Feedforward neural network

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https://en.wikipedia.org/wiki/Feedforward_neural_network 

In a feedforward network, information always moves one direction; it never goes backwards.

Simplified example of training a neural network in object detection: The network is trained by multiple images that are known to depict starfish and sea urchins, which are correlated with "nodes" that represent visual features. 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): 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 a false 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.

A feedforward neural network (FNN) is one of the two broad types of artificial neural network, characterized by direction of the flow of information between its layers. Its flow is uni-directional, meaning that the information in the model flows in only one direction—forward—from the input nodes, through the hidden nodes (if any) and to the output nodes, without any cycles or loops, in contrast to recurrent neural networks, which have a bi-directional flow. Modern feedforward networks are trained using the backpropagation method and are colloquially referred to as the "vanilla" neural networks.

Timeline

• In 1958, a layered network of perceptrons, consisting of an input layer, a hidden layer with randomized weights that did not learn, and an output layer with learning connections, was introduced already by Frank Rosenblatt in his book Perceptron. This extreme learning machine was not yet a deep learning network.

• In 1965, the first deep-learning feedforward network, not yet using stochastic gradient descent, was published by Alexey Grigorevich Ivakhnenko and Valentin Lapa, at the time called the Group Method of Data Handling.

• In 1967, a deep-learning network, using stochastic gradient descent for the first time, was able to classify non-linearily separable pattern classes, as reported Shun'ichi Amari. Amari's student Saito conducted the computer experiments, using a five-layered feedforward network with two learning layers.

• In 1970, modern backpropagation method, an efficient application of a chain-rule-based supervised learning, was for the first time published by the Finnish researcher Seppo Linnainmaa. The term (i.e. "back-propagating errors") itself has been used by Rosenblatt himself, but he did not know how to implement it, although a continuous precursor of backpropagation was already used in the context of control theory in 1960 by Henry J. Kelley. It is known also as a reverse mode of automatic differentiation.

• In 1982, backpropagation was applied in the way that has become standard, for the first time by Paul Werbos.

• In 1985, an experimental analysis of the technique was conducted by David E. Rumelhart et al. Many improvements to the approach have been made in subsequent decades.

• In 1987, using a stochastic gradient descent within a (wide 12-layer nonlinear) feed-forward network, Matthew Brand has trained it to reproduce logic functions of nontrivial circuit depth, using small batches of random input/output samples. He, however, concluded that on hardware (sub-megaflop computers) available at the time it was impractical, and proposed using fixed random early layers as an input hash for a single modifiable layer.

• In 1990s, an (much simpler) alternative to using neural networks, although still related support vector machine approach was developed by Vladimir Vapnik and his colleagues. In addition to performing linear classification, they were able to efficiently perform a non-linear classification using what is called the kernel trick, using high-dimensional feature spaces.

• In 2003, interest in backpropagation networks returned due to the successes of deep learning being applied to language modelling by Yoshua Bengio with co-authors.

• In 2017, modern transformer architectures has been introduced.

• In 2021, a very simple NN architecture combining two deep MLPs with skip connections and layer normalizations was designed and called MLP-Mixer; its realizations featuring 19 to 431 millions of parameters were shown to be comparable to vision transformers of similar size on ImageNet and similar image classification tasks.

Mathematical foundations

Activation function

The two historically common activation functions are both sigmoids, and are described by

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

The first is a hyperbolic tangent that ranges from -1 to 1, while the other is the logistic 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 the rectifier and softplus functions. More specialized activation functions include radial basis functions (used in radial basis networks, another class of supervised neural network models).

In recent developments of deep learning the rectified linear unit (ReLU) is more frequently used as one of the possible ways to overcome the numerical problems related to the sigmoids.

Learning

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 of supervised learning, and is carried out through backpropagation.

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

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

Using gradient 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 the learning 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.

History

Linear neural network

The simplest kind of feedforward neural network is a linear network, which consists of a single layer of output nodes; the inputs are fed directly to the outputs via a series of weights. The sum of the products of the weights and the inputs is calculated in each node. The mean squared errors between these calculated outputs and a given target values are minimized by creating an adjustment to the weights. This technique has been known for over two centuries as the method of least squares or linear regression. It was used as a means of finding a good rough linear fit to a set of points by Legendre (1805) and Gauss (1795) for the prediction of planetary movement.

Perceptron

Main article: Perceptron

If using a threshold, i.e. a linear activation function, the resulting linear threshold unit is called a perceptron. (Often the term is used to denote just one of these units.) Multiple parallel linear units are able to approximate 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. This result can be found in Peter Auer, Harald Burgsteiner and Wolfgang Maass "A learning rule for very simple universal approximators consisting of a single layer of perceptrons".

Perceptrons can be trained by a simple learning algorithm that is usually called the delta 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 of gradient descent.

Multilayer perceptron

A two-layer neural network capable of calculating XOR. The numbers within the neurons represent each neuron's explicit threshold. The numbers that annotate arrows represent the weight of the inputs. Note that If the threshold of 2 is met then a value of 1 is used for the weight multiplication to the next layer. Not meeting the threshold results in 0 being used. The bottom layer of inputs is not always considered a real neural network layer

A multilayer perceptron (MLP) is a misnomer for a modern feedforward artificial neural network, consisting of fully connected neurons with a nonlinear kind of activation function, organized in at least three layers, notable for being able to distinguish data that is not linearly separable. It is a misnomer because the original perceptron used a Heaviside step function, instead of a nonlinear kind of activation function (used by modern networks).

Other feedforward networks

Examples of other feedforward networks include convolutional neural networks and radial basis function networks, which use a different activation function.