# AI News, Artificial Neural Networks/Neural Network Basics

## Artificial Neural Networks/Neural Network Basics

Artificial Neural Networks, also known as “Artificial neural nets”, “neural nets”, or ANN for short, are a computational tool modeled on the interconnection of the neuron in the nervous systems of the human brain and that of other organisms.

Artificial neural networks are very different from biological networks, although many of the concepts and characteristics of biological systems are faithfully reproduced in the artificial systems. Artificial

neural nets are a type of non-linear processing system that is ideally suited for a wide range of tasks, especially tasks where there is no existing algorithm for task completion.

With proper training, ANN are capable of generalization, the ability to recognize similarities among different input patterns, especially patterns that have been corrupted by noise.

The term “Neural Net” refers to both the biological and artificial variants, although typically the term is used to refer to artificial systems only.

Each neuron is a multiple-input, multiple-output (MIMO) system that receives signals from the inputs, produces a resultant signal, and transmits that signal to all outputs.

However, to reproduce the effect of the synapse, the connections between PE are assigned multiplicative weights, which can be calibrated or “trained” to produce the proper system output.

Where ζ is the weighted sum of the inputs (the inner product of the input vector and the tap-weight vector), and σ(ζ) is a function of the weighted sum.

If we recognize that the weight and input elements form vectors w and x, the ζ weighted sum becomes a simple dot product:

The dotted line in the center of the neuron represents the division between the calculation of the input sum using the weight vector, and the calculation of the output value using the activation function.

Neural networks tend to have one input per degree of freedom in the input space, and one output per degree of freedom in the output space.

Expert systems, by contrast, are used in situations where there is insufficient data and theoretical background to create any kind of a reliable problem model.

Expert systems emulate the deduction processes of a human expert, by collecting information and traversing the solution space in a directed manner.

Though such assumptions are not required, it has been found that the addition of such a priori information as the statistical distribution of the input space can help to speed training.

During training, the neural network performs the necessary analytical work, which would require non-trivial effort on the part of the analyst if other methods were to be used.

During training, care must be taken not to provide too many input examples and different numbers of training examples could produce very different results in the quality and robustness of the network.

Some of the more important parameters in terms of training and network capacity are the number of hidden neurons, the learning rate and the momentum parameter.

These neurons are essentially hidden from view, and their number and organization can typically be treated as a black box to people who are interfacing with the system.

## Artificial neural network

Artificial neural networks (ANN) or connectionist systems are computing systems vaguely inspired by the biological neural networks that constitute animal brains.&#91;1&#93;

The neural network itself isn't an algorithm, but rather a framework for many different machine learning algorithms to work together and process complex data inputs.&#91;2&#93;

For example, in image recognition, they might learn to identify images that contain cats by analyzing example images that have been manually labeled as 'cat' or 'no cat' and using the results to identify cats in other images.

An ANN is based on a collection of connected units or nodes called artificial neurons, which loosely model the neurons in a biological brain.

In common ANN implementations, the signal at a connection between artificial neurons is a real number, and the output of each artificial neuron is computed by some non-linear function of the sum of its inputs.

Signals travel from the first layer (the input layer), to the last layer (the output layer), possibly after traversing the layers multiple times.

Artificial neural networks have been used on a variety of tasks, including computer vision, speech recognition, machine translation, social network filtering, playing board and video games and medical diagnosis.

(1943) created a computational model for neural networks based on mathematics and algorithms called threshold logic.

With mathematical notation, Rosenblatt described circuitry not in the basic perceptron, such as the exclusive-or circuit that could not be processed by neural networks at the time.&#91;9&#93;

In 1959, a biological model proposed by Nobel laureates Hubel and Wiesel was based on their discovery of two types of cells in the primary visual cortex: simple cells and complex cells.&#91;10&#93;

Much of artificial intelligence had focused on high-level (symbolic) models that are processed by using algorithms, characterized for example by expert systems with knowledge embodied in if-then rules, until in the late 1980s research expanded to low-level (sub-symbolic) machine learning, characterized by knowledge embodied in the parameters of a cognitive model.&#91;citation needed&#93;

key trigger for renewed interest in neural networks and learning was Werbos's (1975) backpropagation algorithm that effectively solved the exclusive-or problem by making the training of multi-layer networks feasible and efficient.

Support vector machines and other, much simpler methods such as linear classifiers gradually overtook neural networks in machine learning popularity.

The vanishing gradient problem affects many-layered feedforward networks that used backpropagation and also recurrent neural networks (RNNs).&#91;22&#93;&#91;23&#93;

As errors propagate from layer to layer, they shrink exponentially with the number of layers, impeding the tuning of neuron weights that is based on those errors, particularly affecting deep networks.

