Artificial Neural Network_module_3_ppt.pptx

MODULE -3
ARTIFICIAL NEURAL NETWORKS
7/22/2024 BHAVANI B H, Asst. Prof., Dept. of CSE, JSSATEB 1

• Introduction
• Neural Network Representation
• Appropriate Problems for Neural Network Learning
• Perceptrons
• Multilayer Networks and BACKPROPAGATION Algorithms
• Remarks on the BACKPROPAGATION Algorithms
BHAVANI B H, Asst. Prof., Dept. of CSE, JSSATEB 2
CONTENT
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INTRODUCTION
Artificial neural networks (ANNs) provide a general, practical method
for learning real-valued, discrete-valued, and vector-valued target
functions from examples.
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Biological Motivation
• The study of artificial neural networks (ANNs) has been inspired by the
observation that biological learning systems are built of very complex webs of
interconnected Neurons
• Human information processing system consists of brain neuron: basic building
block cell that communicates information to and from various parts of body
• Simplest model of a neuron: considered as a threshold unit –a processing element
(PE)
• Collects inputs & produces output if the sum of the input exceeds an internal
threshold value
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cell body
synapse
nucleus
axon
dendrites
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Facts of Human Neurobiology
• Number of neurons ~ 1011
• Connection per neuron ~ 10 4 – 5
• Neuron switching time ~ 0.001 second or 10 -3
• Scene recognition time ~ 0.1 second
• 100 inference steps doesn’t seem like enough
• Highly parallel computation based on distributed representation
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Properties of Neural Networks
• Many neuron-like threshold switching units
• Many weighted interconnections among units
• Highly parallel, distributed process
• Emphasis on tuning weights automatically
• Input is a high-dimensional discrete or real-valued (e.g, sensor input)
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When to consider Neural Networks ?
• Input is a high-dimensional discrete or real-valued (e.g., sensor input)
• Output is discrete or real-valued
• Output is a vector of values
• Possibly noisy data
• Form of target function is unknown
• Human readability of result is unimportant
Examples:
1. Speech phoneme recognition
2. Image classification
3. Financial perdition
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Neuron
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NEURAL NETWORK REPRESENTATIONS
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• A prototypical example of ANN learning is provided by Pomerleau's (1993)
system ALVINN, which uses a learned ANN to steer an autonomous vehicle
driving at normal speeds on public highways.
• The input to the neural network is a 30x32 grid of pixel intensities obtained from
a forward-pointed camera mounted on the vehicle.
• The network output is the direction in which the vehicle is steered.
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• Figure illustrates the neural network representation.
• The network is shown on the left side of the figure, with the input camera image
depicted below it.
• Each node (i.e., circle) in the network diagram corresponds to the output of a
single network unit, and the lines entering the node from below are its inputs.
• There are four units that receive inputs directly from all of the 30 x 32 pixels in
the image. These are called "hidden" units because their output is available only
within the network and is not available as part of the global network output. Each
of these four hidden units computes a single real-valued output based on a
weighted combination of its 960 inputs
• These hidden unit outputs are then used as inputs to a second layer of 30 "output"
units.
• Each output unit corresponds to a particular steering direction, and the output
values of these units determine which steering direction is recommended most
strongly.
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• The diagrams on the right side of the figure depict the learned weight values
associated with one of the four hidden units in thisANN.
• The large matrix of black and white boxes on the lower right depicts the weights
from the 30 x 32 pixel inputs into the hidden unit. Here, a white box indicates a
positive weight, a black box a negative weight, and the size of the box indicates
the weight magnitude.
• The smaller rectangular diagram directly above the large matrix shows the
weights from this hidden unit to each of the 30 output units.
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APPROPRIATE PROBLEMS FOR
NEURAL NETWORK LEARNING
ANN is appropriate for problems with the following characteristics :
• Instances are represented by many attribute-value pairs.
• The target function output may be discrete-valued, real-valued, or a vector of
several real- or discrete-valued attributes.
• The training examples may contain errors.
• Long training times are acceptable.
