Neural network for machine learning

 Its very hard to write programs that solve problems like recognizing a 3D
object from a novel viewpoint.
 Even if we could have written such a program, it would have been very complicated
 Its hard to write a program that computes credit card fraudulent
 There are no specific rules that are simple and reliable. We need to combine a large
number of weak rules
 Fraud is a moving target. The program needs to keep changing

 Instead of writing programs for a specific task, we collect a lot of examples
that specify the correct output for a given input
 The machine learning algorithm then takes these examples and produces a
program that does the job
 Looks very different from a typical program
 Program works well for new cases and the ones that we train it on
 If the data changes, the program can change by training on the new data
 Massive amount of computation is now cheaper than paying for the code

 To study how the brain actually works
 It very big and complicated. So, we need the use of computer simulation
 To understand the style of parallel computation inspired by neurons and
their adaptive connection
 Very different from sequential connection
 Should be good at things that the brain is good at. Ex. Vision
 Should be bad at thing the brain is bad at. Ex. Computation 24 X 44
 To solve practical problems by using novel learning algorithms inspired by
brain

 Revolutionary Idea: think of neural tissue as circuits performing mathematical
computation

 Linear weighted sum of inputs
 Non-linear, possibly stochastic transfer function
 Learning rule

 Gross physical structure
 One axon that branches
 There is a dendritic tree that collects inputs from other neurons
 Axons typically contact dendritic tree at synapses
 A spike of activity in axon causes charge to be injected into post-synaptic neuron
 Spike generation
 There is an axon hillock that generates outgoing spikes whenever enough charge has flowed in at
synapse to depolarize the cell membrane

 To model things we have to idealize them (ex. Atoms)
 Idealization removes complicated details that are not essential for understanding the
main principle
 Allows to apply mathematics and make analogies to other familiar systems
 Its worth understanding models that are known to be wrong
 Neurons that communicate real values rather than discrete spikes of activity

 These are simple but computationally limited
 If we can make them learn, we may get insight into more complicated neurons

 First compute the weighted sum of the inputs
 Then send out the fixed spike of activity if the weighted sum exceeds a threshold
 There are two equivalent ways to write the equations for a binary threshold neuron

 Also called threshold linear neuron
 Computes a linear weighted sum of their inputs
 The output is a non-linear function of the total input

 Gives a real valued output that is smooth and bounded function of their total input
 Typically, they use logistic function
 They have derivatives, which make the learning easy

 They use the same equation as logistic unit
 They treat the output of the logistic as probability of producing a spike in short time window.

 Supervised Learning
 Learn to predict the output when given an input vector
 Reinforcement Learning
 Learn to select an action to maximize payoff
 Unsupervised Learning
 Discover a good internal representation of the input

 Each training case consist of an input vector x and a target output t
 Regression: the target output is a real number or a whole vector of a real number
 The price of stock in six month time
 The temperature at noon tomorrow
 Classification: the target output is a class label
 The simplest case is between 1 and 0
 We can have multiple alternative label
 Working - We start by choosing a model class: y=f(x, w)
 A model class f, is a way of using some numerical parameters, w to match each input vector, x
into a predicted output y

 In reinforcement learning, the output is an action or sequence of actions and the only
supervisory signal is the occasional scalar reward.
 The goal in selecting each action is to maximize the expected sum of the future rewards.
 Reinforcement learning is difficult
 The rewards are typically delayed and its hard to know where we went wrong.
 A scalar reward does not supply much information

Architecture of a neural network means way in which the neurons are connected
to each other

 Most common type
 First layer is input and the last layer is output
 If there is more than one hidden layer, we call it
“deep” neural network
 They compute a series of transformations that
change the similarities between cases
 Activities of neuron in each layer are a non-linear
function of the activities in the layer below

 These have directed cycles in their connection graph
 This means that you can sometimes get back to where you started by
following the arrows
 They can have complicated dynamics and this can make them very
difficult to train
 They are more biologically realistic
 They have a natural way to model sequential data
 Equivalent to deep nets with one hidden layer per time slice
 They use same weights at every time slice and get input at every time slice
 They have the ability to remember information in their hidden state
for a long time
 Its hard to train to use this potential

 Like recurrent network, but the connections between units are symmetrical (have same
weights in both direction)
 Much easier to analyze than recurrent networks
 More restricted in what they can do because they are restricted by energy function
 Ex. Cannot model cycles
 Symmetrically connected nets without hidden units are called Hopfield nets

 The space has one dimension for each
weight
 A point in the space represents a
particular setting of the weight
 Each training case represents a
hyperplane
 The weights must lie on one side of a
hyperplane to get the answer correct

Theorem:
If a problem is linearly separable, then
A perceptron will learn it
In a finite number of steps

Works fine for single layer of trainable weights, but what about multi-layer neurons?