To overcome this problem, Schmidhuber adopted a multi-level hierarchy of networks (1992) pre-trained one level at a time by unsupervised learning and fine-tuned by backpropagation.&#91;24&#93;

(2006) proposed learning a high-level representation using successive layers of binary or real-valued latent variables with a restricted Boltzmann machine&#91;26&#93;

Once sufficiently many layers have been learned, the deep architecture may be used as a generative model by reproducing the data when sampling down the model (an 'ancestral pass') from the top level feature activations.&#91;27&#93;&#91;28&#93;

In 2012, Ng and Dean created a network that learned to recognize higher-level concepts, such as cats, only from watching unlabeled images taken from YouTube videos.&#91;29&#93;

Earlier challenges in training deep neural networks were successfully addressed with methods such as unsupervised pre-training, while available computing power increased through the use of GPUs and distributed computing.

for very large scale principal components analyses and convolution may create a new class of neural computing because they are fundamentally analog rather than digital (even though the first implementations may use digital devices).&#91;31&#93;

in Schmidhuber's group showed that despite the vanishing gradient problem, GPUs makes back-propagation feasible for many-layered feedforward neural networks.

Between 2009 and 2012, recurrent neural networks and deep feedforward neural networks developed in Schmidhuber's research group won eight international competitions in pattern recognition and machine learning.&#91;33&#93;&#91;34&#93;

Researchers demonstrated (2010) that deep neural networks interfaced to a hidden Markov model with context-dependent states that define the neural network output layer can drastically reduce errors in large-vocabulary speech recognition tasks such as voice search.

A team from his lab won a 2012 contest sponsored by Merck to design software to help find molecules that might identify new drugs.&#91;47&#93;

As of 2011&#91;update&#93;, the state of the art in deep learning feedforward networks alternated between convolutional layers and max-pooling layers,&#91;42&#93;&#91;48&#93;

Artificial neural networks were able to guarantee shift invariance to deal with small and large natural objects in large cluttered scenes, only when invariance extended beyond shift, to all ANN-learned concepts, such as location, type (object class label), scale, lighting and others.

An artificial neural network is a network of simple elements called artificial neurons, which receive input, change their internal state (activation) according to that input, and produce output depending on the input and activation.

An artificial neuron mimics the working of a biophysical neuron with inputs and outputs, but is not a biological neuron model.

The network forms by connecting the output of certain neurons to the input of other neurons forming a directed, weighted graph.

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Sometimes a bias term added to total weighted sum of inputs to serve as threshold to shift the activation function.&#91;53&#93;

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The learning rule is a rule or an algorithm which modifies the parameters of the neural network, in order for a given input to the network to produce a favored output.

A common use of the phrase 'ANN model' is really the definition of a class of such functions (where members of the class are obtained by varying parameters, connection weights, or specifics of the architecture such as the number of neurons or their connectivity).

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(commonly referred to as the activation function&#91;55&#93;) is some predefined function, such as the hyperbolic tangent or sigmoid function or softmax function or rectifier function.

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is an important concept in learning, as it is a measure of how far away a particular solution is from an optimal solution to the problem to be solved.

For applications where the solution is data dependent, the cost must necessarily be a function of the observations, otherwise the model would not relate to the data.

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While it is possible to define an ad hoc cost function, frequently a particular cost (function) is used, either because it has desirable properties (such as convexity) or because it arises naturally from a particular formulation of the problem (e.g., in a probabilistic formulation the posterior probability of the model can be used as an inverse cost).

In 1970, Linnainmaa finally published the general method for automatic differentiation (AD) of discrete connected networks of nested differentiable functions.&#91;64&#93;&#91;65&#93;

In 1986, Rumelhart, Hinton and Williams noted that this method can generate useful internal representations of incoming data in hidden layers of neural networks.&#91;71&#93;

The choice of the cost function depends on factors such as the learning type (supervised, unsupervised, reinforcement, etc.) and the activation function.

For example, when performing supervised learning on a multiclass classification problem, common choices for the activation function and cost function are the softmax function and cross entropy function, respectively.

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The network is trained to minimize L2 error for predicting the mask ranging over the entire training set containing bounding boxes represented as masks.

the cost function is related to the mismatch between our mapping and the data and it implicitly contains prior knowledge about the problem domain.&#91;79&#93;

commonly used cost is the mean-squared error, which tries to minimize the average squared error between the network's output,

Minimizing this cost using gradient descent for the class of neural networks called multilayer perceptrons (MLP), produces the backpropagation algorithm for training neural networks.

Tasks that fall within the paradigm of supervised learning are pattern recognition (also known as classification) and regression (also known as function approximation).

The supervised learning paradigm is also applicable to sequential data (e.g., for hand writing, speech and gesture recognition).

This can be thought of as learning with a 'teacher', in the form of a function that provides continuous feedback on the quality of solutions obtained thus far.

The cost function is dependent on the task (the model domain) and any a priori assumptions (the implicit properties of the model, its parameters and the observed variables).

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whereas in statistical modeling, it could be related to the posterior probability of the model given the data (note that in both of those examples those quantities would be maximized rather than minimized).

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The aim is to discover a policy for selecting actions that minimizes some measure of a long-term cost, e.g., the expected cumulative cost.

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because of the ability of Artificial neural networks to mitigate losses of accuracy even when reducing the discretization grid density for numerically approximating the solution of the original control problems.

