pattern classification - michigan state university

Pattern Pattern

ClassificationClassification

All materials in these slides were taken All materials in these slides were taken

from from

Pattern Classification (2nd ed) by R. O. Pattern Classification (2nd ed) by R. O.

Duda, P. E. Hart and D. G. Stork, John Duda, P. E. Hart and D. G. Stork, John

Wiley & Sons, 2000Wiley & Sons, 2000

with the permission of the authors and with the permission of the authors and

the publisherthe publisher

Chapter 6: Multilayer Neural Networks Chapter 6: Multilayer Neural Networks

(Sections 6.1(Sections 6.1--6.3)6.3)

• Introduction

• Feedforward Operation and Classification

• Backpropagation Algorithm

Artificial Neural Net ModelsArtificial Neural Net Models

• Massive parallelism is essential for high

performance cognitive tasks (speech & image

recognition). Humans need only a few msec. for

most cognitive tasks

• Design a massively parallel architecture

composed of many simple processing elements

interconnected to achieve certain collective

computational capabilities

• Also known as connectionist models and parallel

distributed processing (PDP) models

• Derive inspiration from “knowledge” of biological

neural nets

•

Natural Neural Net ModelsNatural Neural Net Models

• Human brain consists of very large number of

neurons (between 10**10 to 10**12)

• No. of interconnections per neuron is between 1K

to 10K

• Total number of interconnections is about 10**14

• Damage to a few neurons or synapse (links)

does not appear to impair overall performance

significantly (robustness)

Artificial Neural Net ModelsArtificial Neural Net Models

• Artificial Neural nets are specified by

• Net topology

• Node (processor) characteristics

• Training/learning rules

• We consider only feedforward multilayer

networks

• These networks essentially implement a non-

parametric non-linear classifier

Linear Discriminant Functions• A discriminant function that is a linear combination of

input features can be written as

Weight

vector

Weight

vector

Bias or Threshold

weight

Bias or Threshold

weight

Sign of the

function

value gives

the class

label

Sign of the

function

value gives

the class

label

Pattern Classification, Chapter 6

6

IntroductionIntroduction

• Goal: Classify objects by learning nonlinearity

• There are many problems for which linear discriminants are not sufficient for minimum error

• The central difficulty is the choice of the appropriate

nonlinear functions

• A “brute” approach might be to select a complete basis set such as all polynomials; such a classifier

would require too many parameters to be determined

from a limited number of training samples


7

• There is no automatic method for determining the

nonlinearities when no information is provided to the classifier

• In using the multilayer neural Networks or multilayer

Perceptrons, the form of the nonlinearity is learned from the training data

• Multilayer networks can, in principle, provide the optimal solution to an arbitrary classification problem

• Nothing “magical” about multilayer neural networks; they implement linear discriminants but in a space where the

inputs have been mapped nonlinearly


8

Feedforward Operation and Feedforward Operation and

ClassificationClassification

• A three-layer neural network consists of an

input layer, a hidden layer and an output layer

interconnected by modifiable weights

represented by links between layers

• Multilayer neural network implements linear

discriminants, but in a space where the inputs

have been mapped nonlinearly

• Figure 6.1 shows a simple three-layer

network


9


10


11

• A single “bias unit” is connected to each unit in addition to the input units

• Net activation:

where the subscript i indexes units in the input layer, j in the hidden layer; wji denotes the input-to-hidden layer weights at the hidden unit j. (In neurobiology, such weights or connections are called “synapses”)

• Each hidden unit emits an output that is a nonlinear function of its activation, that is: yj = f(netj)

∑∑∑∑ ∑∑∑∑==== ====

≡≡≡≡====++++====d

1i

d

0i

t

jjii0jjiij ,x.wwxwwxnet


12Figure 6.1 shows a simple threshold function

• The function f(.) is also called the activation function or “nonlinearity” of a unit. There are more general activation functions with desirables properties

• Each output unit similarly computes its net activation based on the hidden unit signals as:

where the subscript k indexes units in the ouput layer and nH denotes the number of hidden units

<<<<−−−−

≥≥≥≥≡≡≡≡====

0net if 1

0net if 1)netsgn()net(f

∑∑∑∑ ∑∑∑∑==== ====

========++++====H Hn

1j

n

0j

t

kkjj0kkjjk ,y.wwywwynet


13

• The output units are referred as zk. An output unit

computes the nonlinear function of its net input,

emitting

zk = f(netk)

