’s and Hadoop) An algorithm implementable by a parallel framework

6=

You can easily have efficient implementations

The paper (Chu et al., 2007) has the following title Map-Reduce for Machine Learning on Multicore The authors show that many machine learning algorithms can be implemented by MapReduce

### An Introduction of Distributed Systems IX

These algorithms include linear regression, k-means, logistic regression, naive Bayes, SVM, ICA, PCA, EM, Neural networks, etc

But their implementations are on shared-memory machines; see the word “multicore” in their title Many wrongly think that their paper implies that these methods can be efficiently implemented in a distributed environment. But this is wrong

### Evaluation I

Traditionally a parallel program is evaluated by scalability

(64, 530,474) (128, 1,060,938) (256, 2,121,863)

(number of machines, data size)

Speedup

Total time
Similarity matrix
Eigendecomposition
K−means
2^{8}

2^{7}

2^{5}
2^{6}

### Evaluation II

We hope that when (machines, data size) doubled, the speedup also doubled.

64 machines, 500k data ⇒ ideal speedup is 64 128 machines, 1M data ⇒ ideal speedup is 128 That is, a linear relationship in the above figure But in some situations we can simply check throughput.

For example, # documents per hour.

### Data Locality I

Transferring data across networks is slow.

We should try to access data from local disk Hadoop tries to move computation to the data.

If data in node A, try to use node A for computation But most machine learning algorithms are not

designed to achieve good data locality.

Traditional parallel machine learning algorithms distribute computation to nodes

This works well in dedicated parallel machines with fast communication among nodes

### Data Locality II

But in data-center environments this may not work

⇒ communication cost is very high

Now go back to machine learning algorithms

### Classification and Clustering I

They are the two major types of machine learning methods

Classification Clustering

Distributed system are more useful for which one?

### Classification and Clustering II

The answer is clustering

Clustering: if you have l instances, you need cluster all of them

Classification: you may not need to use all your training data

Many training data + a so so method may not be better than

Some training data + an advanced method Usually it is easier to play with advanced methods on one computer

### Classification and Clustering III

The difference between clustering and classification can also be seen on Apache Mahout, a machine learning library on Hadoop.

It has more clustering implementations than classification

See http://mahout.apache.org/

Indeed, some classification implementations in Mahout are sequential rather than parallel.

### A Brief Summary Now

Going to distributed or not is sometimes a difficult decision

There are many considerations

Data already in distributed file systems or not

The availability of distributed learning algorithms for your problems

The efforts for writing a distributed code The selection of parallel frameworks And others

We use some simple examples to illustrate why the decision is not easy.

### Example: A Multi-class Classification Problem I

At eBay (I am currently a visitor there), I need to train

55M documents in 29 classes

The number of features ranges from 3M to 100 M, depending on the settings

I can tell you that I don’t want to run it in a distributed environment

Reasons

### Example: A Multi-class Classification Problem II

- I can access machines with 75G RAM to run the data without problem

- Training is not too slow. Using one core, for 55M documents and 3M features, training multi-class SVM by LIBLINEAR takes only 20 minutes

- On one computer I can easily try various features.

From 3M to 100M, accuracy is improved. It won’t be easy to achieve this by using more data in a distributed cluster.

### Example: A Bagging Implementation I

Assume data is large, say 1TB. You have 10 machines with 100GB RAM each.

One way to train this large data is a bagging approach

machine 1 trains 1/10 data

2 1/10

... ...

10 1/10

Then use 10 models for prediction and combine results

### Example: A Bagging Implementation II

Reasons of doing so is obvious: parallel data loading and parallel computation

But it is not that simple if using MapReduce and Hadoop.

Hadoop file system is not designed so we can easily copy a subset of data to a node

That is, you cannot say: block 10 goes to node 75 A possible way is

1. Copy all data to HDFS

### Example: A Bagging Implementation III

2. Let each n/p points to have the same key (assume p is # of nodes). The reduce phase

collects n/p points to a node. Then we can do the parallel training

As a result, we may not get 1/10 loading time In Hadoop, data are transparent to users We don’t know details of data locality and communication

Here is an interesting communication between me and a friend (called D here)

### Example: A Bagging Implementation IV

Me: If I have data in several blocks and would like to copy them to HDFS, it’s not easy to specifically assign them to different machines

D: yes, that’s right.