Tasks that fall within the paradigm of reinforcement learning are control problems, games and other sequential decision making tasks.

Training a neural network model essentially means selecting one model from the set of allowed models (or, in a Bayesian framework, determining a distribution over the set of allowed models) that minimizes the cost.

This is done by simply taking the derivative of the cost function with respect to the network parameters and then changing those parameters in a gradient-related direction.

convolutional neural network (CNN) is a class of deep, feed-forward networks, composed of one or more convolutional layers with fully connected layers (matching those in typical Artificial neural networks) on top.

recent development has been that of Capsule Neural Network (CapsNet), the idea behind which is to add structures called capsules to a CNN and to reuse output from several of those capsules to form more stable (with respect to various perturbations) representations for higher order capsules.&#91;103&#93;

can find an RNN weight matrix that maximizes the probability of the label sequences in a training set, given the corresponding input sequences.

provide a framework for efficiently trained models&#160;for hierarchical processing of temporal data, while enabling the investigation of the inherent role of RNN layered composition.&#91;clarification needed&#93;

This is particularly helpful when training data are limited, because poorly initialized weights can significantly hinder model performance.

that integrate the various and usually different filters (preprocessing functions) into its many layers and to dynamically rank the significance of the various layers and functions relative to a given learning task.

This grossly imitates biological learning which integrates various preprocessors (cochlea, retina, etc.) and cortexes (auditory, visual, etc.) and their various regions.

Its deep learning capability is further enhanced by using inhibition, correlation and its ability to cope with incomplete data, or 'lost' neurons or layers even amidst a task.

The link-weights allow dynamic determination of innovation and redundancy, and facilitate the ranking of layers, of filters or of individual neurons relative to a task.

LAMSTAR had a much faster learning speed and somewhat lower error rate than a CNN based on ReLU-function filters and max pooling, in 20 comparative studies.&#91;140&#93;

These applications demonstrate delving into aspects of the data that are hidden from shallow learning networks and the human senses, such as in the cases of predicting onset of sleep apnea events,&#91;132&#93;

The whole process of auto encoding is to compare this reconstructed input to the original and try to minimize the error to make the reconstructed value as close as possible to the original.

with a specific approach to good representation, a good representation is one that can be obtained robustly from a corrupted input and that will be useful for recovering the corresponding clean input.

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of the first denoising auto encoder is learned and used to uncorrupt the input (corrupted input), the second level can be trained.&#91;146&#93;

Once the stacked auto encoder is trained, its output can be used as the input to a supervised learning algorithm such as support vector machine classifier or a multi-class logistic regression.&#91;146&#93;

It formulates the learning as a convex optimization problem with a closed-form solution, emphasizing the mechanism's similarity to stacked generalization.&#91;150&#93;

Each block estimates the same final label class y, and its estimate is concatenated with original input X to form the expanded input for the next block.

Thus, the input to the first block contains the original data only, while downstream blocks' input adds the output of preceding blocks.

It offers two important improvements: it uses higher-order information from covariance statistics, and it transforms the non-convex problem of a lower-layer to a convex sub-problem of an upper-layer.&#91;152&#93;

TDSNs use covariance statistics in a bilinear mapping from each of two distinct sets of hidden units in the same layer to predictions, via a third-order tensor.

The need for deep learning with real-valued inputs, as in Gaussian restricted Boltzmann machines, led to the spike-and-slab RBM (ssRBM), which models continuous-valued inputs with strictly binary latent variables.&#91;156&#93;

One of these terms enables the model to form a conditional distribution of the spike variables by marginalizing out the slab variables given an observation.

However, these architectures are poor at learning novel classes with few examples, because all network units are involved in representing the input (a distributed representation) and must be adjusted together (high degree of freedom).

It is a full generative model, generalized from abstract concepts flowing through the layers of the model, which is able to synthesize new examples in novel classes that look 'reasonably' natural.

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deep predictive coding network (DPCN) is a predictive coding scheme that uses top-down information to empirically adjust the priors needed for a bottom-up inference procedure by means of a deep, locally connected, generative model.

DPCNs predict the representation of the layer, by using a top-down approach using the information in upper layer and temporal dependencies from previous states.&#91;174&#93;

For example, in sparse distributed memory or hierarchical temporal memory, the patterns encoded by neural networks are used as addresses for content-addressable memory, with 'neurons' essentially serving as address encoders and decoders.

Preliminary results demonstrate that neural Turing machines can infer simple algorithms such as copying, sorting and associative recall from input and output examples.

Approaches that represent previous experiences directly and use a similar experience to form a local model are often called nearest neighbour or k-nearest neighbors methods.&#91;189&#93;

Unlike sparse distributed memory that operates on 1000-bit addresses, semantic hashing works on 32 or 64-bit addresses found in a conventional computer architecture.

These models have been applied in the context of question answering (QA) where the long-term memory effectively acts as a (dynamic) knowledge base and the output is a textual response.&#91;194&#93;

While training extremely deep (e.g., 1 million layers) neural networks might not be practical, CPU-like architectures such as pointer networks&#91;196&#93;

overcome this limitation by using external random-access memory and other components that typically belong to a computer architecture such as registers, ALU and pointers.