• In the case of c outputs (classes), we can view the

network as computing c discriminant functions

zk = gk(x); the input x is classified according to the

largest discriminant function gk(x) ∀ k = 1, …,c

• The three-layer network with the weights listed in

fig. 6.1 solves the XOR problem


14

• The hidden unit y1 computes the boundary:

≥ 0 ⇒ y1 = +1

x1 + x2 + 0.5 = 0

< 0 ⇒ y1 = -1

• The hidden unit y2 computes the boundary:

≤ 0 ⇒ y2 = +1

x1 + x2 -1.5 = 0

< 0 ⇒ y2 = -1

• Output unit emits z1 = +1 if and only if y1 = +1 and y2 = +1

Using the terminology of computer logic, the units are

behaving like gates, where the first hidden unit is an OR gate, the second hidden unit is an AND gate, and the output

unit implements

zk = y1 AND NOT y2 = (x1 OR x2) and NOT(x1 AND x2) =

x1 XOR x2

which provides the nonlinear decision of fig. 6.1


15

• General Feedforward Operation – case of c output units

• Hidden units enable us to express more complicated

nonlinear functions and extend classification capability

• Activation function does not have to be a sign function, it

is often required to be continuous and differentiable

• We can allow the activation in the output layer to be

different from the activation function in the hidden layer or

have different activation for each individual unit

• Assume for now that all activation functions are identical

c)1,...,(k

(1) wwxwfwfz)x(gHn

1j

0k

d

1i

0jijikjkk

====

++++

++++====≡≡≡≡ ∑∑∑∑ ∑∑∑∑

==== ====


16

• Expressive Power of multi-layer Networks

Question: Can every decision boundary be implemented by a three-layer network described by equation (1)?

Answer: Yes (due to A. Kolmogorov)

“Any continuous function from input to output can be implemented in a three-layer net, given sufficient number of hidden units nH, proper nonlinearities, and weights.”

Any continuous function g(x) defined on the unit cube can be represented in the following form

for properly chosen functions δj and βij

(((( )))) )2n];1,0[I(Ix )x()x(g n1n2

1j

iijj ≥≥≥≥====∈∈∈∈∀∀∀∀==== ∑∑∑∑++++

====

ββββΣΣΣΣδδδδ


17

• Eq (8) can be expressed in neural network terminology as follows:

• Each of the (2n+1) hidden units δj takes as input a sum of dnonlinear functions, one for each input feature xi

• Each hidden unit emits a nonlinear function δj of its total input

• The output unit emits the sum of the contributions of the hidden units

Unfortunately, Kolmogorov’s theorem tells us very little about

how to find the nonlinear functions based on data; this is the

central problem in network-based pattern recognition

Another question: how many hidden nodes we should have?


18


19

Neural Network FunctionsNeural Network Functions


20


21

Backpropagation AlgorithmBackpropagation Algorithm

• Any function from input to output can be

implemented as a three-layer neural network

• These results are of greater theoretical interest

than practical, since the construction of such a

network requires the nonlinear functions and the

weight values which are unknown!


22

• Our goal is to learn the interconnection weights based on the training patterns and the desired outputs

• In a three-layer network, it is a straightforward matter to understand how the output, and thus the error, depends

on the hidden-to-output layer weights

• The power of backpropagation is that it enables us to compute an effective error for each hidden unit, and thus

derive a learning rule for the input-to-hidden weights. This

is known as:

The credit assignment problem

Network Learning


23

• Network has two modes of operation:

• Feedforward

The feedforward operations consists of presenting a pattern to the input units and passing (or feeding) the

signals through the network in order to yield a

decision from the outputs units

• Learning

The supervised learning consists of presenting an

input pattern and modifying the network parameters (weights) to bring the actual outputs closer to the

desired target values


24


25Network Learning

• Start with an untrained network, present a training

pattern to the input layer, pass the signal through the network and determine the output.

• Let tk be the k-th target (or desired) output and zk be the k-th computed output with k = 1, …, c. Let w

represent all the weights of the network

• The training error:

• The backpropagation learning rule is based on

gradient descent

• The weights are initialized with random values and are changed in a direction that will reduce the error:

∑∑∑∑====

−−−−====−−−−====c

1k

22

kk zt2

1)zt(

2

1)w(J

w

Jw

∂∂∂∂∂∂∂∂

−−−−==== ηηηη∆∆∆∆


26where η is the learning rate which indicates the relative size of the change in weights

w(m +1) = w(m) + ∆w(m)

where m is the m-th training pattern presented

• Error on the hidden–to-output weights

where the sensitivity of unit k is defined as:

and describes how the overall error changes with the activation of the unit’s net activation

kj

kk

kj

k

kkj w

net

w

net.