Me: So probably using a poor-man’s approach is easier. I use USB to copy block/code to 10 machines and hit return 10 times

D: Yes, but you can do better by scp and ssh.

Indeed that’s usually how I do “parallel programming”

This example is a bit extreme

### Example: A Bagging Implementation V

We are not saying that Hadoop or MapReduce are not useful

The point is that they are not designed in particular for machine learning applications.

We need to know when and where they are suitable to be used.

Also whether your data are already in distributed systems or not is important

### Resources of Distributes Machine Learning

There are many books about Hadoop and MapReduce. I don’t list them here.

For things related to machine learning, a collection of recent works is in the following book

Scaling Up Machine Learning, edited by Bekkerman, Bilenko, and John Langford, 2011.

This book covers materials using various parallel environments. Many of them use distributed clusters.

### Outline

1 Why distributed machine learning?

2 Distributed classification algorithms Kernel support vector machines Linear support vector machines Parallel tree learning

3 Distributed clustering algorithms k-means

Spectral clustering Topic models

4 Discussion and conclusions

### Support Vector Machines I

A popular classification method developed in the past two decades (Boser et al., 1992; Cortes and Vapnik, 1995)

Training data {y_{i}, x_{i}}, x_{i} ∈ R^{n}, i = 1, . . . , l , y_{i} = ±1
l : # of data, n: # of features

SVM solves the following optimization problem min

w,b

w^{T}w
2 + C

l

X

i =1

max(0, 1 − y_{i}(w^{T}x_{i} + b))

w^{T}w/2: regularization term

### Support Vector Machines II

C : regularization parameter Decision function

sgn(w^{T}φ(x) + b)

φ(x): data mapped to a higher dimensional space

### Finding the Decision Function

w: maybe infinite variables

The dual problem: finite number of variables minα

1

2α^{T}Qα − e^{T}α

subject to 0 ≤ α_{i} ≤ C , i = 1, . . . , l
y^{T}α = 0,

where Q_{ij} = y_{i}y_{j}φ(x_{i})^{T}φ(x_{j}) and e = [1, . . . , 1]^{T}
At optimum

w =Pl

i =1α_{i}y_{i}φ(x_{i})

A finite problem: #variables = #training data

### Kernel Tricks

Q_{ij} = y_{i}y_{j}φ(x_{i})^{T}φ(x_{j}) needs a closed form
Example: x_{i} ∈ R^{3}, φ(x_{i}) ∈ R^{10}

φ(x_{i}) = [1,√

2(x_{i})_{1},√

2(x_{i})_{2},√

2(x_{i})_{3}, (x_{i})^{2}_{1},
(x_{i})^{2}_{2}, (x_{i})^{2}_{3},√

2(x_{i})_{1}(x_{i})_{2},√

2(x_{i})_{1}(x_{i})_{3},√

2(x_{i})_{2}(x_{i})_{3}]^{T}
Then φ(x_{i})^{T}φ(x_{j}) = (1 + x^{T}_{i} x_{j})^{2}.

Kernel: K (x, y) = φ(x)^{T}φ(y); common kernels:

e^{−γkx}^{i}^{−x}^{j}^{k}^{2}, (Gaussian and Radial Basis Function)
(x^{T}_{i} x_{j}/a + b)^{d} (Polynomial kernel)

### Computational and Memory Bottleneck I

The square kernel matrix. Assume the Gaussian kernel is taken

e^{−γkx}^{i}^{−x}^{j}^{k}^{2}
Then

O(l^{2}) memory and O(l^{2}n) computation
If l = 10^{6}, then

10^{12} × 8 bytes = 8TB

Existing methods (serial or parallel) try not to use the whole kernel matrix at the same time

### Computational and Memory Bottleneck II

Distributed implementations include, for example, Chang et al. (2008); Zhu et al. (2009)

We will look at ideas of these two implementations Because the computational cost is high (not linear), the data loading and communication cost is less a concern.