The key characteristic of these models is that their depth, the size of their short-term memory, and the number of parameters can be altered independently – unlike models like LSTM, whose number of parameters grows quadratically with memory size.

In that work, an LSTM RNN or CNN was used as an encoder to summarize a source sentence, and the summary was decoded using a conditional RNN language model to produce the translation.&#91;201&#93;

For the sake of dimensionality reduction of the updated representation in each layer, a supervised strategy selects the best informative features among features extracted by KPCA.

The main idea is to use a kernel machine to approximate a shallow neural net with an infinite number of hidden units, then use stacking to splice the output of the kernel machine and the raw input in building the next, higher level of the kernel machine.

The basic search algorithm is to propose a candidate model, evaluate it against a dataset and use the results as feedback to teach the NAS network.&#91;205&#93;

game-playing and decision making (backgammon, chess, poker), pattern recognition (radar systems, face identification, signal classification,&#91;208&#93;

object recognition and more), sequence recognition (gesture, speech, handwritten and printed text recognition), medical diagnosis, finance&#91;209&#93;

models of how the dynamics of neural circuitry arise from interactions between individual neurons and finally to models of how behavior can arise from abstract neural modules that represent complete subsystems.

These include models of the long-term, and short-term plasticity, of neural systems and their relations to learning and memory from the individual neuron to the system level.

specific recurrent architecture with rational valued weights (as opposed to full precision real number-valued weights) has the full power of a universal Turing machine,&#91;223&#93;

but also in statistical learning theory, where the goal is to minimize over two quantities: the 'empirical risk' and the 'structural risk', which roughly corresponds to the error over the training set and the predicted error in unseen data due to overfitting.

Supervised neural networks that use a mean squared error (MSE) cost function can use formal statistical methods to determine the confidence of the trained model.

A confidence analysis made this way is statistically valid as long as the output probability distribution stays the same and the network is not modified.

By assigning a softmax activation function, a generalization of the logistic function, on the output layer of the neural network (or a softmax component in a component-based neural network) for categorical target variables, the outputs can be interpreted as posterior probabilities.

Potential solutions include randomly shuffling training examples, by using a numerical optimization algorithm that does not take too large steps when changing the network connections following an example and by grouping examples in so-called mini-batches.

No neural network has solved computationally difficult problems such as the n-Queens problem, the travelling salesman problem, or the problem of factoring large integers.

Sensor neurons fire action potentials more frequently with sensor activation and muscle cells pull more strongly when their associated motor neurons receive action potentials more frequently.&#91;226&#93;

Other than the case of relaying information from a sensor neuron to a motor neuron, almost nothing of the principles of how information is handled by biological neural networks is known.

The motivation behind Artificial neural networks is not necessarily to strictly replicate neural function, but to use biological neural networks as an inspiration.

Alexander Dewdney commented that, as a result, artificial neural networks have a 'something-for-nothing quality, one that imparts a peculiar aura of laziness and a distinct lack of curiosity about just how good these computing systems are.

argued that the brain self-wires largely according to signal statistics and therefore, a serial cascade cannot catch all major statistical dependencies.

While the brain has hardware tailored to the task of processing signals through a graph of neurons, simulating even a simplified neuron on von Neumann architecture may compel a neural network designer to fill many millions of database rows for its connections&#160;&#8211;&#32;

Schmidhuber notes that the resurgence of neural networks in the twenty-first century is largely attributable to advances in hardware: from 1991 to 2015, computing power, especially as delivered by GPGPUs (on GPUs), has increased around a million-fold, making the standard backpropagation algorithm feasible for training networks that are several layers deeper than before.&#91;231&#93;

Arguments against Dewdney's position are that neural networks have been successfully used to solve many complex and diverse tasks, ranging from autonomously flying aircraft&#91;233&#93;

Neural networks, for instance, are in the dock not only because they have been hyped to high heaven, (what hasn't?) but also because you could create a successful net without understanding how it worked: the bunch of numbers that captures its behaviour would in all probability be 'an opaque, unreadable table...valueless as a scientific resource'.

In spite of his emphatic declaration that science is not technology, Dewdney seems here to pillory neural nets as bad science when most of those devising them are just trying to be good engineers.

Although it is true that analyzing what has been learned by an artificial neural network is difficult, it is much easier to do so than to analyze what has been learned by a biological neural network.

Furthermore, researchers involved in exploring learning algorithms for neural networks are gradually uncovering general principles that allow a learning machine to be successful.

Advocates of hybrid models (combining neural networks and symbolic approaches), claim that such a mixture can better capture the mechanisms of the human mind.&#91;236&#93;&#91;237&#93;

The simplest, static types have one or more static components, including number of units, number of layers, unit weights and topology.

The various types of neural networks are explained and demonstrated, applications of neural networks like ANNs in medicine are described, and a detailed historical background is provided.

It is composed of a large number of highly interconnected processing elements (neurones) working in unison to solve specific problems.