net

J

w

J

∂∂∂∂∂∂∂∂

−−−−====∂∂∂∂∂∂∂∂

∂∂∂∂∂∂∂∂

====∂∂∂∂∂∂∂∂

δδδδ

k

knet

J

∂∂∂∂∂∂∂∂

−−−−====δδδδ

)net('f)zt(net

z.

z

J

net

Jkkk

k

k

kk

k −−−−====∂∂∂∂∂∂∂∂

∂∂∂∂∂∂∂∂

−−−−====∂∂∂∂

∂∂∂∂−−−−====δδδδ


27

Since netk = wkt.y, therefore:

Conclusion: the weight update (or learning rule) for the

hidden-to-output weights is:

∆wkj = ηδkyj = η(tk – zk) f’ (netk)yj

• Learning rule for the input-to-hiden units is more subtle

and is the crux of the credit assignment problem

• Error on the input-to-hidden units: Using the chain rule

j

kj

k yw

net====

∂∂∂∂∂∂∂∂

ji

j

j

j

jji w

net.

net

y.

y

J

w

J

∂∂∂∂

∂∂∂∂

∂∂∂∂

∂∂∂∂

∂∂∂∂∂∂∂∂

====∂∂∂∂∂∂∂∂


28However,

Similarly as in the preceding case, we define the sensitivity of a hidden unit:

Above equation is the core of the “credit assignment”

problem: “The sensitivity at a hidden unit is simply the sum of the individual sensitivities at the output units weighted by

the hidden-to-output weights wkj, all multiplied by f’(netj)”; see fig 6.5

Conclusion: Learning rule for the input-to-hidden weights:

∑∑∑∑ ∑∑∑∑

∑∑∑∑∑∑∑∑

==== ====

========

−−−−−−−−====∂∂∂∂

∂∂∂∂∂∂∂∂∂∂∂∂

−−−−−−−−====

∂∂∂∂∂∂∂∂

−−−−−−−−====

−−−−

∂∂∂∂∂∂∂∂

====∂∂∂∂∂∂∂∂

c

1k

c

1k

kjkkk

j

k

k

kkk

c

1k j

kkk

2

k

c

1k

k

jj

w)net('f)zt(y

net.

net

z)zt(

y

z)zt()zt(

2

1

yy

J

∑∑∑∑====

≡≡≡≡c

1k

kkjjj w)net('f δδδδδδδδ

[[[[ ]]]] ijkkjjiji x)net('f wxw

j

��

δδδδ

δδδδΣΣΣΣηηηηδδδδηηηη∆∆∆∆ ========

Sensitivity at Hidden NodeSensitivity at Hidden Node

Backpropagation AlgorithmBackpropagation Algorithm

• More specifically, the “backpropagation of

errors” algorithm

• During training, an error must be propagated

from the output layer back to the hidden layer to

learn the input-to-hidden weights

• It is gradient descent in a layered network

• Exact behavior of the learning algorithm

depends on the starting point

• Start the process with random values of weights;

in practice you learn many networks with

different initializations


30


31• Training protocols:

• Stochastic: patterns are chosen randomly from training set; network weights are updated for each pattern

• Batch: Present all patterns before updating weights

• On-line: present each pattern once & only once (no memory for storing patterns)

• Stochastic backpropagation algorithm:

Begin initialize nH; w, criterion θ, η, m ← 0

do m ← m + 1

xm ← randomly chosen pattern

wji ← wji + ηδjxi; wkj ← wkj + ηδkyjuntil ||∇J(w)|| < θreturn w

End


32

• Stopping criterion

• The algorithm terminates when the change in the

criterion function J(w) is smaller than some preset

value θ; other stopping criteria that lead to better performance than this one

• A weight update may reduce the error on the single

pattern being presented but can increase the error on the full training set

• In stochastic backpropgation and batch propagation, we must make several passes through the training

data


33

• Learning Curves

• Before training starts, the error on the training set is high;

as the learning proceeds, error becomes smaller

• Error per pattern depends on the amount of training data

and the expressive power (such as the number of weights) in the network

• Average error on an independent test set is always higher than on the training set, and it can decrease as well as

increase

• A validation set is used in order to decide when to stop training ; we do not want to overfit the network and

decrease the power of the classifier’s generalization

“Stop training when the error on the validation set is

minimum”


34

Representation at the Hidden LayerRepresentation at the Hidden Layer

• What do the learned weights mean?

