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Simplified example of training a neural network for object detection: The network is trained on multiple images depicting eitherstarfish orsea urchins, which are correlated with "nodes" that represent visualfeatures. The starfish match with a ringed texture and a star outline, whereas most sea urchins match with a striped texture and oval shape. However, the instance of a ring-textured sea urchin creates a weakly weighted association between them.

Subsequent run of the network on an input image (left):^[1] The network correctly detects the starfish. However, the weakly weighted association between ringed texture and sea urchin also confers a weak signal to the latter from one of two intermediate nodes. In addition, a shell that was not included in the training gives a weak signal for the oval shape, also resulting in a weak signal for the sea urchin output. These weak signals may result in afalse positive result for sea urchin.
In reality, textures and outlines would not be represented by single nodes, but rather by associated weight patterns of multiple nodes.

Feedforward refers to recognition-inference architecture of neural networks.Artificial neural network architectures are based on inputs multiplied by weights to obtain outputs (inputs-to-output): feedforward.^[2]Recurrent neural networks, or neural networks with loops allow information from later processing stages to feed back to earlier stages for sequence processing.^[3] However, at every stage of inference a feedforward multiplication remains the core, essential for backpropagation^{[4][5][6][7][8]} or backpropagation through time. Thus neural networks cannot contain feedback likenegative feedback orpositive feedback where the outputs feed back to thevery same inputs and modify them, because this forms an infinite loop which is not possible to rewind in time to generate an error signal through backpropagation. This issue and nomenclature appear to be a point of confusion between some computer scientists and scientists in other fields studying brain networks.^[9]

The two historically commonactivation functions are bothsigmoids, and are described by

${\displaystyle y(v_i)=\tanh (v_i)\text{and}y(v_i)=(1+e^{-v_i}{)}^{-1}}$.

The first is ahyperbolic tangent that ranges from -1 to 1, while the other is thelogistic function, which is similar in shape but ranges from 0 to 1. Here${\displaystyle y_i}$ is the output of the${\displaystyle i}$th node (neuron) and${\displaystyle v_i}$ is the weighted sum of the input connections. Alternative activation functions have been proposed, including therectifier and softplus functions. More specialized activation functions includeradial basis functions (used inradial basis networks, another class of supervised neural network models).

In recent developments ofdeep learning therectified linear unit (ReLU) is more frequently used as one of the possible ways to overcome the numericalproblems related to the sigmoids.

Learning occurs by changing connection weights after each piece of data is processed, based on the amount of error in the output compared to the expected result. This is an example ofsupervised learning, and is carried out throughbackpropagation.

We can represent the degree of error in an output node${\displaystyle j}$ in the${\displaystyle n}$th data point (training example) by${\displaystyle e_j(n)=d_j(n)-y_j(n)}$, where${\displaystyle d_j(n)}$ is the desired target value for${\displaystyle n}$th data point at node${\displaystyle j}$, and${\displaystyle y_j(n)}$ is the value produced at node${\displaystyle j}$ when the${\displaystyle n}$th data point is given as an input.

The node weights can then be adjusted based on corrections that minimize the error in the entire output for the${\displaystyle n}$th data point, given by

${\displaystyle \mathcal{E}(n)=\frac{1}{2}\sum _{\text{output node }j}{e}_j^2(n)}$.

Usinggradient descent, the change in each weight${\displaystyle w_{ij}}$ is

${\displaystyle \mathrm{\Delta }{w}_{ji}(n)=-\eta \frac{\mathrm{\partial }\mathcal{E}(n)}{\mathrm{\partial }{v}_j(n)}{y}_i(n)}$

where${\displaystyle y_i(n)}$ is the output of the previous neuron${\displaystyle i}$, and${\displaystyle \eta }$ is thelearning rate, which is selected to ensure that the weights quickly converge to a response, without oscillations. In the previous expression,${\displaystyle \frac{\mathrm{\partial }\mathcal{E}(n)}{\mathrm{\partial }{v}_j(n)}}$ denotes the partial derivate of the error${\displaystyle \mathcal{E}(n)}$ according to the weighted sum${\displaystyle v_j(n)}$ of the input connections of neuron${\displaystyle i}$.