Minsky and Papert, published a book (in 1969) in which they summed up a general feeling of frustration (against neural networks) among researchers, and was thus accepted by most without further analysis.

For a more detailed description of the history click here The first artificial neuron was produced in 1943 by the neurophysiologist Warren McCulloch and the logician Walter Pits.

Neural networks, with their remarkable ability to derive meaning from complicated or imprecise data, can be used to extract patterns and detect trends that are too complex to be noticed by either humans or other computer techniques.

The network is composed of a large number of highly interconnected processing elements(neurones) working in parallel to solve a specific problem.

Even more, a large number of tasks, require systems that use a combination of the two approaches (normally a conventional computer is used to supervise the neural network) in order to perform at maximum efficiency.

In the human brain, a typical neuron collects signals from others through a host of fine structures called dendrites.

The neuron sends out spikes of electrical activity through a long, thin stand known as an axon, which splits into thousands of branches.

At the end of each branch, a structure called a synapse converts the activity from the axon into electrical effects that inhibit or excite activity from the axon into electrical effects that inhibit or excite activity in the connected neurones.

When a neuron receives excitatory input that is sufficiently large compared with its inhibitory input, it sends a spike of electrical activity down its axon.

However because our knowledge of neurones is incomplete and our computing power is limited, our models are necessarily gross idealisations of real networks of neurones.

In the using mode, when a taught input pattern is detected at the input, its associated output becomes the current output.

If the input pattern does not belong in the taught list of input patterns, the firing rule is used to determine whether to fire or not.

The rule goes as follows: Take a collection of training patterns for a node, some of which cause it to fire (the 1-taught set of patterns) and others which prevent it from doing so (the 0-taught set).

Then the patterns not in the collection cause the node to fire if, on comparison , they have more input elements in common with the 'nearest' pattern in the 1-taught set than with the 'nearest' pattern in the 0-taught set.

For example, a 3-input neuron is taught to output 1 when the input (X1,X2 and X3) is 111 or 101 and to output 0 when the input is 000 or 001.

It differs from 000 in 1 element, from 001 in 2 elements, from 101 in 3 elements and from 111 in 2 elements.

Therefore the firing rule gives the neuron a sense of similarity and enables it to respond 'sensibly' to patterns not seen during training.

If we represent black squares with 0 and white squares with 1 then the truth tables for the 3 neurones after generalisation are;

Top neuron Middle neuron Bottom neuron &#160;From the tables it can be seen the following associasions can be extracted:

Feedback architectures are also referred to as interactive or recurrent, although the latter term is often used to denote feedback connections in single-layer organisations.

The commonest type of artificial neural network consists of three groups, or layers, of units: a layer of &quot;input&quot;

The activity of each hidden unit is determined by the activities of the input units and the weights on the connections between the input and the hidden units.

The weights between the input and hidden units determine when each hidden unit is active, and so by modifying these weights, a hidden unit can choose what it represents.

The single-layer organisation, in which all units are connected to one another, constitutes the most general case and is of more potential computational power than hierarchically structured multi-layer organisations.

The perceptron (figure 4.4) turns out to be an MCP model ( neuron with weighted inputs ) with some additional, fixed, pre--processing.

Units labelled A1, A2, Aj , Ap are called association units and their task is to extract specific, localised featured from the input images.

The book was very well written and showed mathematically that single layer perceptrons could not do some basic pattern recognition operations like determining the parity of a shape or determining whether a shape is connected or not.

associative mapping in which the network learns to produce a particular pattern on the set of input units whenever another particular pattern is applied on the set of input units.

This is used to provide pattern completition, ie to produce a pattern whenever a portion of it or a distorted pattern is presented.

nearest-neighbour recall, where the output pattern produced corresponds to the input pattern stored, which is closest to the pattern presented, and interpolative recall, where the output pattern is a similarity dependent interpolation of the patterns stored corresponding to the pattern presented.

Yet another paradigm, which is a variant associative mapping is classification, ie when there is a fixed set of categories into which the input patterns are to be classified.

Whereas in asssociative mapping the network stores the relationships among patterns, in regularity detection the response of each unit has a particular 'meaning'.

Supervised learning which incorporates an external teacher, so that each output unit is told what its desired response to input signals ought to be.

important issue conserning supervised learning is the problem of error convergence, ie the minimisation of error between the desired and computed unit values.

For threshold units, the output is set at one of two levels, depending on whether the total input is greater than or less than some threshold value.

Sigmoid units bear a greater resemblance to real neurones than do linear or threshold units, but all three must be considered rough approximations.

To make a neural network that performs some specific task, we must choose how the units are connected to one another (see figure 4.1), and we must set the weights on the connections appropriately.

We can teach a three-layer network to perform a particular task by using the following procedure: Assume that we want a network to recognise hand-written digits.

The network would therefore need 256 input units (one for each sensor), 10 output units (one for each kind of digit) and a number of hidden units.

For each kind of digit recorded by the sensors, the network should produce high activity in the appropriate output unit and low activity in the other output units.