• The weights connecting hidden layer to output

layer form a linear discriminant

• The weights connecting input layer to hidden

layer represent a mapping from the input

feature space to a latent feature space

• For each hidden unit, the weights from input

layer describe the input pattern that leads to

the maximum activation of that node

Backpropagation as Feature MappingBackpropagation as Feature Mapping

• 64-2-3 sigmoidal network for classifying three characters (E,F,L)

• Non-linear interactions between the features may cause the features of the pattern to not manifest in a single hidden node (in contrary to the example shown above)

• It may be difficult to draw similar interpretations in large networks and caution must be exercised while analyzing weights

Input layer to hidden layer weights for a character recognition task

Weights at two hidden nodes

represented as 8x8 patterns

Left gets activated for F, right gets

activated for L, and both get

activated for E

Practical Techniques for Improving Practical Techniques for Improving

BackpropagationBackpropagation

• A naïve application of backpropagation procedures may lead to slow convergence and poor performance

• Some practical suggestions; no theoretical results

• Activation Function f(.)• Must be non-linear (otherwise, 3-layer network is just a linear

discriminant) and saturate (have max and min value) to keep weights and activation functions bounded

• Activation function and its derivative must be continuous and smooth; optionally monotonic

• Choice may depend on the problem. Eg. Gaussian activation if the data comes from a mixture of Gaussians

• Eg: sigmoid (most popular), polynomial, tanh, sign function

• Parameters of activation function (e.g. Sigmoid)• Centered at 0, odd function f(-net) = -f(net) (anti-symmetric);

leads to faster learning

• Depend on the range of the input values

Activation FunctionActivation Function

The anti-symmetric sigmoid function: f(-x) = -f(x).

a = 1.716, b = 2/3.

First & second derivative

Practical ConsiderationsPractical Considerations• Scaling inputs (important not just for neural networks)

• Large differences in scale of different features due to the choice of units is compensated by normalizing them to be in the same range, [0,1] or [-1,1]; without normalization, error will hardly depend on feature with very small values

• Standardization: Shift the inputs to have zero mean and unit variance

• Target Values• One-of-C representation for the target vector (C is no. of classes).

Better to use +1 and –1 that lie well within the range of sigmoid function saturation values (+1.716, -1.716)

• Higher values (e.g. 1.716, saturation point of a sigmoid) may require the weights to go to infinity to minimize the error

• Training with Noise• For small training sets, it is better to add noise to the input patterns

and generate new “virtual” training patterns

Practical ConsiderationsPractical Considerations• Number of Hidden units (nH)

• Governs the expressive power of the network

• The easier the task, the fewer the nodes needed

• Rule of thumb: total no. of weights must be less than the number of training examples (preferably 10 times less); no. of hidden units determines the total no. of weights

• A more principled method is to adjust the network complexity in response to training data; e.g. start with a “large” no. of hidden units and “decay”, prune, or eliminate weights

• Initializing weights

• We cannot initialize weights to zero, otherwise learning cannot take place

• Choose initial weights w such that |w| < w’

• w’ too small – slow learning; too large – early saturation and no learning

• w’ is chosen to be 1/√d for input layer, and 1/ √nH for hidden layer

Total no. of WeightsTotal no. of Weights

Error per pattern with the increase in number of hidden nodes.

• 2-nH-1 network (with bias) trained on 90 2D-Gaussian patterns (n = 180) from each class (sampled from mixture of 3 Gaussians)

• Minimum test error occurs at 17-21 weights in total (4-5 hidden nodes). This illustrates the rule of thumb that n/10 weights often gives lowest error

Practical ConsiderationsPractical Considerations

• Learning Rate

• Small learning rate: slow convergence

• Large learning rate: high oscillation and slow convergence


• Momentum

• Prevents the algorithm from getting stuck at plateaus and local minima

• Weight decay

• Avoid overfitting by imposing the condition that weights must be small

• After each update, weights are decayed by some factor

• Related to regularization (also used in SVM)

• Hints

• Additional input nodes added to NN that are only used during training. Help learn better feature

representation.


• Training setup

• Online, stochastic, batch-mode

• Stop training

• Halt when validation error reaches (first) minimum

• Number of hidden layers

• More layers -> more complex

• Networks with more hidden layers are more prone to

get caught in local minima

• Smaller the better (KISS)

• Criterion function

• We talked about squared error, but there are others Pattern Classification, Chapter 6

44

pattern classification - michigan state university

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