• Fast evaluation of the learned target function may be required
• The ability of humans to understand the learned target function is not important
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Architectures of Artificial Neural Networks
An artificial neural network can be divided into three parts (layers), which are
known as:
• Input layer: This layer is responsible for receiving information (data), signals,
features, or measurements from the external environment. These inputs are usually
normalized within the limit values produced by activation functions
• Hidden, intermediate, or invisible layers: These layers are composed of neurons
which are responsible for extracting patterns associated with the process or system
being analysed. These layers perform most of the internal processing from a
network.
• Output layer : This layer is also composed of neurons, and thus is responsible for
producing and presenting the final network outputs, which result from the
processing performed by the neurons in the previous layers.
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Architectures of Artificial Neural Networks
The main architectures of artificial neural networks, considering the neuron
disposition, how they are interconnected and how its layers are composed, can be
divided as follows:
1. Single-layer feedforward network
2. Multi-layer feedforward networks
3. Recurrent or Feedback networks
4. Mesh networks
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Single-Layer Feedforward Architecture
• This artificial neural network has just one input layer and a single neural layer, which is also the
output layer.
• Figure illustrates a simple-layer feedforward network composed of n inputs and m outputs.
• The information always flows in a single direction (thus, unidirectional), which is from the input
layer to the output layer
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Multi-Layer Feedforward Architecture
• This artificial neural feedforward networks with multiple layers are composed of one or more
hidden neural layers.
• Figure shows a feedforward network with multiple layers composed of one input layer with n
sample signals, two hidden neural layers consisting of n1 and n2 neurons respectively, and, finally,
one output neural layer composed of m neurons representing the respective output values of the
problem being analyzed.
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Recurrent or Feedback Architecture
• In these networks, the outputs of the neurons are used as feedback inputs for other neurons.
• Figure illustrates an example of a Perceptron network with feedback, where one of its output
signals is fed back to the middle layer.
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Mesh Architectures
• The main features of networks with mesh structures reside in considering the spatial arrangement
of neurons for pattern extraction purposes, that is, the spatial localization of the neurons is directly
related to the process of adjusting their synaptic weights and thresholds.
• Figure illustrates an example of the Kohonen network where its neurons are arranged within a two-
dimensional space
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PERCEPTRONS
• Perceptron is a single layer neural network.
• A perceptron takes a vector of real-valued inputs, calculates a linear combination
of these inputs, then outputs a 1 if the result is greater than some threshold and -1
otherwise
• Given inputs x1 through xn, the output O(x1, . . . , xn) computed by the perceptron
is
• where each wi is a real-valued constant, or weight, that determines the contribution
of input xi to the perceptron output.
• -w0 is a threshold that the weighted combination of inputs w1x1 + . . . + wnxn must
surpass in order for the perceptron to output a 1.
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Sometimes, the perceptron function is written as,
Learning a perceptron involves choosing values for the weights w0 , . . . , wn .
Therefore, the space H of candidate hypotheses considered in perceptron learning is
the set of all possible real-valued weight vectors
Why do we need Weights and Bias?
Weights shows the strength of the particular node.
A bias value allows you to shift the activation function curve up or down
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Representational Power of Perceptrons
*The perceptron can be
viewed as representing a
hyperplane decision surface
in the n-dimensional space
of instances.
*The perceptron outputs a
1 for instances lying on one
side of the hyperplane and
outputs a -1 for instances
lying on the other side
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A single perceptron can be used to represent many Boolean functions
AND function
• IfA=0 & B=0 → 0*0.6 + 0*0.6 = 0.
This is not greater than the threshold of 1, so the output = 0.
• IfA=0 & B=1 → 0*0.6 + 1*0.6 = 0.6.
This is not greater than the threshold, so the output = 0.
• IfA=1 & B=0 → 1*0.6 + 0*0.6 = 0.6.
This is not greater than the threshold, so the output = 0.
• IfA=1 & B=1 → 1*0.6 + 1*0.6 = 1.2.
This exceeds the threshold, so the output = 1.
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The Perceptron Training Rule
The learning problem is to determine a weight vector that causes the perceptron to
produce the correct + 1 or - 1 output for each of the given training examples.
To learn an acceptable weight vector
• Begin with random weights, then iteratively apply the perceptron to each training
example, modifying the perceptron weights whenever it misclassifies an example.
• This process is repeated, iterating through the training examples as many times as
needed until the perceptron classifies all training examples correctly.