### The Approach by Chang et al. (2008) I

Kernel matrix approximation.

Original matrix Q with

Qij = yiyjK (xi, xj) Consider

Q = ¯¯ Φ^{T}Φ ≈ Q.¯

Φ ≡ [¯¯ x_{1}, . . . , ¯x_{l}] becomes new training data
Φ ∈ R¯ ^{d ×l}, d l . # features # data

Testing is an issue, but let’s not worry about it here

### The Approach by Chang et al. (2008) II

They follow Fine and Scheinberg (2001) to use incomplete Cholesky factorization

What is Cholesky factorization?

Any symmetric positive definite Q can be factorized as

Q = LL^{T},
where L ∈ R^{l ×l} is lower triangular

### The Approach by Chang et al. (2008) III

There are several ways to do Cholesky factorization.

If we do it columnwisely

L_{11}
L_{21}
L31

L_{41}
L_{51}

⇒

L_{11}
L_{21} L_{22}
L31 L32

L_{41} L_{42}
L_{51} L_{52}

⇒

L_{11}
L_{21} L_{22}
L31 L32 L33

L_{41} L_{42} L_{43}
L_{51} L_{52} L_{53}

and stop before it’s fully done, then we get

incomplete Cholesky factorization

### The Approach by Chang et al. (2008) IV

To get one column, we need to use previous columns:

L_{43}
L_{53}

needs Q_{43}
Q_{53}

−L_{41} L_{42}
L_{51} L_{52}

L_{31}
L_{32}

This matrix-vector product is parallelized. Each machine is responsible for several rows

Using d = √

l , they report the following training time

### The Approach by Chang et al. (2008) V

Nodes Image (200k) CoverType (500k) RCV (800k)

10 1,958 16,818 45,135

200 814 1,655 2,671

We can see that communication cost is a concern The reason they can get speedup is because the complexity of the algorithm is more than linear They implemented MPI in Google distributed environments

If MapReduce is used, scalability will be worse

### A Primal Method by Zhu et al. (2009) I

They consider stochastic gradient descent methods (SGD)

SGD is popular for linear SVM (i.e., kernels not used).

At the tth iteration, a training instance x_{i}_{t} is chosen
and w is updated by

w ← w − η_{t}∇^{S} 1

2kwk^{2}_{2} + C max(0, 1 − y_{i}_{t}w^{T}x_{i}_{t}),

∇^{S}: a sub-gradient operator; η: learning rate.

### A Primal Method by Zhu et al. (2009) II

The update rule becomes

If 1 − y_{i}_{t}w^{T}x_{i}_{t} > 0, then
w ← (1 − η_{t})w + η_{t}Cy_{i}_{t}x_{i}_{t}.

For kernel SVM, w cannot be stored. So we need to
store all η_{1}, . . . , η_{t}

The calculation of

w^{T}x_{i}_{t}
becomes

t−1

X

s=1

(some coefficient)K (x_{i}_{s}, x_{i}_{t}) (1)

### A Primal Method by Zhu et al. (2009) III

Parallel implementation.

If x_{i}_{1}, . . . , x_{i}_{t} distributedly stored, then (1) can be
computed in parallel

Two challenges

1. x_{i}_{1}, . . . , x_{i}_{t} must be evenly distributed to nodes,
so (1) can be fast.

2. The communication cost can be high
– Each node must have x_{i}_{t}

– Results from (1) must be summed up

Zhu et al. (2009) propose some ways to handle these two problems

### A Primal Method by Zhu et al. (2009) IV

Note that Zhu et al. (2009) use a more

sophisticated SGD by Shalev-Shwartz et al. (2011), though concepts are similar.

