AI News, What is the difference between a Perceptron, Adaline, and neural network model?

What is the difference between a Perceptron, Adaline, and neural network model?

learning algorithms can actually be summarized by 4 simple steps – given that we use stochastic gradient descent for Adaline: We write the weight update in each iteration as:

Here, the activation function is not linear (like in Adaline), but we use a non-linear activation function like the logistic sigmoid (the one that we use in logistic regression) or the hyperbolic tangent, or a piecewise-linear activation function such as the rectifier linear unit (ReLU).

In addition, we often use a softmax function (a generalization of the logistic sigmoid for multi-class problems) in the output layer, and a threshold function to turn the predicted probabilities (by the softmax) into class labels.

By connecting the artificial neurons in this network through non-linear activation functions, we can create complex, non-linear decision boundaries that allow us to tackle problems where the different classes are not linearly separable.

What is the difference between a Perceptron, Adaline, and neural network model?

learning algorithms can actually be summarized by 4 simple steps – given that we use stochastic gradient descent for Adaline: We write the weight update in each iteration as:

Here, the activation function is not linear (like in Adaline), but we use a non-linear activation function like the logistic sigmoid (the one that we use in logistic regression) or the hyperbolic tangent, or a piecewise-linear activation function such as the rectifier linear unit (ReLU).

In addition, we often use a softmax function (a generalization of the logistic sigmoid for multi-class problems) in the output layer, and a threshold function to turn the predicted probabilities (by the softmax) into class labels.

By connecting the artificial neurons in this network through non-linear activation functions, we can create complex, non-linear decision boundaries that allow us to tackle problems where the different classes are not linearly separable.

Single-Layer Neural Networks and Gradient Descent

This article offers a brief glimpse of the history and basic concepts of machine learning.

We will take a look at the first algorithmically described neural network and the gradient descent algorithm in context of adaptive linear neurons, which will not only introduce the principles of machine learning but also serve as the basis for modern multilayer neural networks in future articles.

Thanks to machine learning, we enjoy robust email spam filters, convenient text and voice recognition, reliable web search engines, challenging chess players, and, hopefully soon, safe and efficient self-driving cars.

The perceptron is not only the first algorithmically described learning algorithm [1], but it is also very intuitive, easy to implement, and a good entry point to the (re-discovered) modern state-of-the-art machine learning algorithms: Artificial neural networks (or “deep learning” if you like).

To put the perceptron algorithm into the broader context of machine learning: The perceptron belongs to the category of supervised learning algorithms, single-layer binary linear classifiers to be more specific.

Next, we define an activation function g(\mathbf{z}) that takes a linear combination of the input values \mathbf{x} and weights \mathbf{w} as input (\mathbf{z} = w_1x_{1} + \dots + w_mx_{m}), and if g(\mathbf{z}) is greater than a defined threshold \theta we predict 1 and -1 otherwise;

in this case, this activation function g is an alternative form of a simple “unit step function,” which is sometimes also called “Heaviside step function.” (Please note that the unit step is classically defined as being equal to 0 if % <![CDATA[ z

To summarize the main points from the previous section: A perceptron receives multiple input signals, and if the sum of the input signals exceed a certain threshold it either returns a signal or remains “silent” otherwise.

What made this a “machine learning” algorithm was Frank Rosenblatt’s idea of the perceptron learning rule: The perceptron algorithm is about learning the weights for the input signals in order to draw linear decision boundary that allows us to discriminate between the two linearly separable classes +1 and -1.

Rosenblatt’s initial perceptron rule is fairly simple and can be summarized by the following steps: The output value is the class label predicted by the unit step function that we defined earlier (output =g(\mathbf{z})) and the weight update can be written more formally as w_j := w_j + \Delta w_j.

The value for updating the weights at each increment is calculated by the learning rule where \eta is the learning rate (a constant between 0.0 and 1.0), “target” is the true class label, and the “output” is the predicted class label.

w_j = \eta(1^{(i)} - 1^{(i)})\;x^{(i)}_{j} = 0 However, in case of a wrong prediction, the weights are being “pushed” towards the direction of the positive or negative target class, respectively: \Delta w_j = \eta(1^{(i)} - -1^{(i)})\;x^{(i)}_{j} = \eta(2)\;x^{(i)}_{j} \Delta

If the two classes can’t be separated by a linear decision boundary, we can set a maximum number of passes over the training dataset (“epochs”) and/or a threshold for the number of tolerated misclassifications.

Our intuition tells us that a decision boundary with a large margin between the classes (as indicated by the dashed line in the figure below) likely has a better generalization error than the decision boundary of the perceptron.

In contrast to the perceptron rule, the delta rule of the adaline (also known as Widrow-Hoff” rule or Adaline rule) updates the weights based on a linear activation function rather than a unit step function;

(The fraction \frac{1}{2} is just used for convenience to derive the gradient as we will see in the next paragraphs.) In order to minimize the SSE cost function, we will use gradient descent, a simple yet useful optimization algorithm that is often used in machine learning to find the local minimum of linear systems.

mentioned above, each update is updated by taking a step into the opposite direction of the gradient \Delta \mathbf{w} = - \eta \nabla J(\mathbf{w}), thus, we have to compute the partial derivative of the cost function for each weight in the weight vector: \Delta w_j = - \eta \frac{\partial J}{\partial w_j}.