The derivative to be calculated depends on the induced local field${\displaystyle v_j}$, which itself varies. It is easy to prove that for an output node this derivative can be simplified to

${\displaystyle -\frac{\mathrm{\partial }\mathcal{E}(n)}{\mathrm{\partial }{v}_j(n)}=e_j(n){\varphi }^{\mathrm{\prime }}(v_j(n))}$

where${\displaystyle {\varphi }^{\mathrm{\prime }}}$ is the derivative of the activation function described above, which itself does not vary. The analysis is more difficult for the change in weights to a hidden node, but it can be shown that the relevant derivative is

${\displaystyle -\frac{\mathrm{\partial }\mathcal{E}(n)}{\mathrm{\partial }{v}_j(n)}={\varphi }^{\mathrm{\prime }}(v_j(n))\sum _k-\frac{\mathrm{\partial }\mathcal{E}(n)}{\mathrm{\partial }{v}_k(n)}{w}_{kj}(n)}$.

This depends on the change in weights of the${\displaystyle k}$th nodes, which represent the output layer. So to change the hidden layer weights, the output layer weights change according to the derivative of the activation function, and so this algorithm represents a backpropagation of the activation function.^[10]

• Circa 1800,Legendre (1805) andGauss (1795) created the simplest feedforward network which consists of a single weight layer with linear activation functions. It was trained by theleast squares method for minimisingmean squared error, also known aslinear regression. Legendre and Gauss used it for the prediction of planetary movement from training data.^{[11][12][13][14][15]}

• In 1943,Warren McCulloch andWalter Pitts proposed the binaryartificial neuron as a logical model of biological neural networks.^[16]
• In 1958,Frank Rosenblatt proposed the multilayeredperceptron model, consisting of an input layer, a hidden layer with randomized weights that did not learn, and an output layer with learnable connections.^[17] R. D. Joseph (1960)^[18] mentions an even earlier perceptron-like device:^[13] "Farley and Clark of MIT Lincoln Laboratory actually preceded Rosenblatt in the development of a perceptron-like device." However, "they dropped the subject."

• In 1960, Joseph^[18] also discussedmultilayer perceptrons with an adaptive hidden layer. Rosenblatt (1962)^{[19]: section 16} cited and adopted these ideas, also crediting work by H. D. Block and B. W. Knight. Unfortunately, these early efforts did not lead to a working learning algorithm for hidden units, i.e.,deep learning.

• In 1965,Alexey Grigorevich Ivakhnenko and Valentin Lapa publishedGroup Method of Data Handling, the first workingdeep learning algorithm, a method to train arbitrarily deep neural networks.^[20][21] It is based on layer by layer training through regression analysis. Superfluous hidden units are pruned using a separate validation set. Since the activation functions of the nodes are Kolmogorov-Gabor polynomials, these were also the first deep networks with multiplicative units or "gates."^[13] It was used to train an eight-layer neural net in 1971.

• In 1967,Shun'ichi Amari reported^[22] the first multilayered neural network trained bystochastic gradient descent, which was able to classify non-linearily separable pattern classes. Amari's student Saito conducted the computer experiments, using a five-layered feedforward network with two learning layers.^[13]

• In 1970,Seppo Linnainmaa published the modern form ofbackpropagation in his master thesis (1970).^[23][24][13] G.M. Ostrovski et al. republished it in 1971.^[25][26]Paul Werbos applied backpropagation to neural networks in 1982^[7][27] (his 1974 PhD thesis, reprinted in a 1994 book,^[28] did not yet describe the algorithm^[26]). In 1986,David E. Rumelhart et al. popularised backpropagation but did not cite the original work.^[29][8]

• In 2003, interest in backpropagation networks returned due to the successes ofdeep learning being applied tolanguage modelling byYoshua Bengio with co-authors.^[30]

If using a threshold, i.e. a linearactivation function, the resultinglinear threshold unit is called aperceptron. (Often the term is used to denote just one of these units.) Multiple parallel non-linear units are able toapproximate any continuous function from a compact interval of the real numbers into the interval [−1,1] despite the limited computational power of single unit with a linear threshold function.^[31]

Perceptrons can be trained by a simple learning algorithm that is usually called thedelta rule. It calculates the errors between calculated output and sample output data, and uses this to create an adjustment to the weights, thus implementing a form ofgradient descent.

Amultilayer perceptron (MLP) is amisnomer for a modern feedforward artificial neural network, consisting of fully connected neurons (hence the synonym sometimes used offully connected network (FCN)), often with a nonlinear kind of activation function, organized in at least three layers, notable for being able to distinguish data that is notlinearly separable.^[32]

Examples of other feedforward networks includeconvolutional neural networks andradial basis function networks, which use a different activation function.

• Hopfield network
• Feed-forward
• Backpropagation
• Rprop

• Feedforward neural networks tutorial
• Feedforward Neural Network: Example
• Feedforward Neural Networks: An Introduction

• Neural network architectures

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