MPI rather than MapReduce is used

Again, if they use MapReduce, the communication cost will be a big concern

### Discussion: Parallel Kernel SVM

An attempt to use MapReduce is by Liu (2010) As expected, the speedup is not good

From both Chang et al. (2008); Zhu et al. (2009), we know that algorithms must be carefully designed so that time saved on computation can compensate communication/loading

### Outline

1 Why distributed machine learning?

2 Distributed classification algorithms Kernel support vector machines Linear support vector machines Parallel tree learning

3 Distributed clustering algorithms k-means

Spectral clustering Topic models

4 Discussion and conclusions

### Linear Support Vector Machines

By linear we mean kernels are not used

For certain problems, accuracy by linear is as good as nonlinear

But training and testing are much faster Especially document classification

Number of features (bag-of-words model) very large Recently linear classification is a popular research topic. Sample works in 2005-2008: Joachims (2006); Shalev-Shwartz et al. (2007); Hsieh et al.

(2008)

There are many other recent papers and software

### Comparison Between Linear and Nonlinear (Training Time & Testing Accuracy)

Linear RBF Kernel

Data set Time Accuracy Time Accuracy

MNIST38 0.1 96.82 38.1 99.70

ijcnn1 1.6 91.81 26.8 98.69

covtype 1.4 76.37 46,695.8 96.11

news20 1.1 96.95 383.2 96.90

real-sim 0.3 97.44 938.3 97.82

yahoo-japan 3.1 92.63 20,955.2 93.31 webspam 25.7 93.35 15,681.8 99.26 Size reasonably large: e.g., yahoo-japan: 140k instances and 830k features

### Comparison Between Linear and Nonlinear (Training Time & Testing Accuracy)

Linear RBF Kernel

Data set Time Accuracy Time Accuracy

MNIST38 0.1 96.82 38.1 99.70

ijcnn1 1.6 91.81 26.8 98.69

covtype 1.4 76.37 46,695.8 96.11

news20 1.1 96.95 383.2 96.90

real-sim 0.3 97.44 938.3 97.82

yahoo-japan 3.1 92.63 20,955.2 93.31 webspam 25.7 93.35 15,681.8 99.26 Size reasonably large: e.g., yahoo-japan: 140k instances and 830k features

### Comparison Between Linear and Nonlinear (Training Time & Testing Accuracy)

Linear RBF Kernel

Data set Time Accuracy Time Accuracy

MNIST38 0.1 96.82 38.1 99.70

ijcnn1 1.6 91.81 26.8 98.69

covtype 1.4 76.37 46,695.8 96.11

news20 1.1 96.95 383.2 96.90

real-sim 0.3 97.44 938.3 97.82

yahoo-japan 3.1 92.63 20,955.2 93.31 webspam 25.7 93.35 15,681.8 99.26 Size reasonably large: e.g., yahoo-japan: 140k instances and 830k features

### Parallel Linear SVM I

It is known that linear SVM or logistic regression can easily train millions of data in a few seconds on one machine

This is a figure shown earlier

### Parallel Linear SVM II

Training linear SVM is faster than kernel SVM because w can be maintained

Recall that SGD’s update rule is

If 1 − y_{i}_{t}w^{T}x_{i}_{t} > 0, then

w ← (1 − η_{t})w + η_{t}Cy_{i}_{t}x_{i}_{t}. (2)
For linear, we directly calculate

w^{T}x_{i}_{t}

### Parallel Linear SVM III

For kernel, w cannot be stored. So we need to store
all η_{1}, . . . , η_{t−1}

t−1

X

s=1

(some coefficient)K (x_{i}_{s}, x_{i}_{t})
For linear SVM, each iteration is cheap.