 In perceptron, the weights are always getting closer to a good set of weights
 In a linear neuron, the outputs are always getting closer to the target output
 Why perceptron convergence procedure cannot be generalized to hidden layers?
 The perceptron learning algorithm works by ensuring that every time the weights change, they
get closer to a generously feasible set of weights
 This type of extension cannot be extended to more complex network
 We hence show that the actual output values get closer to the target values while this may
not be the case with perceptron i.e. the outputs may get away from the target outputs

 Does the learning procedure eventually get the right answer?
 There may be no perfect answer
 By making the learning rate slow, we get very close to the desired answer.
 How quickly do the weights converge?
 Can be very slow if the input dimensions are highly correlated

 Selection of parameter values which are optimal in some desired sense
 Ex. Minimize the object function over a dataset
 Parameters are weights and biases
 Training the neural nets is iterative and time consuming and hence its in our interest to
reduce training time
 Methods
 Gradient descent
 Line search
 Conjugate gradient search

 Horizontal axis corresponds
to weight and vertical axis for
error
 For linear neuron with squared
error, it is a quadratic bowl
 Vertical cross-sections are
parabolas
 Horizontal cross-sections are
ellipses

 The gradient is big in the direction in which we only want to travel a small distance
 The gradient is small in the direction in which we want to travel large distance

 If the learning rate is big, the weight slosh to and
fro across the ravine.
 If the learning rate is too big, this oscillation
diverges
 What we would like to achieve
 Move quickly in directions with small and
consistent gradients
 Move slowly in direction with big inconsistent
gradients

 Straightforward, iterative, tractable, locally optimal descent in error
 Cannot avoid local minima and cannot escape them – my overshoot them
 Cannot guarantee a scalable bound on time complexity
 Search direction only locally optimal

 Local minima is possible by random perturbation
 Stochastic gradient descent is a form of injecting randomness into gradient descent

 When applying machine leaning to sequences, we often want to turn an input sequence to
output sequence that lives in different domain
 Ex. Turn a sequence of sound pressure into a sequence of word identities
 When there is no separate target sequence, we get a teaching sequence by trying to
predict the next term in the input sequence
 Target output sequence is the input output sequence with an advance of 1 step
 Its like predicting one pixel of an image from the other pixel, or one patch of the image from
other

 Autoregressive models
 Output depends linearly on its own previous
values
 Take previous terms and predicts the next
 Weighted average of previous terms
 Feed-forward neural network
 Take in a few terms, put them through some
hidden units and predict the next term
 Connection between units do not form a
directed cycle

 Recurrent means feeding back on itself
 They are powerful because they combine
two properties:
 Have distributes hidden states – means several
different units can be active at once. Hence, they
can remember multiple values at once
 Non-linear Dynamics – allows the dynamics to be
updated in complicated way

 They can oscillate – good for motor control
 They can settle to point attractors – good for retrieving memories
 They behave chaotically – bad for information processing
 Implement small programs in parallel

 Recurrent backpropogation network
 Discrete time
 Simple Recurrent Network – Elamn net
 Jodan net
 Fixed point attractor network
 Continuous time
 Spin – Glass Model – Hopfield, Boltzmann
 Interactive – Activation Model: cognitive modeling
 Competitive networks – self-organizing feature maps

 Assume that there is a time delay of
one in using each connection
 The recurrent net is just a layered
net that keeps reusing the same
weights

 We can specify inputs in several ways:
 Specify the initial subsets of all the units
 Specify the initial states of a subset of
units
 Specify the states of the same subset of
the units at every time step
 Specify desired final activities of all the
units
 Specify desired activities of all units for
the last few steps
 Specify the desired activity of a subset of
unit

 LIMITATIONS
 Maximum number of digits must be decided in
advance
 This cannot be generalized for long numbers
because it use different weights

 The network has two input units and one output unit
 Given two input unit each time
 Desired output for each step is the output for column
that was provided as input two time steps ago
 Takes one time step to update the hidden units based on
the input
 It takes another time step for the hidden unit to cause the
output

 There is big difference between the
forward pass and the backward pass
 In forward pass, we use squashing
function (like logistic) to prevent the
activity vectors from exploding
 The backward pass is completely linear. If
you double the error derivatives at the
final layer, all the error derivatives will
double

 What happens to the magnitude of the gradient as we backpropogate?
 If the weights are small, the gradients shrinks exponentially
 If the weights are big, the gradients grows exponentially
 Typical feed-forward nets can cope with these factors because they have a few hidden
units
 In the RNN trained on long sequences, the gradients can explode or vanish
 Can be avoided by initializing the weights exponentially

 Long Short Term Memory:
 Make RNN of the little modules that are designed to hold values for a long time
 Hessian Free Optimization:
 Deals with vanishing gradient problem
 Ex. HF optimizer
 Echo State Networks
 Initialize connections so that the hidden state has a huge reservoir of weakly coupled oscillator
 Good initialization with momentum
 Initialize like in echo-state networks but learn all connections using momentum

 Dynamic state of the neural network is a short term memory which has to be converted to
long term to make the data last
 Very successful for task like recognizing handwriting
 Example considered – getting a RNN to remember things for long time (like hundred of
time steps)
 Uses logistic and linear units
 Write gate - Information gets in
 Keep gate - Information stored
 Read gate - Information is extracted

 Circuit implements analog memory cell
 Linear unit with self link and weight of 1 will
maintain state
 Activate write gate to store information
 Activate read gate for retrieving information
 Backprop is possible because logistic has nice
derivatives

