Feedforward neural network
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A feedforward neural network (FNN) is one of the two broad types of
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.[10][11][12] This extreme learning machine[13][12] was not yet a deep learning network.
- In 1965, the first Alexey Grigorevich Ivakhnenko and Valentin Lapa, at the time called the Group Method of Data Handling.[14][15][12]
- 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.[16] 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,[17][18] was for the first time published by the Finnish researcher Seppo Linnainmaa.[4][19][12] The term (i.e. "back-propagating errors") itself has been used by Rosenblatt himself,[11] but he did not know how to implement it,[12] although a continuous precursor of backpropagation was already used in the context of control theory in 1960 by Henry J. Kelley.[5][12] 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.[7][12]
- In 1985, an experimental analysis of the technique was conducted by
- 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.[20]
- In 1990s, an (much simpler) alternative to using neural networks, although still relatedfeature 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.[22]
- In 2017, modern transformer architectures were introduced.[23]
Mathematical foundations
Activation function
The two historically common
- .
The first is a
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 in the th data point (training example) by , where is the desired target value for th data point at node , and is the value produced at node when the 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 th data point, given by
- .
Using gradient descent, the change in each weight is
where is the output of the previous neuron , and is the learning rate, which is selected to ensure that the weights quickly converge to a response, without oscillations. In the previous expression, denotes the partial derivate of the error according to the weighted sum of the input connections of neuron .
The derivative to be calculated depends on the induced local field , which itself varies. It is easy to prove that for an output node this derivative can be simplified to
where 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
- .
This depends on the change in weights of the 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.[24]
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
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".[29]
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 multilayer perceptron (MLP) is a misnomer for a modern feedforward artificial neural network, consisting of fully connected neurons (hence the synonym sometimes used of fully 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 not linearly separable.[30].
Other feedforward networks
Examples of other feedforward networks include convolutional neural networks and radial basis function networks, which use a different activation function.
See also
References
- ISBN 1492671207.)
{{cite book}}
: CS1 maint: multiple names: authors list (link - ^ ISBN 3-89319-554-8.
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- ^ a b Linnainmaa, Seppo (1970). The representation of the cumulative rounding error of an algorithm as a Taylor expansion of the local rounding errors (Masters) (in Finnish). University of Helsinki. p. 6–7.
- ^ doi:10.2514/8.5282.
- ^ Rosenblatt, Frank. x. Principles of Neurodynamics: Perceptrons and the Theory of Brain Mechanisms. Spartan Books, Washington DC, 1961
- ^ a b Werbos, Paul (1982). "Applications of advances in nonlinear sensitivity analysis" (PDF). System modeling and optimization. Springer. p. 762–770. Archived (PDF) from the original on 14 April 2016. Retrieved 2 July 2017.
- ^ a b Rumelhart, David E., Geoffrey E. Hinton, and R. J. Williams. "Learning Internal Representations by Error Propagation". David E. Rumelhart, James L. McClelland, and the PDP research group. (editors), Parallel distributed processing: Explorations in the microstructure of cognition, Volume 1: Foundation. MIT Press, 1986.
- ^ Hastie, Trevor. Tibshirani, Robert. Friedman, Jerome. The Elements of Statistical Learning: Data Mining, Inference, and Prediction. Springer, New York, NY, 2009.
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- ^ a b Rosenblatt, Frank (1962). Principles of Neurodynamics. Spartan, New York.
- ^ arXiv:2212.11279 [cs.NE].
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- Ivakhnenko, A. G. (1973). Cybernetic Predicting Devices. CCM Information Corporation.
- Ivakhnenko, A. G.; Grigorʹevich Lapa, Valentin (1967). Cybernetics and forecasting techniques. American Elsevier Pub. Co.
- ^ Amari, Shun'ichi (1967). "A theory of adaptive pattern classifier". IEEE Transactions. EC (16): 279-307.
- S2CID 29739148. Retrieved 2019-08-04.
- ISBN 9780598818461.
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- ^ Matthew Brand (1988) Machine and Brain Learning. University of Chicago Tutorial Studies Bachelor's Thesis, 1988. Reported at the Summer Linguistics Institute, Stanford University, 1987
- ^ R. Collobert and S. Bengio (2004). Links between Perceptrons, MLPs and SVMs. Proc. Int'l Conf. on Machine Learning (ICML).
- ^ Bengio, Yoshua; Ducharme, Réjean; Vincent, Pascal; Janvin, Christian (March 2003). "A neural probabilistic language model". The Journal of Machine Learning Research. 3: 1137–1155.
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- ISBN 0-13-273350-1.
- ^ Mansfield Merriman, "A List of Writings Relating to the Method of Least Squares"
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- ^ Bretscher, Otto (1995). Linear Algebra With Applications (3rd ed.). Upper Saddle River, NJ: Prentice Hall.
- ^
ISBN 0-674-40340-1.
- PMID 18249524. Archived from the original(PDF) on 2011-07-06. Retrieved 2009-09-08.
- ^ Cybenko, G. 1989. Approximation by superpositions of a sigmoidal function Mathematics of Control, Signals, and Systems, 2(4), 303–314.