It is difficult to parallelize the code Issues for parallelization

- Many methods (e.g., stochastic gradient descent or coordinate descent) are inherently sequential - Communication cost is a concern

### Simple Distributed Linear Classification I

Bagging: train several subsets and ensemble results;

we mentioned this approach in earlier discussion - Useful in distributed environments; each node ⇒ a subset

- Example: Zinkevich et al. (2010) Some results by averaging models

yahoo-korea kddcup10 webspam epsilson

Using all 87.29 89.89 99.51 89.78

Avg. models 86.08 89.64 98.40 88.83

### Simple Distributed Linear Classification II

Using all: solves a single linear SVM

Avg. models: each node solves a linear SVM on a subset

Slightly worse but in general OK

### ADMM by Boyd et al. (2011) I

Recall the SVM problem (bias term b omitted) minw

w^{T}w
2 + C

l

X

i =1

max(0, 1 − yiw^{T}xi)
An equivalent optimization problem

w1,...,wminm,z

1

2z^{T}z + C

m

X

j =1

X

i ∈B_{j}

max(0, 1 − y_{i}w^{T}_{j} x_{i})+

ρ 2

m

X

j =1

kw_{j} − zk^{2}
subject to w_{j} − z = 0, ∀j

### ADMM by Boyd et al. (2011) II

The key is that

z = w_{1} = · · · = w_{m}
are all optimal w

This optimization problem was proposed in 1970s,
but is now applied to distributed machine learning
Each node has B_{j} and updates w_{j}

Only w_{1}, . . . , w_{m} must be collected
Data not moved

Still, communication cost at each iteration is a concern

### ADMM by Boyd et al. (2011) III

We cannot afford too many iterations

An MPI implementation is by Zhang et al. (2012) I am not aware of any MapReduce implementation yet

### Vowpal Wabbit (Langford et al., 2007) I

It started as a linear classification package on a single computer

After version 6.0, Hadoop support has been provided Parallel strategies SGD initially and then LBFGS (quasi Newton)

The interesting point is that it argues that AllReduce is a more suitable operation than MapReduce

What is AllReduce?

Every node starts with a value and ends up with the sum at all nodes

### Vowpal Wabbit (Langford et al., 2007) II

In Agarwal et al. (2012), the authors argue that many machine learning algorithms can be

implemented using AllReduce LBFGS is an example

In the following talk

Scaling Up Machine Learning

the authors train 17B samples with 16M features on 1K nodes ⇒ 70 minutes

### The Approach by Pechyony et al. (2011) I

They consider the following SVM dual minα

1

2α^{T}Qα − e^{T}α

subject to 0 ≤ α_{i} ≤ C , i = 1, . . . , l
This is the SVM dual without considering the bias
term “b”

Ideas similar to ADMM
Data split to B_{1}, . . . , B_{m}

Each node responsible for one block

### The Approach by Pechyony et al. (2011) II

If a block of variables B is updated and others B ≡ {1, . . . , l }\B are fixed, then the sub-problem is¯

1

2(α + d)^{T}Q(α + d) − e^{T}(α + d)

= 1

2d^{T}_{B}Q_{BB}d_{B} + (Q_{B,:}α)^{T}d_{B} − e^{T}d_{B} + const
(3)

If

w =Pl

i =1α_{i}y_{i}x_{i}

is maintained during iterations, then (3) becomes 1

2d^{T}_{B}QBBdB + w^{T}X_{B,:}^{T} dB − e^{T}dB

### The Approach by Pechyony et al. (2011) III

They solve 1

2d^{T}_{B}_{i}Q_{B}_{i}_{B}_{i}d_{B}_{i} + w^{T}X_{B}^{T}_{i}_{,:}d_{B}_{i} − e^{T}d_{B}_{i}, ∀i
in parallel

They need to collect all d_{B}_{i} and then update w
They have a MapReduce implementation

Issues:

No convergence proof yet

### Outline

1 Why distributed machine learning?

3 Distributed clustering algorithms k-means

Spectral clustering Topic models

4 Discussion and conclusions

### Parallel Tree Learning I

We describe the work by Panda et al. (2009) It considers two parallel tasks

- single tree generation - tree ensembles

The main procedure of constructing a tree is to decide how to split a node

This becomes difficult if data are larger than a machine’s memory

Basic idea:

### Parallel Tree Learning II

A B

C D

If A and B are finished, then we can generate C and D in parallel

But a more careful design is needed. If data for C can fit in memory, we should generate all

subsequent nodes on a machine

### Parallel Tree Learning III

That is, when we are close to leaf nodes, no need to use parallel programs

If you have only few samples, a parallel

implementation is slower than one single machine The concept looks simple, but generating a useful code is not easy

The authors mentioned that they face some challenges

- “MapReduce was not intended ... for highly

iterative process .., MapReduce start and tear down costs were primary bottlenecks”

### Parallel Tree Learning IV

- “cost ... in determining split points ... higher than expected”

- “... though MapReduce offers graceful handling of failures within a specific MapReduce ..., since our computation spans multiple MapReduce ...”

