Perceptron

"Perceptrons" redirects here. For the book of that title, see Perceptrons (book).

The perceptron is a type of artificial neural network invented in 1957 at the Cornell Aeronautical Laboratory by Frank Rosenblatt.^[1] It can be seen as the simplest kind of feedforward neural network: a linear classifier.

Definition

The perceptron is a binary classifier which maps its input x (a real-valued vector) to an output value f(x) (a single binary value):

$f(x) = \begin{cases}1 & \text{if }w \cdot x + b > 0\\0 & \text{otherwise}\end{cases}$

where w is a vector of real-valued weights, $w \cdot x$ is the dot product (which here computes a weighted sum), and b is the 'bias', a constant term that does not depend on any input value.

The value of f(x) (0 or 1) is used to classify x as either a positive or a negative instance, in the case of a binary classification problem. If b is negative, then the weighted combination of inputs must produce a positive value greater than | b | in order to push the classifier neuron over the 0 threshold. Spatially, the bias alters the position (though not the orientation) of the decision boundary. The perceptron learning algorithm does not terminate if the learning set is not linearly separable.

The perceptron is considered the simplest kind of feed-forward neural network.

Learning algorithm

Below is an example of a learning algorithm for a single-layer (no hidden layer) perceptron. For multilayer perceptrons, more complicated algorithms such as backpropagation must be used. Alternatively, methods such as the delta rule can be used if the function is non-linear and differentiable, although the one below will work as well.

The learning algorithm we demonstrate is the same across all the output neurons, therefore everything that follows is applied to a single neuron in isolation. We first define some variables:

• $y = f(\mathbf{z}) \,$ denotes the output from the perceptron for an input vector $\mathbf{z}$.
• $b \,$ is the bias term, which in the example below we take to be 0.
• $D = \{(\mathbf{x}_1,d_1),\dots,(\mathbf{x}_s,d_s)\} \,$ is the training set of s samples, where:
  • $\mathbf{x}_j$ is the n-dimensional input vector.
  • $d_j \,$ is the desired output value of the perceptron for that input.

We show the values of the nodes as follows:

• $x_{j,i} \,$ is the value of the ith node of the jth training input vector.
• $x_{j,0} = 1 \,$.

To represent the weights:

• $w_i \,$ is the ith value in the weight vector, to be multiplied by the value of the ith input node.

An extra dimension, with index n + 1, can be added to all input vectors, with $x_{j,n+1}=1 \,$, in which case $w_{n+1} \,$ replaces the bias term. To show the time-dependence of $\mathbf{w}$, we use:

• $w_i(t) \,$ is the weight i at time t.
• $\alpha \,$ is the learning rate, where $0 < \alpha \leq 1$.

Too high a learning rate makes the perceptron periodically oscillate around the solution unless additional steps are taken.

The appropriate weights are applied to the inputs, and the resulting weighted sum passed to a function that produces the output y.

Learning algorithm steps

1. Initialise weights and threshold. Note that weights may be initialised by setting each weight node $w_i(0) \,$ to 0 or to a small random value. In the example below, we choose the former.

2. For each sample $j \,$ in our training set $D \,$, perform the following steps over the input $\mathbf{x}_j \,$ and desired output $d_j \,$:

2a. Calculate the actual output:

$y_j(t) = f[\mathbf{w}(t)\cdot\mathbf{x}_j] = f[w_0(t) + w_1(t)x_{j,1} + w_2(t)x_{j,2} + \dotsb + w_n(t)x_{j,n}]$

2b. Adapt weights:

$w_i(t+1) = w_i(t) + \alpha (d_j - y_j(t)) x_{j,i} \,$, for all nodes $0 \leq i \leq n$.

Step 2 is repeated until the iteration error $d_j - y_j(t) \,$ is less than a user-specified error threshold $\gamma \,$, or a predetermined number of iterations have been completed. Note that the algorithm adapts the weights immediately after steps 2a and 2b are applied to a pair in the training set rather than waiting until all pairs in the training set have undergone these steps.