To train the network, we present an image of a digit and compare the actual activity of the 10 output units with the desired activity.

Next we change the weight of each connection so as to reduce the error.We repeat this training process for many different images of each different images of each kind of digit until the network classifies every image correctly.

To implement this procedure we need to calculate the error derivative for the weight (EW) in order to change the weight by an amount that is proportional to the rate at which the error changes as the weight is changed.

It was developed independently by two teams, one (Fogelman-Soulie, Gallinari and Le Cun) in France, the other (Rumelhart, Hinton and Williams) in U.S. In order to train a neural network to perform some task, we must adjust the weights of each unit in such a way that the error between the desired output and the actual output is reduced.

To compute the EA for a hidden unit in the layer just before the output layer, we first identify all the weights between that hidden unit and the output units to which it is connected.

After calculating all the EAs in the hidden layer just before the output layer, we can compute in like fashion the EAs for other layers, moving from layer to layer in a direction opposite to the way activities propagate through the network.

Since neural networks are best at identifying patterns or trends in data, they are well suited for prediction or forecasting needs including:

Neural networks are ideal in recognising diseases using scans since there is no need to provide a specific algorithm on how to identify the disease.

Diagnosis can be achieved by building a model of the cardiovascular system of an individual and comparing it with the real time physiological measurements taken from the patient.

If this routine is carried out regularly, potential harmful medical conditions can be detected at an early stage and thus make the process of combating the disease much easier.

model of an individual's cardiovascular system must mimic the relationship among physiological variables (i.e., heart rate, systolic and diastolic blood pressures, and breathing rate) at different physical activity levels.

Sensor fusion enables the ANNs to learn complex relationships among the individual sensor values, which would otherwise be lost if the values were individually analysed.

In medical modelling and diagnosis, this implies that even though each sensor in a set may be sensitive only to a specific physiological variable, ANNs are capable of detecting complex medical conditions by fusing the data from the individual biomedical sensors.

trained an autoassociative memory neural network to store a large number of medical records, each of which includes information on symptoms, diagnosis, and treatment for a particular case.

There is also a strong potential for using neural networks for database mining, that is, searching for patterns implicit within the explicitly stored information in databases.

A feedforward neural network is integrated with the AMT and was trained using back-propagation to assist the marketing control of airline seat allocations.

Stephens, 1987] While it is significant that neural networks have been applied to this problem, it is also important to see that this intelligent technology can be integrated with expert systems and other approaches to make a functional system.

Finally, I would like to state that even though neural networks have a huge potential we will only get the best of them when they are intergrated with computing, AI, fuzzy logic and related subjects.

## Supervised learning in spiking neural networks with FORCE training

We explored the potential of the FORCE method in training spiking neural networks to perform an arbitrary task.

The parameters G and Q control the relative balance between the chaos inducing static weight matrix and the learned feedback term, respectively.

The goal of RLS is to minimize the squared error between the network dynamics $$\left( {\hat x(t)} \right)$$ and the target dynamics (i.e., the task, x(t)) (Fig.1a)29,30.

We considered three types of spiking integrate-and-fire model: the theta model, the leaky integrate-and-fire (LIF), and the Izhikevich model (see Methods section for a more detailed explanation).

All networks considered were constrained to be intermediate in size (1000–5000 neurons) and have low post-training average firing rates (&lt;60 Hz).

To demonstrate the basic principle of this method and to compare with our spiking network simulations, we applied FORCE training to a network of rate equations (Fig.1a) as in ref.

To ascertain how these networks can learn to perform the target dynamics, we computed the eigenvalues of the resulting weight matrix before and after learning (Fig.1e).

We sought to determine whether the FORCE method could train spiking neural networks given the impressive capabilities of this technique in training smooth systems of rate equations (Fig.1) previously demonstrated in ref.

We implemented the FORCE method in different spiking neural networks of integrate-and-fire neurons in order to compare the robustness of the method across neuronal models and potential supervisors.

To determine where the onset of chaos occurred as a function of G, we simulated networks of these neurons over a range of G parameters and computed the coefficients of variation and the interspike-interval (ISI) distributions (Supplementary Fig.1).

For the Izhikevich and Theta neurons, there was an immediate onset to chaotic spiking from quiescence (G ≈ 103, G ≈ 0.02, respectively) as the bias currents for these models were placed at rheobase or threshold value.

All neuron models exhibited bimodal interspike-interval distributions indicative of a possible transition to rate chaos for sufficiently high G-values27,28.

With these modifications to the FORCE method, we successfully trained networks of spiking theta neurons to mimic various types of oscillators.

Examples include sinusoids at different frequencies, sawtooth oscillations, Van der Pol oscillators, in addition to teaching signals with noise present (Fig.2b).

With a 20 ms decay time constant, the product of sines oscillator presented a challenge to the theta neuron to learn.

Oscillators with higher (lower) frequencies are learned over larger (G, Q) parameter regions in networks with faster (slower) synaptic decay time constants, τ

For spiking networks, we observe that in some cases, systems without dominant eigenvalues performed better than systems with dominant eigenvalues while in other cases the opposite was true.