The authors address these issues using engineering techniques.

In some places they even need RPCs (Remote Procedure Calls) rather than standard MapReduce For 314 million instances (> 50G storage), in 2009 they report

### Parallel Tree Learning V

nodes time (s) 25 ≈ 400 200 ≈ 1,350

This is good in 2009. At least they trained a set where one single machine cannot handle at that time

The running time does not decrease from 200 to 400 nodes

This study shows that

### Parallel Tree Learning VI

- Implementing a distributed learning algorithm is not easy. You may need to solve certain engineering issues

- But sometimes you must do it because of handling huge data

### Outline

1 Why distributed machine learning?

3 Distributed clustering algorithms k-means

Spectral clustering Topic models

4 Discussion and conclusions

### k-means I

One of the most basic and widely used clustering algorithms

The idea is very simple.

Finding k cluster centers and assign each data to the cluster of its closest center

### k-means II

Algorithm 1 k-means procedure

1 Find initial k centers

2 While not converge

- Find each point’s closest center

- Update centers by averaging all its members

We discuss difference between MPI and MapReduce implementations of k-means

### k-means: MPI implementation I

Broadcast initial centers to all machines While not converged

Each node assigns its data to k clusters and compute local sum of each cluster

An MPI AllReduce operation obtains sum of all k clusters to find new centers

Communication versus computation:

If x ∈ R^{n}, then

transfer kn elements after kn × l /p operations, l : total number of data and p: number of nodes.

### k-means: MapReduce implementation I

We describe one implementation by Thomas Jungblut

http:

//codingwiththomas.blogspot.com/2011/05/

k-means-clustering-with-mapreduce.html You don’t specifically assign data to nodes

That is, data has been stored somewhere at HDFS Each instance: a (key, value) pair

key: its associated cluster center value: the instance

### k-means: MapReduce implementation II

Map:

Each (key, value) pair find the closest center and update the key

Reduce:

For instances with the same key (cluster), calculate the new cluster center

As we said earlier, you don’t control where data points are.

Therefore, it’s unclear how expensive loading and communication is.

### Outline

1 Why distributed machine learning?

3 Distributed clustering algorithms k-means

Spectral clustering Topic models

4 Discussion and conclusions

### Spectral Clustering I

Input: Data points x_{1}, . . . , x_{n}; k: number of desired
clusters.

1 Construct similarity matrix S ∈ R^{n×n}.

2 Modify S to be a sparse matrix.

3 Compute the Laplacian matrix L by
L = I − D^{−1/2}SD^{−1/2},

4 Compute the first k eigenvectors of L; and construct
V ∈ R^{n×k}, whose columns are the k eigenvectors.

### Spectral Clustering II

5 Compute the normalized matrix U of V by
Uij = V_{ij}

q Pk

r =1V_{ir}^{2}

, i = 1, . . . , n, j = 1, . . . , k.

6 Use k-means algorithm to cluster n rows of U into k groups.

Early studies of this method were by, for example, Shi and Malik (2000); Ng et al. (2001)

We discuss the parallel implementation by Chen et al.

