The Perceptron

A Learning Program

For decades there have been attempts to create computer programs that can learn like people - Artificial Intelligence. For example, how do you teach a child to recognize what a chair is? You show him examples telling him "This is a chair ; That one is not a chair" until the child learns the concept of what a chair is. In this stage, the child can look at the examples we have shown him and answer correctly to the question "Is this object a chair?". Furthermore, if we show to the child new objects, that he didn't see before, we could expect him to recognize correctly whether the new object is a chair or not, providing that we've given him enough positive and negative examples. This is exactly the idea behind the perceptron.

The Perceptron

The perceptron is a program that learn concepts, i.e. it can learn to respond with True (1) or False (0) for inputs we present to it, by repeatedly "studying" examples presented to it. The Perceptron is a single layer neural network whose weights and biases could be trained to produce a correct target vector when presented with the corresponding input vector. The training technique used is called the perceptron learning rule. The perceptron generated great interest due to its ability to generalize from its training vectors and work with randomly distributed connections. Perceptrons are especially suited for simple problems in pattern classification. Our perceptron network consists of a single neuron connected to two inputs through a set of 2 weights, with an additional bias input. The perceptron calculates its output using the following equation:

P * W + b > 0
        

where P is the input vector presented to the network, W is the vector of weights and b is the bias.

The Learning Rule

The perceptron is trained to respond to each input vector with a corresponding target output of either 0 or 1. The learning rule has been proven to converge on a solution in finite time if a solution exists. The learning rule can be summarized in the following two equations:

For all i:

                W(i) = W(i) + [ T - A ] * P(i)

                b    = b + [ T - A ]
        

where W is the vector of weights, P is the input vector presented to the network, T is the correct result that the neuron should have shown, A is the actual output of the neuron, and b is the bias.

Training

Vectors from a training set are presented to the network one after another. If the network's output is correct, no change is made. Otherwise, the weights and biases are updated using the perceptron learning rule. An entire pass through all of the input training vectors is called an epoch. When such an entire pass of the training set has occured without error, training is complete. At this time any input training vector may be presented to the network and it will respond with the correct output vector. If a vector P not in the training set is presented to the network, the network will tend to exhibit generalization by responding with an output similar to target vectors for input vectors close to the previously unseen input vector P.

Limitations

Perceptron networks have several limitations. First, the output values of a perceptron can take on only one of two values (True or False). Second, perceptrons can only classify linearly separable sets of vectors. If a straight line or plane can be drawn to seperate the input vectors into their correct categories, the input vectors are linearly separable and the perceptron will find the solution. If the vectors are not linearly separable learning will never reach a point where all vectors are classified properly. The most famous example of the perceptron's inability to solve problems with linearly nonseparable vectors is the boolean exclusive-or problem.

Our Implementation

We implemented a single neuron perceptron with 2 inputs. The input for the neuron can be taken from a graphic user interface, by clicking on points in a board. A click with the left mouse button generates a '+' sign on the board, marking that it's a point where the perceptron should respond with 'True'. A click with the right mouse button generates a '-' sign on the board, marking that it's a point where the perceptron should respond with 'False'. When enough points have been entered, the user can click on 'Start', which will introduce these points as inputs to the perceptron, have it learn these input vectors and show a line which corresponds to the linear division of the plane into regions of opposite neuron response. Here's an example of a screen shot: