ADALINE

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:

• ${\displaystyle x}$ is the input vector
• ${\displaystyle w}$ is the weight vector
• ${\displaystyle n}$ is the number of inputs
• ${\displaystyle \theta }$ some constant
• ${\displaystyle y}$ is the output of the model

then we find that the output is ${\displaystyle y=\sum _{j=1}^{n}x_{j}w_{j}+\theta }$. If we further assume that

• ${\displaystyle x_{0}=1}$
• ${\displaystyle w_{0}=\theta }$

then the output further reduces to: ${\displaystyle y=\sum _{j=0}^{n}x_{j}w_{j}}$

The learning rule used by ADALINE is the LMS ("least mean squares") algorithm, a special case of gradient descent.

Define the following notations:

• ${\displaystyle \eta }$ is the learning rate (some positive constant)
• ${\displaystyle y}$ is the output of the model
• ${\displaystyle o}$ is the target (desired) output
• ${\displaystyle E=(o-y)^{2}}$ is the square of the error.

The LMS algorithm updates the weights by

$${\displaystyle w\leftarrow w+\eta (o-y)x.}$$

This update rule minimizes ${\displaystyle E}$, the square of the error,[6] and 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.

MADALINE Rule 1 (MRI) - The first of these dates back to 1962 and cannot adapt the weights of the hidden-output connection.[10]

MADALINE Rule 2 (MRII) - The second training algorithm improved on Rule I and was described in 1988.[8] 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:

• finds the hidden layer unit (ADALINE classifier) with the lowest confidence in its prediction,
• tentatively flips the sign of the unit,
• accepts or rejects the change based on whether the network's error is reduced,
• stops when the error is zero.

MADALINE Rule 3 - The third "Rule" applied to a modified network with sigmoid activations instead of signum; it was later found to be equivalent to backpropagation.[11]

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]

• Multilayer perceptron

• "Delta Learning Rule: ADALINE". Artificial Neural Networks. Universidad Politécnica de Madrid. Archived from the original on 2002-06-15.
• "Memristor-Based Multilayer Neural Networks With Online Gradient Descent Training". Implementation of the ADALINE algorithm with memristors in analog computing.