(2011)

### MPI and MapReduce

Similarity matrix

Only done once: suitable for MapReduce
But size grows in O(n^{2})

First k Eigenvectors

An iterative algorithm called implicitly restarted Arnoldi

Iterative: not suitable for MapReduce MPI is used but no fault tolerance

### Sample Results I

2,121,863 points and 1,000 classes

(64, 530,474) (128, 1,060,938) (256, 2,121,863)

(number of machines, data size)

Speedup

Total time
Similarity matrix
Eigendecomposition
K−means
2^{8}

2^{7}

2^{5}
2^{6}

### Sample Results II

We can see that scalability of eigen decomposition is not good

Nodes Similarity Eigen kmeans Total Speedup 16 752542s 25049s 18223s 795814s 16.00 32 377001s 12772s 9337s 399110s 31.90 64 192029s 8751s 4591s 205371s 62.00 128 101260s 6641s 2944s 110845s 114.87 256 54726s 5797s 1740s 62263s 204.50

### How to Scale Up?

We can see two bottlenecks

- computation: O(n^{2}) similarity matrix
- communication: finding eigenvectors

To handle even larger sets we may need to modify the algorithm

For example, we can use only part of the similarity matrix (e.g., Nystr¨om approximation)

Slightly worse performance, but may scale up better The decision relies on your number of data and other considerations

### Outline

1 Why distributed machine learning?

3 Distributed clustering algorithms k-means

Spectral clustering Topic models

4 Discussion and conclusions

### Latent Dirichlet Allocation I

Basic idea

each word wij ⇒ an associated topic z_{ij}
For a query

“ice skating”

LDA (Blei et al., 2003) can infer from “ice” that

“skating” is closer to a topic “sports” rather than a topic “computer”

The LDA model

### Latent Dirichlet Allocation II

p(w, z, Θ, Φ|α, β) =

m

Y

i =1
m_{i}

Y

j =1

p(w_{ij}|z_{ij}, Φ)p(z_{ij}|θ_{i})

" _{m}
Y

i =1

p(θ_{i}|α)

#

k

Y

j =1

p(φ_{j}|β)

w_{ij}: j th word from i th document

z_{ij}: the topic

p(w_{ij}|z_{ij}, Φ) and p(z_{ij}|θ_{i}): multinomial distributions
That is, w_{ij} is drawn from z_{ij}, Φ and z_{ij} is drawn
from θ_{i}

p(θ_{i}|α), p(φ_{j}|β): Dirichlet distributions

### Latent Dirichlet Allocation III

α, β: prior of Θ, Φ, respectively

Maximizing the likelihood is not easy, so Griffiths and Steyvers (2004) propose using Gipps sampling to iteratively estimate the posterior p(z|w)

While the model looks complicated, Θ and Φ can be integrated out to

p(w, z|α, β)

Then at each iteration only a counting procedure is needed

We omit details but essentially the algorithm is

### Latent Dirichlet Allocation IV

Algorithm 2 LDA Algorithm For each iteration

For each document i

For each word j in document i Sampling and counting

Distributed learning seems straightforward - Divide data to several nodes

- Each node counts local data - Models are summed up

### Latent Dirichlet Allocation V

However, an efficient implementation is not that simple

Some existing implementations

Wang et al. (2009): both MPI and MapReduce Newman et al. (2009): MPI

Smola and Narayanamurthy (2010): Something else Smola and Narayanamurthy (2010) claim higher throughputs.

These works all use same algorithm, but implementations are different

### Latent Dirichlet Allocation VI

A direct MapReduce implementation may not be efficient due to I/O at each iteration

Smola and Narayanamurthy (2010) use quite sophisticated techniques to get high throughputs - They don’t partition documents to several machines. Otherwise machines need to wait for synchronization

- Instead, they consider several samplers and synchronize between them

- They use memcached so data stored in memory rather than disk

### Latent Dirichlet Allocation VII

- They use Hadoop streaming so C++ rather than Java is used

- And some other techniques

We can see that a efficient implementation is not easy

### Conclusions

Distributed machine learning is still an active research topic

It is related to both machine learning and systems While machine learning people can’t develop

systems, they need to know how to choose systems An important fact is that existing distributed

systems or parallel frameworks are not particularly designed for machine learning algorithms

Machine learning people can

- help to affect how systems are designed - design new algorithms for existing systems

### Acknowledgments

I thank comments from Wen-Yen Chen Dennis DeCoste Alex Smola

Chien-Chih Wang Xiaoyun Wu Rong Yen

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