moving toward deep learning algorithms on hpcc systems

Moving Towards Deep Learning Algorithms on HPCC Systems

Maryam M. Najafabadi

Overview

• L-BFGS

• HPCC Systems

• Implementation of L-BFGS on HPCC Systems

• SoftMax

• Sparse Autoencoder

• Toward Deep Learning

2

Mathematical optimization

• Minimizing/Maximizing a function

Minimum

3

Optimization Algorithms in Machine Learning

• Linear Regression

Minimize Errors

4


• SVM

Maximize Margin

5


• Collaborative filtering

• K-means

• Maximum likelihood estimation

• Graphical models

• Neural Networks

• Deep Learning

6

Formulate Training as an Optimization Problem

• Training model: finding parameters that minimize some objective function

Define ParametersDefine an Objective

FunctionFind values for the parameters that

minimize the objective function

Cost term Regularization term

Optimization Algorithm

7

How they work

Search Direction

Step Length

8

Gradient Descent

• Step length• Constant value

• Search direction• Negative gradient

9

Gradient Descent



Small Step Length

10

Gradient Descent



Large Step Length

11

Newton Methods

• Step length• Use a line search

• Search direction• Use Curative Information (Inverse of Hessian Matrix)

12

Quasi Newton Methods

• Problem with large n in Newton methods• Calculation of inverse of Hessian matrix too expensive

• Continuously updating an approximation of the inverse of the Hessian matrix in each iteration

13

BFGS

• Broyden, Fletcher, Goldfarb, and Shanno

• Most popular Quasi Newton Method

• Uses Wolfe line search to find step length

• Needs to keep n×n matrix in memory

14

L-BFGS

• Limited-memory: only a few vectors of length n (m×n instead of n×n)

• m << n

• Useful for solving large problems (large n)

• More stable learning

• Uses curvature information to take a more direct route • faster convergence

15

How to use

• Define a function that calculates Objective value and Gradient

ObjectiveFunc (x, ObjectiveFunc_params, TrainData , TrainLabel)

16

Why L-BFGS?

• Toward Deep Learning• Optimization is heart of DL and many other ML algorithms

• Popular

• Advantages over SGD

17

HPCC Systems

• Open source, massive parallel-processing computing platform for big data processing and analytics

• LexisNexis Risk Solutions

• Uses commodity clusters of hardware running on top of the Linux operating system

• Based on DataFlow programming model

• THOR-ROXIE

• ECL

18

THOR

19

RECORD base

DataFlow Analysis

• Main focus is how the data is being changed

• A Graph represents a transformation on the data

• Each node is an operation

• Edges show the data flow

20

A DataFlow example

21

Id value

1 2

1 3

2 5

1 10

3 4

2 9

Id value

1 2

1 3

1 10

2 5

2 9

3 4

Id value

1 10

2 9

3 4

MAX

ECL

• Enterprise Control Language

• Compiled into optimized C++ code

• Declarative Language provides parallel and distributed DataFloworiented processing

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ECL




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Declarative

• What to accomplish, rather than How to accomplish

• You’re describing what you’re trying to achieve, without instructing how to do it

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ECL




25

ECL




26

27

Id value

1 2

1 3

2 5

1 10

3 4

2 9

28

Id value

2 5

2 9

Node 1

Node 2

READ

Id value

1 2

1 3

3 4

1 10

29

Id value

2 5

2 9

Node 1

Node 2

LOCAL SORT

Id value

1 2

1 3

1 10

3 4

Id value

2 5

2 9

Node 1

Node 2

READ

Id value

1 2

1 3

3 4

1 10

30

Id value

2 5

2 9

Node 1

Node 2

LOCAL SORT

Id value

1 2

1 3

1 10

3 4

Id value

2 5

2 9

Node 1

Node 2

READ

Id value

1 2

1 3

3 4

1 10

Id value

1 2

1 3

1 10

3 4

Id value

2 5

2 9

Node 1

Node 2

LOCAL GROUP

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Id value

2 5

2 9

Node 1

Node 2

LOCAL SORT

Id value

1 2

1 3

1 10

3 4

Id value

2 5

2 9

Node 1

Node 2

READ

Id value

1 2

1 3

3 4

1 10

Id value

1 2

1 3

1 10

3 4

Id value

2 5

2 9

Node 1

Node 2

LOCAL GROUP

Id value

1 10

3 4

Id value

2 9

Node 1

Node 2

LOCAL AGG/MAX

Back to L-BFGS

• Minimize f(x)

• Start with an initialized x : x0

• Repeatedly update: xk+1 = xk + αkpk

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Wolfe line search L-BFGS

• If x too large it does not fit in memory of one machine

• Needs m × n memory

• Distribute x on different machines

• Try to do computations locally

• Do global computations as necessary

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34






35

. . .






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. . .

. . .

