Multilayer Perceptron Neural Networks
A Brief History of Neural Networks
Neural networks are predictive models loosely based on
the action of biological neurons.
The selection of the name “neural network” was one of the great PR
successes of the Twentieth Century. It certainly sounds more exciting
than a technical description such as “A network of weighted, additive
values with nonlinear transfer functions”. However, despite the name,
neural networks are far from “thinking machines” or “artificial
brains”.
A typical artifical neural network might have a hundred neurons.
In comparison, the human nervous system is
believed to have about 3x10^{10} neurons.
We are still light years from “Data” on Star Trek.
The original “Perceptron” model was developed by Frank Rosenblatt in
1958. Rosenblatt’s model consisted of three layers, (1) a “retina”
that distributed inputs to the second layer, (2) “association units”
that combine the inputs with weights and trigger a threshold step
function which feeds to the output layer, (3) the output layer which
combines the values. Unfortunately, the use of a step function in the
neurons made the perceptions difficult or impossible to train. A
critical analysis of perceptrons published in 1969 by Marvin Minsky
and Seymore Papert pointed out a number of critical weaknesses of
perceptrons, and, for a period of time, interest in perceptrons waned.
Interest in neural networks was revived in 1986 when David Rumelhart,
Geoffrey Hinton and Ronald Williams published “Learning Internal
Representations by Error Propagation”. They proposed a multilayer
neural network with nonlinear but differentiable transfer functions
that avoided the pitfalls of the original perceptron’s step functions.
They also provided a reasonably effective training algorithm for
neural networks.
Types of Neural Networks
When used without qualification, the terms
“Neural Network” (NN) and “Artificial Neural Network” (ANN) usually
refer to a Multilayer Perceptron Network. However, there are
many other types of neural networks including
Probabilistic Neural Networks,
General Regression Neural Networks,
Radial Basis Function
Networks, Cascade Correlation, Functional Link Networks, Kohonen
networks, GramCharlier networks, Learning Vector Quantization,
Hebb networks, Adaline networks, Heteroassociative networks,
Recurrent Networks and Hybrid Networks.
DTREG implements the most widely used types of neural networks:
Multilayer Perceptron Networks (also known as multilayer feedforward network),
Cascade Correlation Neural Networks,
Probabilistic Neural Networks
(PNN) and General Regression Neural Networks (GRNN). This section
describes Multilayer Perceptron Networks.
Click here for information about
Probabilistic and General Regression neural networks.
Click here for information about
Cascade Correlation neural networks.
The Multilayer Perceptron Neural Network Model
The following diagram
illustrates a perceptron network with three layers:
This network has an input layer (on the left) with three neurons, one
hidden layer (in the middle) with three neurons and an output layer
(on the right) with three neurons.
There is one neuron in the input layer for each predictor variable.
In the case of categorical variables, N1 neurons are used to
represent the N categories of the variable.
Input Layer — A vector of predictor variable values
(x_{1}...x_{p})
is presented
to the input layer. The input layer (or processing before the input layer)
standardizes these values so that the range of each variable is 1 to 1.
The input layer distributes the values to each of the neurons in the hidden layer.
In addition to the predictor
variables, there is a constant input of 1.0, called the bias that is
fed to each of the hidden layers; the bias is multiplied by a weight
and added to the sum going into the neuron.
Hidden Layer — Arriving at a neuron in the hidden layer, the value from
each input neuron is multiplied by a weight (w_{ji}),
and the resulting
weighted values are added together producing a combined value
u_{j}. The
weighted sum (u_{j}) is fed into a transfer function,
σ, which outputs a
value h_{j}.
The outputs from the hidden layer are distributed to the
output layer.
Output Layer — Arriving at a neuron in the output layer, the value from
each hidden layer neuron is multiplied by a weight
(w_{kj}), and the
resulting weighted values are added together producing a combined
value v_{j}.
The weighted sum (v_{j}) is fed into a transfer function, σ,
which outputs a value y_{k}.