• Weights are modified at each step according to the perceptron training rule,
which revises the weight wi associated with input xi according to the rule.
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• The role of the learning rate is to moderate the degree to which weights are
changed at each step. It is usually set to some small value (e.g., 0.1) and is
sometimes made to decay as the number of weight-tuning iterations increases
Drawback: The perceptron rule finds a successful weight vector when the training
examples are linearly separable, it can fail to converge if the examples are not
linearly separable.
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Gradient Descent and the Delta Rule
• If the training examples are not linearly separable, the delta rule converges toward
a best-fit approximation to the target concept.
• The key idea behind the delta rule is to use gradient descent to search the
hypothesis space of possible weight vectors to find the weights that best fit the
training examples.
To understand the delta training rule, consider the task of training an unthresholded
perceptron. That is, a linear unit for which the output O is given by
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To derive a weight learning rule for linear units, specify a measure for the training
error of a hypothesis (weight vector), relative to the training examples.
Where,
• D is the set of training examples,
• td is the target output for training example d,
• od is the output of the linear unit for training example d
• E [ w ] is simply half the squared difference between the target output td and the linear unit output
od, summed over all training examples.
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Visualizing the Hypothesis Space
• To understand the gradient descent algorithm, it is helpful to visualize the entire
hypothesis space of possible weight vectors and their associated E values as
shown in below figure.
• Here the axes w0 and wl represent possible values for the two weights of a simple
linear unit. The w0, wl plane therefore represents the entire hypothesis space.
• The vertical axis indicates the error E relative to some fixed set of training
examples.
• The arrow shows the negated gradient at one particular point, indicating the
direction in the w0, wl plane producing steepest descent along the error surface.
• The error surface shown in the figure thus summarizes the desirability of every
weight vector in the hypothesis space
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• Given the way in which we chose to define E, for linear units this error surface must always be
parabolic with a single global minimum.
Gradient descent search determines a weight vector that minimizes E by starting with an arbitrary
initial weight vector, then repeatedly modifying it in small steps.
At each step, the weight vector is altered in the direction that produces the steepest descent along the
error surface depicted in above figure. This process continues until the global minimum error is
reached.
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Derivation of the Gradient Descent Rule
How to calculate the direction of steepest descent along the error surface?
The direction of steepest can be found by computing the derivative of E with respect
to each component of the vector w . This vector derivative is called the gradient of E
with respect to w , written as
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• The gradient specifies the direction of steepest increase of E, the training rule for
gradient descent is
• Here η is a positive constant called the learning rate, which determines the step
size in the gradient descent search.
• The negative sign is present because we want to move the weight vector in the
direction that decreases E
• This training rule can also be written in its component form
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Calculate the gradient at each step. The vector of
differentiating E from Equation (2), as
𝜕𝐸
𝜕𝑤
𝑖
derivatives that form the gradient can be obtained by
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GRADIENT DESCENT algorithm for training a linear unit
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To summarize, the gradient descent algorithm for training linear units is as follows:
• Pick an initial random weight vector.
• Apply the linear unit to all training examples, then computeΔwi for each weight
according to Equation (7).
• Update each weight wi by addingΔwi, then repeat this process
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Features of Gradient Descent Algorithm
Gradient descent is an important general paradigm for learning. It is a strategy for
searching through a large or infinite hypothesis space that can be applied whenever
1. The hypothesis space contains continuously parameterized hypotheses
2. The error can be differentiated with respect to these hypothesis parameters
The key practical difficulties in applying gradient descent are
1. Converging to a local minimum can sometimes be quite slow
2. If there are multiple local minima in the error surface, then there is no guarantee
that the procedure will find the global minimum
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Stochastic Approximation to Gradient Descent
• The gradient descent training rule presented in Equation (7) computes weight
updates after summing over all the training examples in D
• The idea behind stochastic gradient descent is to approximate this gradient descent
search by updating weights incrementally, following the calculation of the error
for each individual example
where t, o, and xi are the target value, unit output, and ith input for the training
example in question
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One way to view this stochastic gradient descent is to consider a distinct error
function Ed ( w ) for each individual training example d as follows
Where, td and od are the target value and the unit output value for training example
d.