• Dot Product

38

1, 3, 6, 8, 10, 9, 1, 2, 3, 9, 8

2, 3, 3, 8, 3, 11, 1, 2, 5, 9, 5

• Dot Product

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1, 3, 6, 8

3, 11, 1, 2

10, 9, 1, 2 3, 9, 8

Node 1 Node 2 Node 3

5, 9, 5 2, 3, 3, 8

• Dot Product

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1, 3, 6, 8

3, 11, 1, 2

10, 9, 1, 2 3, 9, 8


5, 9, 5 2, 3, 3, 8

LCOAL Dot Product 120 134 136

• Dot Product

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1, 3, 6, 8

3, 11, 1, 2

10, 9, 1, 2 3, 9, 8


5, 9, 5 2, 3, 3, 8

LCOAL Dot Product

Global Summation

120 134 136

390

Using ECL for implementing L-BFGS

42

0.1, 0.3, 0.6, 0.8, 0.2, 0.7, 0.5, 0.5, 0.5, 0.3, 0.4, 0.6, 0.7, 0.7x


43

0.1, 0.3, 0.6, 0.8, 0.2, 0.7, 0.5, 0.5, 0.5, 0.3, 0.4, 0.6, 0.7, 0.7x


44

0.1, 0.3, 0.6, 0.8, 0.2, 0.7, 0.5, 0.5, 0.5, 0.3, 0.4, 0.6, 0.7, 0.7x

Node1 Node2 Node3 Node4

Node_id partition_values1 0.1, 0.3, 0.6, 0.82 0.2, 0.7, 0.5, 0.53 0.5, 0.3, 0.4, 0.64 0.7, 0.7


45

0.1, 0.3, 0.6, 0.8, 0.2, 0.7, 0.5, 0.5, 0.5, 0.3, 0.4, 0.6, 0.7, 0.7x

Node1 Node2 Node3 Node4


Node 1

Node 4

Node 2

Node 3


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0.1, 0.3, 0.6, 0.8, 0.2, 0.7, 0.5, 0.5, 0.5, 0.3, 0.4, 0.6, 0.7, 0.7x


Node 1

Node 4

Node 2

Node 3

Example of LOCAL operations

• Scale

47


• Scale

48


Node 1

Node 4

Node 2

Node 3

x


• Scale

49

Node_id partition_values1 1, 3, 682 2, 7, 5, 53 5, 3, 4, 64 7, 7

Node 1

Node 4

Node 2

Node 3

x_10

Example of Global operation

• Dot Product

50


Node 1

Node 4

Node 2

Node 3

Node_id partition_values1 1, 3, 682 2, 7, 5, 53 5, 3, 4, 64 7, 7

Node 1

Node 4

Node 2

Node 3

x x_10


• Dot Product

51


Node 1

Node 4

Node 2

Node 3

Node_id partition_values1 1, 3, 6, 82 2, 7, 5, 53 5, 3, 4, 64 7, 7

Node 1

Node 4

Node 2

Node 3

x x_10


• Dot Product

52

Node_id dot_value1 2.272 1.393 1.674 0.98

dot_local


• Dot Product

53

Node_id dot_value1 2.272 1.393 1.674 0.98

dot_local

L-BFGS based Implementations

• Softmax

• Sparse Autoencoder

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SoftMax Regression

• Generalizes logistic regression

• More than two classes

• MNIST -> 10 different classes

55

Formulate to an optimization problem

• Parameters• K × f variables

• Objective function• Generalize logistic regression objective function

• Define a function to calculate objective value and Gradient at a give point

56

SoftMax Results

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SoftMax Results

• Lshtc-large• 410 GB• 61 itr, 81 fun• 1 hour

• Wikipedia-medium• 1,048 GB• 12 itr, 21 fun• Half an hour

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400 Nodes

More Examples

• Parameter matrix in SoftMax: K × f

• Data Matrix: f × m

• Multiply these two matrix

• Result is K × m

59

60

K

f

f

m

If parameter matrix is small

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K

f

f

m

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Node1 Node2 Node3

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Node1 Node2 Node3

64

Node1 Node2 Node3

K

f

f

m1 m2 m3

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Node1 Node2 Node3

K

f

f

m1 m2 m3

LOCAL JOIN

K×m1 K×m2 K×m3

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Node1 Node2 Node3

K

f

f

m1 m2 m3

LOCAL JOIN

K×m1 K×m2 K×m3K×m

If both matrices big

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K

f

f

m


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f1

f

m2

K

m1m3

f2 f3


69

f1

f

m

K

f2 f3

f1

f2

f3

K×m


70

f1

f

m

K

f2 f3

f1

f2

f3

K×m K×m


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f1

f

m

K

f2 f3

f1

f2

f3

K×m K×m K×m


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f1

f

m

K

f2 f3

f1

f2

f3

K×m K×m K×m ROLLUP

Sparse Autoencoder

• Autoencoder• Output is the same as the input

• Sparsity• constraint the hidden neurons to be inactive most of the time

• Stacking them up makes a Deep Network

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Formulate to an optimization problem

• Parameters• Weight and bias values

• Objective function• Difference between output and expected output

• Penalty term to impose sparsity

• Define a function to calculate objective value and Gradient at a give point

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Sparse Autoencoder results

• 10,000 samples of randomly 8×8 selected patches

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Sparse Autoencoder results

• MNIST dataset

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Toward Deep Learning

• Provide learned features from one layer to another sparse autoencoder

• …. Stack up to build a deep network

• Fine tuning • Using forward propagation to calculate cost value and back propagation to

calculate gradients

• Use L-BFGS to fine tune

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SUMMARY

• HPCC Systems allows implementation of Large-scale ML algorithms

• Optimization Algorithms an important aspect for advanced machine learning problems

• L-BFGS implemented on HPCC Systems• SoftMax• Sparse Autoencoder

• Implement other algorithms by calculating objective value and gradient

• Toward deep learning

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• HPCC Systems• https://hpccsystems.com/

• ECL-ML Library• https://github.com/hpcc-systems/ecl-ml

• My GitHub• https://github.com/maryamregister

• My Email• [email protected]

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https://hpccsystems.com/

https://github.com/hpcc-systems/ecl-ml

https://github.com/maryamregister

Questions?

Thank You

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moving toward deep learning algorithms on hpcc systems

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