The y values are the outputs of the network.
If a regression analysis is being performed with a continuous target
variable, then there is a single neuron in the output layer, and it
generates a single y value. For classification problems with
categorical target variables, there are N neurons in the output layer
producing N values, one for each of the N categories of the target
variable.
Multilayer Perceptron Architecture
The network diagram shown above is a fullconnected,
three layer, feedforward, perceptron neural network. “Fully connected” means
that the output from each input and hidden neuron is distributed to
all of the neurons in the following layer. “Feed forward” means that
the values only move from input to hidden to output layers; no values
are fed back to earlier layers (a Recurrent Network allows values to
be fed backward).
All neural networks have an input layer and an output layer, but the
number of hidden layers may vary. Here is a diagram of a
perceptron network with two hidden layers and four total layers:
When there is more than one hidden layer, the output from one hidden
layer is fed into the next hidden layer and separate weights are
applied to the sum going into each layer.
Training Multilayer Perceptron Networks
The goal of the training
process is to find the set of weight values that will cause the output
from the neural network to match the actual target values as closely
as possible.
There are several issues involved in designing and training a multilayer
perceptron network:
 Selecting how many hidden layers to use in the network.
 Deciding how many neurons to use in each hidden layer.
 Finding a globally optimal solution that avoids local minima.
 Converging to an optimal solution in a reasonable period of time.
 Validating the neural network to test for overfitting.
Selecting the Number of Hidden Layers
For nearly all problems, one
hidden layer is sufficient. Two hidden layers are required for
modeling data with discontinuities such as a saw tooth wave pattern.
Using two hidden layers rarely improves the model, and it may
introduce a greater risk of converging to a local minima. There is no
theoretical reason for using more than two hidden layers. DTREG can
build models with one or two hidden layers.
Three layer models with one hidden layer are recommended.
Deciding how many neurons to use in the hidden layers
One of the most
important characteristics of a perceptron network is the number of neurons
in the hidden layer(s). If an inadequate number of neurons are used,
the network will be unable to model complex data, and the resulting
fit will be poor.
If too many neurons are used, the training time may become excessively
long, and, worse, the network may over fit the data. When overfitting
occurs, the network will begin to model random noise in the data. The
result is that the model fits the training data extremely well, but it
generalizes poorly to new, unseen data. Validation must be used to
test for this.
DTREG includes an automated feature to find the optimal number of
neurons in the hidden layer. You specify
the minimum and maximum number of neurons you want it to test, and it
will build models using varying numbers of neurons and measure the
quality using either cross validation or holdout data not used for
training. This is a highly effective method for finding the optimal
number of neurons, but it is computationally expensive, because many
models must be built, and each model has to be validated. If you have
a multiprocessor computer, you can configure DTREG to use multiple
CPU’s during the process.
The automated search for the optimal number of neurons only searches
the first hidden layer. If you select a model with two hidden layers,
you must manually specify the number of neurons in the second hidden
layer.
Finding a globally optimal solution
A typical neural network might have a couple of hundred weighs whose
values must be found to produce an optimal solution. If neural
networks were linear models like linear regression, it would be a
breeze to find the optimal set of weights. But the output of a neural
network as a function of the inputs is often highly nonlinear; this
makes the optimization process complex.
If you plotted the error as a function of the weights, you would
likely see a rough surface with many local minima such as this:
This picture is highly simplified because it represents only a single
weight value (on the horizontal axis). With a typical neural network,
you would have a 200dimension, rough surface with many local valleys.
Optimization methods such as steepest descent and conjugate gradient
are highly susceptible to finding local minima if they begin the
search in a valley near a local minimum. They have no ability to see
the big picture and find the global minimum.
Several methods have been tried to avoid local minima. The simplest
is just to try a number of random starting points and use the one with
the best value. A more sophisticated technique called simulated
annealing improves on this by trying widely separated random values
and then gradually reducing (“cooling”) the random jumps in the hope
that the location is getting closer to the global minimum.