This is partially due to the fact that the Izhikevich neuron has spike frequency adaptation which operates on a longer time scale (i.e., 100 ms).

The long time scale affords the reservoir a greater capability for memory, allowing it to learn longer signals.

We wondered how the convergence rates of these networks would vary as a function of the network size, N, for both FORCE trained spiking and rate networks (Supplementary Fig.8).

As oscillators are simple dynamical systems, we wanted to assess if FORCE can train a spiking neural network to perform more complicated tasks.

Thus, we considered two additional tasks: reproducing the dynamics for a low-dimensional chaotic system and statistically classifying inputs applied to a network of neurons.

To quantify the error and compare it with a rate network, we developed an attractor density based metric (Supplementary Materials) for the marginal density functions on the attractor.

The spiking network had comparable performance to the rate network (0.27, 0.30, 0.24 for rate and 0.52, 0.38, 0.3 for spiking).

Further, the spiking and rate networks were both able to regenerate the stereotypical Lorenz tent map, albeit with superior performance in the rate network (Supplementary Fig.9).

We wanted to determine if FORCE training could be adapted to generate weight matrices that respect Dales law, the constraint that a neuron can only be excitatory or inhibitory, not both.

The average firing rate of the network after training was 34 Hz (Fig.3b), with variability in the neuronal responses from replay to replay, yet forming a stable peri-simulus time histogram (Supplementary Figs.14–16).

The errors are primarily located after the two non-unique E-note repetitions that occur in the first bar and the end of the third bar (Supplementary Figs.14, 15) in addition to the non-unique ED sequences that occur at the end of the second bar and beginning of the fourth bar.

These pulses feed into a network of Izhikevich neurons that was successfully FORCE trained to reproduce the spectrogram of a recorded song from an adult zebra finch (Fig.4b, see Supplementary Movie1).

Additionally, by varying the (G, Q) parameters and some Izhikevich model parameters, the spiking statistics of RA neurons are easily reproduced both qualitatively and quantitatively (Fig.4d, e inset, see Fig.2b, c in ref.

We hypothesized that the performance of this network was strongly associated to the precise, clock-like inputs provided by HVC and that similar inputs could aid in the encoding and replay of other types of information.

To test this hypothesis, we removed the HVC input pattern and found that the replay of the learned song was destroyed (not shown), similar to experimental lesioning of this area in adult canaries39.

Due to the temporal information that these high-dimensional signals provide, we will subsequently refer to them as high-dimensional temporal signals (HDTS, see Methods section for further details).

To further explore the benefits that an HDTS might provide, we FORCE trained a network of Izhikevich neurons to internally generate its own HDTS, while simultaneously also training it to reproduce the first bar of Ode to Joy.

Given the improvements that an HDTS confers over supervisor duration, training time, and accuracy, we wanted to know if these input signals would help populations of neurons to learn natural high dimensional signals.

To test this, we trained a network of Izhikevich neurons to learn a 1920 dimensional supervisor that corresponds to the pixels of an 8 s scene from a movie (Fig.5).

As in the long Ode to Joy example, the HDTS could also be learned simultaneously to the movie scene, constituting a 1984 dimensional supervisor (1920 + 64 dimensional HDTS, Supplementary Fig.18c, d).

We were able to successfully train the network to replay the movie scene in both cases (time averaged correlation coefficient of r = 0.98, Fig.5c).

We were surprised to see that despite the complexity and high dimensionality of the encoding signal, the histogram of spike times across the replay network displayed a strong 4 Hz modulation conferred by the HDTS.

The emergence of these slower oscillations corresponds to a sharp decline in replay performance as the scenes are no longer replayed in chronological order (Fig.5f).

We found that the replay performance decreased approximately linearly with the proportion of neurons that were lesioned, with the amplitude of the mean activity also decreasing (Fig.5e, f).

The network was able to replay the movie in compressed time (correlation of r &gt; 0.8) up to a compression factor of 16× with accuracy sharply dropping for further compression.

40, the authors found that a recorded sequence of neuronal cross correlations in rats elicited during a spatial sequence task reappeared in compressed time during sleep.

At lower compression ratios, the mean population activity mirrors the HDTS while at higher compression ratios (≥8×), large synchronized events in the mean activity emerge that repeat with each movie replay.

In the section on linear classification we computed scores for different visual categories given the image using the formula $$s = W x$$, where $$W$$ was a matrix and $$x$$ was an input column vector containing all pixel data of the image.

In the case of CIFAR-10, $$x$$ is a [3072x1] column vector, and $$W$$ is a [10x3072] matrix, so that the output scores is a vector of 10 class scores.

There are several choices we could make for the non-linearity (which we’ll study below), but this one is a common choice and simply thresholds all activations that are below zero to zero.

Notice that the non-linearity is critical computationally - if we left it out, the two matrices could be collapsed to a single matrix, and therefore the predicted class scores would again be a linear function of the input.

three-layer neural network could analogously look like $$s = W_3 \max(0, W_2 \max(0, W_1 x))$$, where all of $$W_3, W_2, W_1$$ are parameters to be learned.