• Stochastic gradient descent iterates over the training examples d in D, at each
iteration altering the weights according to the gradient with respect to Ed( w )
• The sequence of these weight updates, when iterated over all training examples,
provides a reasonable approximation to descending the gradient with respect to
our original error function Ed ( w )
• By making the value of η sufficiently small, stochastic gradient descent can be
made to approximate true gradient descent arbitrarily closely
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The key differences between standard gradient descent and stochastic gradient
descent are
• In standard gradient descent, the error is summed over all examples before
updating weights, whereas in stochastic gradient descent weights are updated upon
examining each training example.
• Summing over multiple examples in standard gradient descent requires more
computation per weight update step. On the other hand, because it uses the true
gradient, standard gradient descent is often used with a larger step size per weight
update than stochastic gradient descent.
• In cases where there are multiple local minima with respect to stochastic gradient
descent can sometimes avoid falling into these local minima because it uses the
various ❑
Ed ( w ) rather than ❑
E ( w ) to guide its search
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MULTILAYER NETWORKS AND THE
BACKPROPAGATION ALGORITHM
Multilayer networks learned by the BACKPROPAGATION algorithm are capable
of expressing a rich variety of nonlinear decision surfaces
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• Decision regions of a multilayer feedforward network. The network shown here was trained to recognize 1 of
10 vowel sounds occurring in the context "h_d" (e.g., "had," "hid"). The network input consists of two
parameters, F1 and F2, obtained from a spectral analysis of the sound. The 10 network outputs correspond to
the 10 possible vowel sounds. The network prediction is the output whose value is highest.
• The plot on the right illustrates the highly nonlinear decision surface represented by the learned network.
Points shown on the plot are test examples distinct from the examples used to train the network.
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A Differentiable Threshold Unit
• Sigmoid unit-a unit very much like a perceptron, but based on a smoothed,
differentiable threshold function.
• The sigmoid unit first computes a linear combination of its inputs, then applies a
threshold to the result. In the case of the sigmoid unit, however, the threshold
output is a continuous function of its input.
• More precisely, the sigmoid unit computes its output O as
σ is the sigmoid function
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The BACKPROPAGATION Algorithm
• The BACKPROPAGATION Algorithm learns the weights for a multilayer network, given a
network with a fixed set of units and interconnections. It employs gradient descent to attempt to
minimize the squared error between the network output values and the target values for these
outputs.
• In BACKPROPAGATION algorithm, we consider networks with multiple output units rather than
single units as before, so we redefine E to sum the errors over all of the network output units.
where,
• outputs - is the set of output units in the network
• tkd and Okd - the target and output values associated with the kth output unit
• d - training example
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Derivation of the BACKPROPAGATION Rule
• Deriving the stochastic gradient descent rule: Stochastic gradient descent involves
iterating through the training examples one at a time, for each training example d
descending the gradient of the error Ed with respect to this single example
• For each training example d every weight wji is updated by adding to it Δwji
Here outputs is the set of output units in the network, tk is the target value of unit k for
training example d, and ok is the output of unit k given training example d.
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The derivation of the stochastic gradient descent rule is conceptually
straightforward, but requires keeping track of a number of subscripts and variables
xji = the ith input to unit j
wji = the weight associated with the ith input to unit j
netj = Σi wjixji (the weighted sum of inputs for unit j )
oj = the output computed by unit j
tj = the target output for unit j
σ = the sigmoid function
outputs = the set of units in the final layer of the network
Downstream(j) = the set of units whose immediate inputs include the output of
unit j
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Consider two cases in turn: the case where unit j is an output unit for the network, and the case where j is an
internal unit (hidden unit).
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Case 1: Training Rule for Output Unit Weights.
• wji can influence the rest of the network only through netj , netj can influence the network only through oj.
Therefore, we can invoke the chain rule again to write
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Case 2: Training Rule for Hidden Unit Weights.
• In the case where j is an internal, or hidden unit in the network, the derivation of
the training rule for wji must take into account the indirect ways in which wji can
influence the network outputs and hence Ed.
• For this reason, we will find it useful to refer to the set of all units immediately
downstream of unit j in the network and denoted this set of units by Downstream(
j).
• netj can influence the network outputs only through the units in Downstream(j).