 Perceptron
 Make early layers random and fixed
 We learn the last layer which is a linear model
 It uses the transformed inputs to predict the output
 Echo state network
 Fix the input->hidden and hidden->hidden connections at random values
 Learn hidden->output connection
 Choose the random connections carefully

In competitive learning, neurons compete among themselves to be activated

 Output units are said to be in competition for input patterns
 During training, the output unit that provides highest activation to a given pattern is
declared the winner and is moved closer to the input pattern
 Unsupervised learning
 Also called winner-takes-all
 One neuron wins over all others
 Only the winning neuron learns
 Hard Learning – weight of only the winner is updated
 Soft Learning – weight of winner and close associates is updated

 Produces a mapping from multi-dimensional input space onto a lattice of clusters
 Mapping is topology-preserving
 Typically organized as 1D or 2D lattice
 Have a strong neurological basis
 Topology is preserved. Ex. If we touch parts of the body that are close together, group of cells will
fire that are also close together
 K-SOM results from synergy of three basic processes
 Competition
 Cooperation
 Adaptation

 Each neuron in SOM is assigned a weight vector
with same dimensionality N as the input space
 Any given input pattern is compared to the
weight vector of each neuron and the closest is
declared the winner
 The Euclidean norm is usually used to measure
distance

 The activation of winning neuron is spread to
neurons in its immediate neighbourhood
 This allows topologically close neurons to become
sensitive to similar patterns
 The size of neighbourhood is initially large, but
shrinks over time
 Large neighbourhood promotes a topology
preserving mapping
 Smaller neighbourhood allows neurons to
specialize in later stages of training

 During training, the winner neuron and its
topological neighbours are adapted to make their
weight vectors more similar to the input pattern
that caused the activation
 Neurons that are closer to the winner will adapt
more heavily than neurons far away
 Magnitude of adaptation is controlled by learning
rate

A neuron learn by shifting its weight from inactive neurons to active neurons
Change Dwij applied to synaptic weight wij as
where xi is the input signal and a is the learning rate parameter
The overall effect relies on moving the synaptic weight vector of the winning neuron
towards the input pattern
Matching criteria is the equivalent Euclidean distance


 
D
ncompetitiothelosesneuronif,0
ncompetitiothewinsneuronif),(
j
jwx
w
iji
ij


Euclidean distance is given by
where xi and wij are the ith elements of the vectors X and Wj, respectively.
To identify the winning neuron, jX, that best matches the input vector X, we may apply the
following condition:
2/1
1
2
)(








 

n
i
ijij wxd WX
,j
j
minj WXX  j = 1, 2, . . .,m

Suppose 2D input vector is presented to three neuron kohenen network
Initial weight vector is given by







12.0
52.0
X







81.0
27.0
1W 






70.0
42.0
2W 






21.0
43.0
3W

 We find the winning neuron using the minimum-distance euclidean criteria
 Neuron 3 is the winner and its weight vector is updated according to the competitive
learning rule
2
212
2
1111 )()( wxwxd  73.0)81.012.0()27.052.0( 22

2
222
2
1212 )()( wxwxd  59.0)70.012.0()42.052.0( 22

2
232
2
1313 )()( wxwxd  13.0)21.012.0()43.052.0( 22

0.01)43.052.0(1.0)( 13113 D wxw
0.01)21.012.0(1.0)( 23223 D wxw

 The updated weight vector with iteration (p+1) is determined as
 The weight vector w3 of the winning neuron 3 becomes closer to the input vector X with
each iteration



















D
20.0
44.0
01.0
0.01
21.0
43.0
)()()1( 333 ppp WWW

 Kohenen network with 100 neurons arranged in the form of 2D lattice with 10 rows and
10 columns
 The network is required to classify 2D input vectors – each neuron should respond to
input vectors occurring in that region only
 The network is trained with 1000 2D input vectors generated randomly in a square
region in the interval between -1 and +1
 Learning rate parameter is 0.1

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
W(2,j)
W(1,j)
Initial random weights

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8-1
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1
W(2,j)
W(1,j)
Network after 100 iterations

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8-1
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1
W(2,j)
W(1,j)

 Serves as Content-Addressable Memory system with binary threshold nodes
 Provides model for understanding human memory
 Used for storing memory as distributed patterns of activity
 Stable states are fixed point attractors

 Two ways of updating
 Asynchronous: picks one neuron, calculate weight sum and updates immediately. Can be done in
fixed order or neurons can be picked at random
 Synchronous: weight sum is calculated without updating neurons. Then all neurons are set to
new values
 Conditions on weight matrix:
 symmetry: wij = wji
 no self connections: wii = 0s

 Global energy depends on one connection weight and the binary state of two neurons
Weight of two neurons
Activity of two
connecting neurons
Bias term

 Memories could be energy minima of a neural net
 The binary threshold decision rule can then be used to clean up incomplete or corrupted
memories
 Using energy minima to represent memories gives a content-addressable memory
 An item can be accessed by just knowing a part of its content

Neural network for machine learning

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