Separability and convergence

The training set D is said to be linearly separable if there exists a positive constant γ and a weight vector $\mathbf{w}$ such that $( \mathbf{w}\cdot\mathbf{x}_j + b ) d_j > \gamma$ for all 1 < j < m. That is, if we say that $\mathbf{w}$ is the weight vector to the perceptron, then the output of the perceptron, $\mathbf{w} \cdot \mathbf{x}_i + b$, multiplied by the desired output of the perceptron, d_i, must be greater than the positive constant, γ, for all input-vector/output-value pairs (x_j,d_j) in D_m.

Novikoff (1962) proved that the perceptron algorithm converges after a finite number of iterations if the data set is linearly separable. The idea of the proof is that the weight vector is always adjusted by a bounded amount in a direction that it has a negative dot product with, and thus can be bounded above by $O(\sqrt{t})$ where t is the number of changes to the weight vector. But it can also be bounded below by O(t) because if there exists an (unknown) satisfactory weight vector, then every change makes progress in this (unknown) direction by a positive amount that depends only on the input vector. This can be used to show that the number t of updates to the weight vector is bounded by (2R / γ)², where R is the maximum norm of an input vector.

However, if the training set is not linearly separable, the above online algorithm will not converge.

Note that the decision boundary of a perceptron is invariant with respect to scaling of the weight vector, i.e. a perceptron trained with initial weight vector $\mathbf{w}$ and learning rate $\alpha \,$ is an identical estimator to a perceptron trained with initial weight vector $\mathbf{w}/\alpha \,$ and learning rate 1. Thus, since the initial weights become irrelevant with increasing number of iterations, the learning rate does not matter in the case of the perceptron and is usually just set to one.

Variants

The pocket algorithm with ratchet (Gallant, 1990) solves the stability problem of perceptron learning by keeping the best solution seen so far "in its pocket". The pocket algorithm then returns the solution in the pocket, rather than the last solution. It can be used also for non-separable data sets, where the aim is to find a perceptron with a small number of misclassifications.

In separable problems, perceptron training can also aim at finding the largest separating margin between the classes. The so-called perceptron of optimal stability can be determined by means of iterative training and optimization schemes, e.g. the Min-Over algorithm (Krauth and Mezard, 1987)^[2] or the AdaTron (Anlauf and Biehl, 1989)) .^[3] The latter exploits the fact that the corresponding quadratic optimization problem is convex. The perceptron of optimal stability is, together with the kernel trick, one of the conceptual foundations of the support vector machine.

The α-perceptron further utilised a preprocessing layer of fixed random weights, with thresholded output units. This enabled the perceptron to classify analogue patterns, by projecting them into a binary space. In fact, for a projection space of sufficiently high dimension, patterns can become linearly separable.

As an example, consider the case of having to classify data into two classes. Here is a small such data set, consisting of two points coming from two Gaussian distributions.

• 

  Two-class Gaussian data

• 

  A linear classifier operating on the original space

• 

  A linear classifier operating on a high-dimensional projection

A linear classifier can only separate things with a hyperplane, so it's not possible to classify all the examples perfectly. On the other hand, we may project the data into a large number of dimensions. In this case a random matrix was used to project the data linearly to a 1000-dimensional space; then each resulting data point was transformed through the hyperbolic tangent function. A linear classifier can then separate the data, as shown in the third figure. However the data may still not be completely separable in this space, in which the perceptron algorithm would not converge. In the example shown, stochastic steepest gradient descent was used to adapt the parameters.

Furthermore, by adding nonlinear layers between the input and output, one can separate all data and indeed, with enough training data, model any well-defined function to arbitrary precision. This model is a generalization known as a multilayer perceptron.

Another way to solve nonlinear problems without the need of multiple layers is the use of higher order networks (sigma-pi unit). In this type of network each element in the input vector is extended with each pairwise combination of multiplied inputs (second order). This can be extended to n-order network.

It should be kept in mind, however, that the best classifier is not necessarily that which classifies all the training data perfectly. Indeed, if we had the prior constraint that the data come from equi-variant Gaussian distributions, the linear separation in the input space is optimal.