Therefore, we can write
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REMARKS ON THE BACKPROPAGATION
ALGORITHM
1. Convergence and Local Minima
• The BACKPROPAGATION multilayer networks is only guaranteed to converge
toward some local minimum in E and not necessarily to the global minimum error.
• Despite the lack of assured convergence to the global minimum error,
BACKPROPAGATION is a highly effective function approximation method in
practice.
• Local minima can be gained by considering the manner in which network weights
evolve as the number of training iterations increases.
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Common heuristics to attempt to alleviate the problem of local minima include:
1. Add a momentum term to the weight-update rule. Momentum can sometimes
carry the gradient descent procedure through narrow local minima
2. Use stochastic gradient descent rather than true gradient descent
3. Train multiple networks using the same data, but initializing each network with
different random weights
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2. Representational Power of Feedforward Networks
What set of functions can be represented by feed-forward networks?
The answer depends on the width and depth of the networks. There are three quite
general results are known about which function classes can be described by which
types of Networks
1. Boolean functions – Every boolean function can be represented exactly by some
network with two layers of units, although the number of hidden units required
grows exponentially in the worst case with the number of network inputs
2. Continuous functions – Every bounded continuous function can be
approximated with arbitrarily small error by a network with two layers of units
3. Arbitrary functions – Any function can be approximated to arbitrary accuracy by
a network with three layers of units.
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3. Hypothesis Space Search and Inductive Bias
• Hypothesis space is the n-dimensional Euclidean space of the n network weights
and hypothesis space is continuous.
• As it is continuous, E is differentiable with respect to the continuous parameters of
the hypothesis, results in a well-defined error gradient that provides a very useful
structure for organizing the search for the best hypothesis.
• It is difficult to characterize precisely the inductive bias of
BACKPROPAGATION algorithm, because it depends on the interplay between
the gradient descent search and the way in which the weight space spans the space
of representable functions. However, one can roughly characterize it as smooth
interpolation between data points.
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4. Hidden Layer Representations
BACKPROPAGATION can define new hidden layer features that are not explicit in the
input representation, but which capture properties of the input instances that are most
relevant to learning the target function.
Consider example, the network shown in below Figure
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• Consider training the network shown in Figure to learn the simple target function f
(x) = x, where x is a vector containing seven 0's and a single 1.
• The network must learn to reproduce the eight inputs at the corresponding eight
output units. Although this is a simple function, the network in this case is
constrained to use only three hidden units. Therefore, the essential information
from all eight input units must be captured by the three learned hidden units.
• When BACKPROPAGATION applied to this task, using each of the eight possible
vectors as training examples, it successfully learns the target function. By
examining the hidden unit values generated by the learned network for each of the
eight possible input vectors, it is easy to see that the learned encoding is similar to
the familiar standard binary encoding of eight values using three bits (e.g.,
000,001,010,. . . , 111). The exact values of the hidden units for one typical run of
shown in Figure.
• This ability of multilayer networks to automatically discover useful
representations at the hidden layers is a key feature ofANN learning
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5. Generalization, Overfitting, and Stopping Criterion
What is an appropriate condition for terminating the weight update loop?
• One choice is to continue training until the error E on the training examples falls
below some predetermined threshold.
• To see the dangers of minimizing the error over the training data, consider how the
error E varies with the number of weight iterations
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• Consider first the top plot in this figure. The lower of the two lines shows the
monotonically decreasing error E over the training set, as the number of gradient
descent iterations grows. The upper line shows the error E measured over a
different validation set of examples, distinct from the training examples. This line
measures the generalization accuracy of the network-the accuracy with which it
fits examples beyond the training data.
• The generalization accuracy measured over the validation examples first
decreases, then increases, even as the error over the training examples continues to
decrease. How can this occur? This occurs because the weights are being tuned to
fit idiosyncrasies of the training examples that are not representative of the general
distribution of examples. The large number of weight parameters in ANNs
provides many degrees of freedom for fitting such idiosyncrasies
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Why does overfitting tend to occur during later iterations, but not during earlier
iterations?
• By giving enough weight-tuning iterations, BACKPROPAGATION will often be
able to create overly complex decision surfaces that fit noise in the training data or
unrepresentative characteristics of the particular training sample.
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