The partial derivative of the SSE cost function for a particular weight can be calculated as follows: (t = target, o = output) And if we plug the results back into the learning rule, we get \Delta w_j = - \eta \frac{\partial J}{\partial w_j} = - \eta \sum_i (t^{(i)} - o^{(i)})(- x^{(i)}_{j}) = \eta \sum_i (t^{(i)} - o^{(i)})x^{(i)}_{j}, Eventually, we can apply a simultaneous weight update similar to the perceptron rule: \mathbf{w} := \mathbf{w} + \Delta \mathbf{w}.

Another advantage of online learning is that the classifier can be immediately updated as new training data arrives, e.g., in web applications, and old training data can be discarded if storage is an issue.

In later articles, we will take a look at different approaches to dynamically adjust the learning rate, the concepts of “One-vs-All” and “One-vs-One” for multi-class classification, regularization to overcome overfitting by introducing additional information, dealing with nonlinear problems and multilayer neural networks, different activation functions for artificial neurons, and related concepts such as logistic regression and support vector machines.

ADALINE

ADALINE (Adaptive Linear Neuron or later Adaptive Linear Element) is an early single-layer artificial neural network and the name of the physical device that implemented this network.[1][2][3][4][5] The network uses memistors.

It was developed by Professor Bernard Widrow and his graduate student Ted Hoff at Stanford University in 1960.

It consists of a weight, a bias and a summation function.

The difference between Adaline and the standard (McCulloch–Pitts) perceptron is that in the learning phase, the weights are adjusted according to the weighted sum of the inputs (the net).

In the standard perceptron, the net is passed to the activation (transfer) function and the function's output is used for adjusting the weights.

Adaline is a single layer neural network with multiple nodes where each node accepts multiple inputs and generates one output.

Given the following variables:as then we find that the output is

If we further assume that then the output further reduces to:

Let us assume: then the weights are updated as follows

.[6] This update rule is in fact the stochastic gradient descent update for linear regression.[7] MADALINE (Many ADALINE[8]) is a three-layer (input, hidden, output), fully connected, feed-forward artificial neural network architecture for classification that uses ADALINE units in its hidden and output layers, i.e.

its activation function is the sign function.[9] The three-layer network uses memistors.

Three different training algorithms for MADALINE networks, which cannot be learned using backpropagation because the sign function is not differentiable, have been suggested, called Rule I, Rule II and Rule III.

The first of these dates back to 1962 and cannot adapt the weights of the hidden-output connection.[10] The second training algorithm improved on Rule I and was described in 1988.[8] The third 'Rule' applied to a modified network with sigmoid activations instead of signum;

it was later found to be equivalent to backpropagation.[10] The Rule II training algorithm is based on a principle called 'minimal disturbance'.

It proceeds by looping over training examples, then for each example, it: Additionally, when flipping single units' signs does not drive the error to zero for a particular example, the training algorithm starts flipping pairs of units' signs, then triples of units, etc.[8]

Machine Learning with scikit-learn

In this tutorial, we'll learn another type of single-layer neural network (still this is also a perceptron) called Adaline (Adaptive linear neuron) rule (also known as the Widrow-Hoff rule).

The perceptron algorithm enables the model automatically learn the optimal weight coefficients that are then multiplied with the input features in order to make the decision of whether a neuron fires or not.

Update of each weight $w_j$ in the weight vector $w$ can be written as: The value of $\Delta w_j$ , which is used to update the weight $w_j$, is calculated as the following: One of the most critical tasks in supervised machine learning algorithms is to minimize cost function.

Errors (SSE) between the calculated outcome and the true class label: COmpared with the unit step function, the advantages of this continuous linear activation function are:

the adaptive linear learning rule looks identical to the perceptron rule, the $\phi(z^{(i)})$ with $z^{(i)}=w^Tx^{(i)}$ is a real number and not an integer class label. Also,

can minimize a cost function by taking a step into the opposite direction of a gradient that is calculated from the whole training set, and this is why this approach is also called as batch gradient descent. Implementation

the perceptron rule and Adaptive Linear Neuron are very similar, we can take the perceptron implementation that we defined earlier and change the fit method so that the weights are updated by minimizing the cost function via gradient descent. Here

errors = y - output

self.weight[0] += self.rate * errors.sum()

Although the adaptive linear learning rule looks identical to the perceptron rule, the $\phi(z^{(i)})$ with $z^{(i)}=w^Tx^{(i)}$ is a real number and not an integer class label.

We can minimize a cost function by taking a step into the opposite direction of a gradient that is calculated from the whole training set, and this is why this approach is also called as batch gradient descent.

Since the perceptron rule and Adaptive Linear Neuron are very similar, we can take the perceptron implementation that we defined earlier and change the fit method so that the weights are updated by minimizing the cost function via gradient descent.

So, to standardize the $j$-th feature, we just need to subtract the sample mean $\mu_j$ from every training sample and divide it by its standard deviation $sigma_j$: where $x_j$ is a vector consisting of the $j$-th feature values of all training samples $n$.

We can standardize by using the NumPy methods mean and std: After the standardization, we will train the Linear model again using the not so small learning rate of $\eta = 0.01$: Here is our new code for the two pictures above: Continued to Single Layer Neural Network : Adaptive Linear Neuron using linear (identity) activation function with stochastic gradient descent (SGD).

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