Other training algorithms for linear classifiers are possible: see, e.g., support vector machine and logistic regression.

Example

A perceptron learns to perform a binary NAND function on inputs $x_1 \,$ and $x_2 \,$.

Inputs: $x_0 \,$, $x_1 \,$, $x_2 \,$, with input $x_0 \,$ held constant at 1.

Threshold: 0.5

Bias: 0

Learning rate: 0.1

Training set, consisting of four samples: $\{((0, 0), 1), ((0, 1), 1), ((1, 0), 1), ((1, 1), 0)\} \,$

In the following, the final weights of one iteration become the initial weights of the next. Each cycle over all the samples in the training set is demarcated with heavy lines.

Input				Initial weights			Output					Error	Correction	Final weights
Sensor values			Desired output				Per sensor			Sum	Network
x0	x1	x2	z	w0	w1	w2	c0	c1	c2	s	n	e	d	w0	w1	w2
							x0 * w0	x1 * w1	x2 * w2	c0 +c1 +c2	if s>t then 1, else 0	z-n	r * e
1	0	0	1	0	0	0	0	0	0	0	0	1	+0.1	0.1	0	0
1	0	1	1	0.1	0	0	0.1	0	0	0.1	0	1	+0.1	0.2	0	0.1
1	1	0	1	0.2	0	0.1	0.2	0	0	0.2	0	1	+0.1	0.3	0.1	0.1
1	1	1	0	0.3	0.1	0.1	0.3	0.1	0.1	0.5	0	0	0	0.3	0.1	0.1
1	0	0	1	0.3	0.1	0.1	0.3	0	0	0.3	0	1	+0.1	0.4	0.1	0.1
1	0	1	1	0.4	0.1	0.1	0.4	0	0.1	0.5	0	1	+0.1	0.5	0.1	0.2
1	1	0	1	0.5	0.1	0.2	0.5	0.1	0	0.6	1	0	0	0.5	0.1	0.2
1	1	1	0	0.5	0.1	0.2	0.5	0.1	0.2	0.8	1	-1	-0.1	0.4	0	0.1
1	0	0	1	0.4	0	0.1	0.4	0	0	0.4	0	1	+0.1	0.5	0	0.1
1	0	1	1	0.5	0	0.1	0.5	0	0.1	0.6	1	0	0	0.5	0	0.1
1	1	0	1	0.5	0	0.1	0.5	0	0	0.5	0	1	+0.1	0.6	0.1	0.1
1	1	1	0	0.6	0.1	0.1	0.6	0.1	0.1	0.8	1	-1	-0.1	0.5	0	0
1	0	0	1	0.5	0	0	0.5	0	0	0.5	0	1	+0.1	0.6	0	0
1	0	1	1	0.6	0	0	0.6	0	0	0.6	1	0	0	0.6	0	0
1	1	0	1	0.6	0	0	0.6	0	0	0.6	1	0	0	0.6	0	0
1	1	1	0	0.6	0	0	0.6	0	0	0.6	1	-1	-0.1	0.5	-0.1	-0.1
1	0	0	1	0.5	-0.1	-0.1	0.5	0	0	0.5	0	1	+0.1	0.6	-0.1	-0.1
1	0	1	1	0.6	-0.1	-0.1	0.6	0	-0.1	0.5	0	1	+0.1	0.7	-0.1	0
1	1	0	1	0.7	-0.1	0	0.7	-0.1	0	0.6	1	0	0	0.7	-0.1	0
1	1	1	0	0.7	-0.1	0	0.7	-0.1	0	0.6	1	-1	-0.1	0.6	-0.2	-0.1
1	0	0	1	0.6	-0.2	-0.1	0.6	0	0	0.6	1	0	0	0.6	-0.2	-0.1
1	0	1	1	0.6	-0.2	-0.1	0.6	0	-0.1	0.5	0	1	+0.1	0.7	-0.2	0
1	1	0	1	0.7	-0.2	0	0.7	-0.2	0	0.5	0	1	+0.1	0.8	-0.1	0
1	1	1	0	0.8	-0.1	0	0.8	-0.1	0	0.7	1	-1	-0.1	0.7	-0.2	-0.1
1	0	0	1	0.7	-0.2	-0.1	0.7	0	0	0.7	1	0	0	0.7	-0.2	-0.1
1	0	1	1	0.7	-0.2	-0.1	0.7	0	-0.1	0.6	1	0	0	0.7	-0.2	-0.1
1	1	0	1	0.7	-0.2	-0.1	0.7	-0.2	0	0.5	0	1	+0.1	0.8	-0.1	-0.1
1	1	1	0	0.8	-0.1	-0.1	0.8	-0.1	-0.1	0.6	1	-1	-0.1	0.7	-0.2	-0.2
1	0	0	1	0.7	-0.2	-0.2	0.7	0	0	0.7	1	0	0	0.7	-0.2	-0.2
1	0	1	1	0.7	-0.2	-0.2	0.7	0	-0.2	0.5	0	1	+0.1	0.8	-0.2	-0.1
1	1	0	1	0.8	-0.2	-0.1	0.8	-0.2	0	0.6	1	0	0	0.8	-0.2	-0.1
1	1	1	0	0.8	-0.2	-0.1	0.8	-0.2	-0.1	0.5	0	0	0	0.8	-0.2	-0.1
1	0	0	1	0.8	-0.2	-0.1	0.8	0	0	0.8	1	0	0	0.8	-0.2	-0.1
1	0	1	1	0.8	-0.2	-0.1	0.8	0	-0.1	0.7	1	0	0	0.8	-0.2	-0.1