DTREG uses the NguyenWidrow algorithm to select the initial range of
starting weight values. It then uses the conjugate gradient algorithm
to optimize the weights. Conjugate gradient usually finds the optimum
weights quickly, but there is no guarantee that the weight values it finds
are globally optimal. So it is useful to allow DTREG to try the
optimization multiple times with different sets of initial random weight values.
The number of tries allowed is specified on the Multilayer Perceptron property page.
Converging to the Optimal Solution — Conjugate Gradient
Given a set of randomlyselected starting weight values, DTREG uses the
conjugate gradient algorithm to optimize the weight values.
Most training algorithms follow this cycle to refine the weight
values: (1) run a set of predictor variable values through the network
using a tentative set of weights, (2) compute the difference between
the predicted target value and the actual target value for this case,
(3) average the error information over the entire set of training
cases, (4) propagate the error backward through the network and
compute the gradient (vector of derivatives) of the change in error
with respect to changes in weight values, (5) make adjustments to the
weights to reduce the error. Each cycle is called an epoch.
Because the error information is propagated backward through the
network, this type of training method is called backward propagation.
The backpropagation training algorithm was first described by
Rumelhart and McClelland in 1986; it was the first practical method
for training neural networks. The original procedure used the
gradient descent algorithm to adjust the weights toward convergence
using the gradient. Because of this history, the term
“backpropagation” or “backprop” often is used to denote a neural
network training algorithm using gradient descent as the core
algorithm. That is somewhat unfortunate since backward propagation of
error information through the network is used by nearly all training
algorithms, some of which are much better than gradient descent.
Backpropagation using gradient descent often converges very slowly or
not at all. On largescale problems its success depends on
userspecified learning rate and momentum parameters.
There is no
automatic way to select these parameters, and if incorrect values are
specified the convergence may be exceedingly slow, or it may not
converge at all. While backpropagation with gradient descent is still
used in many neural network programs, it is no longer considered to be
the best or fastest algorithm.
DTREG uses the conjugate gradient algorithm to adjust weight values
using the gradient during the backward propagation of errors through
the network. Compared to gradient descent, the conjugate gradient
algorithm takes a more direct path to the optimal set of weight
values. Usually, conjugate gradient is significantly faster and more
robust than gradient descent. Conjugate gradient also does not
require the user to specify learning rate and momentum parameters.
The traditional conjugate gradient algorithm uses the gradient to
compute a search direction. It then uses a line search algorithm such
as Brent’s Method to find the optimal step size along a line in the
search direction. The line search avoids the need to compute the
Hessian matrix of second derivatives, but it requires computing the
error at multiple points along the line. The conjugate gradient
algorithm with line search (CGL) has been used successfully in many
neural network programs, and is considered one of the best methods yet
invented.
DTREG provides the traditional conjugate gradient algorithm with line
search, but it also offers a newer algorithm, Scaled Conjugate
Gradient developed in 1993 by Martin Fodslette Moller.
The scaled conjugate gradient algorithm uses a numerical
approximation for the second derivatives (Hessian matrix), but it
avoids instability by combining the modeltrust region approach from
the LevenbergMarquardt algorithm with the conjugate gradient
approach. This allows scaled conjugate gradient to compute the optimal
step size in the search direction without having to perform the
computationally expensive line search used by the traditional
conjugate gradient algorithm. Of course, there is a cost involved in
estimating the second derivatives.
Tests performed by Moller show the scaled conjugate gradient algorithm
converging up to twice as fast as traditional conjugate gradient and
up to 20 times as fast as backpropagation using gradient descent.
Moller’s tests also showed that scaled conjugate gradient failed to
converge less often than traditional conjugate gradient or
backpropagation using gradient descent.
The Multilayer Perceptron Property Page
Controls for multilayer perceptron analyses are provided on a screen
in DTREG that has the following image:
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