This example can be implemented in the following Python code.

threshold = 0.5
learning_rate = 0.1
weights = [0, 0, 0]
training_set = [((1, 0, 0), 1), ((1, 0, 1), 1), ((1, 1, 0), 1), ((1, 1, 1), 0)]
 
def sum_function(values):
    return sum(value * weights[index] for index, value in enumerate(values))
 
while True:
    print '-' * 60
    error_count = 0
    for input_vector, desired_output in training_set:
        print weights
        result = 1 if sum_function(input_vector) > threshold else 0
        error = desired_output - result
        if error != 0:
            error_count += 1
            for index, value in enumerate(input_vector):
                weights[index] += learning_rate * error * value
    if error_count == 0:
        break

Multiclass perceptron

Like most other techniques for training linear classifiers, the perceptron generalizes naturally to multiclass classification. Here, the input x and the output y are drawn from arbitrary sets. A feature representation function f(x,y) maps each possible input/output pair to a finite-dimensional real-valued feature vector. As before, the feature vector is multiplied by a weight vector w, but now the resulting score is used to choose among many possible outputs:

$\hat y = \mathrm{argmax}_y f(x,y) \cdot w.$

Learning again iterates over the examples, predicting an output for each, leaving the weights unchanged when the predicted output matches the target, and changing them when it does not. The update becomes:

$w_{t+1} = w_t + f(x, y) - f(x,\hat y).$

This multiclass formulation reduces to the original perceptron when x is a real-valued vector, y is chosen from {0,1}, and f(x,y) = yx.

For certain problems, input/output representations and features can be chosen so that $\mathrm{argmax}_y f(x,y) \cdot w$ can be found efficiently even though y is chosen from a very large or even infinite set.

In recent years, perceptron training has become popular in the field of natural language processing for such tasks as part-of-speech tagging and syntactic parsing (Collins, 2002).

History

See also: History of artificial intelligence, AI winter and Frank Rosenblatt

Although the perceptron initially seemed promising, it was eventually proved that perceptrons could not be trained to recognise many classes of patterns. This led to the field of neural network research stagnating for many years, before it was recognised that a feedforward neural network with two or more layers (also called a multilayer perceptron) had far greater processing power than perceptrons with one layer (also called a single layer perceptron). Single layer perceptrons are only capable of learning linearly separable patterns; in 1969 a famous book entitled Perceptrons by Marvin Minsky and Seymour Papert showed that it was impossible for these classes of network to learn an XOR function. It is often believed that they also conjectured (incorrectly) that a similar result would hold for a multi-layer perceptron network. However, this is not true, as both Minsky and Papert already knew that multi-layer perceptrons were capable of producing an XOR Function. (See the page on Perceptrons for more information.) Three years later Stephen Grossberg published a series of papers introducing networks capable of modelling differential, contrast-enhancing and XOR functions. (The papers were published in 1972 and 1973, see e.g.: Grossberg, Contour enhancement, short-term memory, and constancies in reverberating neural networks. Studies in Applied Mathematics, 52 (1973), 213-257, online [1]). Nevertheless the often-miscited Minsky/Papert text caused a significant decline in interest and funding of neural network research. It took ten more years until neural network research experienced a resurgence in the 1980s. This text was reprinted in 1987 as "Perceptrons - Expanded Edition" where some errors in the original text are shown and corrected.

More recently, interest in the perceptron learning algorithm increased again after Freund and Schapire (1998) showed that presented a voted formulation of the original algorithm (attaining a large margin) to which the kernel trick can be applied. Subsequent studies have shown its applicability to a class of more complex tasks, later called as structured learning, than binary classification (Collins, 2002), and to large-scale machine learning problems in a distributed computing setting (McDonald, Hall and Mann, 2010).

References

1. ^ Rosenblatt, Frank (1957), The Perceptron--a perceiving and recognizing automaton. Report 85-460-1, Cornell Aeronautical Laboratory.
2. ^ W. Krauth and M. Mezard. Learning algorithms with optimal stablilty in neural networks. J. of Physics A: Math. Gen. 20: L745-L752 (1987)
3. ^ J.K. Anlauf and M. Biehl. The AdaTron: an Adaptive Perceptron algorithm. Europhysics Letters 10: 687-692 (1989)

• Rosenblatt, Frank (1958), The Perceptron: A Probabilistic Model for Information Storage and Organization in the Brain, Cornell Aeronautical Laboratory, Psychological Review, v65, No. 6, pp. 386–408. doi:10.1037/h0042519.
• Rosenblatt, Frank (1962), Principles of Neurodynamics. Washington, DC:Spartan Books.
• Minsky M. L. and Papert S. A. 1969. Perceptrons. Cambridge, MA: MIT Press.
• Freund, Y. and Schapire, R. E. 1998. Large margin classification using the perceptron algorithm. In Proceedings of the 11th Annual Conference on Computational Learning Theory (COLT' 98). ACM Press.
• Freund, Y. and Schapire, R. E. 1999. Large margin classification using the perceptron algorithm. In Machine Learning 37(3):277-296, 1999.
• Gallant, S. I. (1990). Perceptron-based learning algorithms. IEEE Transactions on Neural Networks, vol. 1, no. 2, pp. 179–191.
• Mcdonald, R., Hall, K., & Mann, G. (2010). Distributed Training Strategies for the Structured Perceptron. pp. 456-464. Association for Computational Linguistics.
• Novikoff, A. B. (1962). On convergence proofs on perceptrons. Symposium on the Mathematical Theory of Automata, 12, 615-622. Polytechnic Institute of Brooklyn.
• Widrow, B., Lehr, M.A., "30 years of Adaptive Neural Networks: Perceptron, Madaline, and Backpropagation," Proc. IEEE, vol 78, no 9, pp. 1415–1442, (1990).
• Collins, M. 2002. Discriminative training methods for hidden Markov models: Theory and experiments with the perceptron algorithm in Proceedings of the Conference on Empirical Methods in Natural Language Processing (EMNLP '02)
• Yin, Hongfeng (1996), Perceptron-Based Algorithms and Analysis, Spectrum Library, Concordia University, Canada

External links

• SergeiAlderman-ANN.rtf
• Chapter 3 Weighted networks - the perceptron and chapter 4 Perceptron learning of Neural Networks - A Systematic Introduction by Raúl Rojas (ISBN 978-3-540-60505-8)
• Pithy explanation of the update rule by Charles Elkan
• C# implementation of a perceptron
• History of perceptrons
• Mathematics of perceptrons
• Perceptron demo applet and an introduction by examples
• Perceptron simple Java applet