The area of Neural Networks has originally been primarily inspired by the goal of modeling biological neural systems, but has since diverged and become a matter of engineering and achieving good results in Machine Learning tasks.

Approximately 86 billion neurons can be found in the human nervous system and they are connected with approximately 10^14 - 10^15 synapses.

The idea is that the synaptic strengths (the weights $$w$$) are learnable and control the strength of influence (and its direction: excitory (positive weight) or inhibitory (negative weight)) of one neuron on another.

Based on this rate code interpretation, we model the firing rate of the neuron with an activation function $$f$$, which represents the frequency of the spikes along the axon.

Historically, a common choice of activation function is the sigmoid function $$\sigma$$, since it takes a real-valued input (the signal strength after the sum) and squashes it to range between 0 and 1.

An example code for forward-propagating a single neuron might look as follows: In other words, each neuron performs a dot product with the input and its weights, adds the bias and applies the non-linearity (or activation function), in this case the sigmoid $$\sigma(x) = 1/(1+e^{-x})$$.

As we saw with linear classifiers, a neuron has the capacity to “like” (activation near one) or “dislike” (activation near zero) certain linear regions of its input space.

With this interpretation, we can formulate the cross-entropy loss as we have seen in the Linear Classification section, and optimizing it would lead to a binary Softmax classifier (also known as logistic regression).

The regularization loss in both SVM/Softmax cases could in this biological view be interpreted as gradual forgetting, since it would have the effect of driving all synaptic weights $$w$$ towards zero after every parameter update.

The sigmoid non-linearity has the mathematical form $$\sigma(x) = 1 / (1 + e^{-x})$$ and is shown in the image above on the left.

The sigmoid function has seen frequent use historically since it has a nice interpretation as the firing rate of a neuron: from not firing at all (0) to fully-saturated firing at an assumed maximum frequency (1).

Also note that the tanh neuron is simply a scaled sigmoid neuron, in particular the following holds: $$\tanh(x) = 2 \sigma(2x) -1$$.

Other types of units have been proposed that do not have the functional form $$f(w^Tx + b)$$ where a non-linearity is applied on the dot product between the weights and the data.

TLDR: “What neuron type should I use?” Use the ReLU non-linearity, be careful with your learning rates and possibly monitor the fraction of “dead” units in a network.

For regular neural networks, the most common layer type is the fully-connected layer in which neurons between two adjacent layers are fully pairwise connected, but neurons within a single layer share no connections.

Working with the two example networks in the above picture: To give you some context, modern Convolutional Networks contain on orders of 100 million parameters and are usually made up of approximately 10-20 layers (hence deep learning).

The full forward pass of this 3-layer neural network is then simply three matrix multiplications, interwoven with the application of the activation function: In the above code, W1,W2,W3,b1,b2,b3 are the learnable parameters of the network.

Notice also that instead of having a single input column vector, the variable x could hold an entire batch of training data (where each input example would be a column of x) and then all examples would be efficiently evaluated in parallel.

Neural Networks work well in practice because they compactly express nice, smooth functions that fit well with the statistical properties of data we encounter in practice, and are also easy to learn using our optimization algorithms (e.g.

Similarly, the fact that deeper networks (with multiple hidden layers) can work better than a single-hidden-layer networks is an empirical observation, despite the fact that their representational power is equal.

As an aside, in practice it is often the case that 3-layer neural networks will outperform 2-layer nets, but going even deeper (4,5,6-layer) rarely helps much more.

We could train three separate neural networks, each with one hidden layer of some size and obtain the following classifiers: In the diagram above, we can see that Neural Networks with more neurons can express more complicated functions.

For example, the model with 20 hidden neurons fits all the training data but at the cost of segmenting the space into many disjoint red and green decision regions.

The subtle reason behind this is that smaller networks are harder to train with local methods such as Gradient Descent: It’s clear that their loss functions have relatively few local minima, but it turns out that many of these minima are easier to converge to, and that they are bad (i.e.

Conversely, bigger neural networks contain significantly more local minima, but these minima turn out to be much better in terms of their actual loss.

In practice, what you find is that if you train a small network the final loss can display a good amount of variance - in some cases you get lucky and converge to a good place but in some cases you get trapped in one of the bad minima.

## THE NATURE OF CODE

The constructor could receive an argument indicating the number of inputs (in this case three: x, y, and a bias) and size the array accordingly.

In this example, we’ll have the perceptron receive its inputs as an array (which should be the same length as the array of weights) and return the output as an integer.

To train a neural network to answer correctly, we’re going to employ the method of supervised learning that we described in section 10.1.

The current velocity acts as a guess and the error (the steering force) tells us how to adjust the velocity in the right direction.

In a moment, we’ll see how adjusting the vehicle’s velocity to follow a target is just like adjusting the weights of a neural network to arrive at the right answer.

For any given weight, what we are looking to calculate is the change in weight, often called Δweight (or “delta” weight, delta being the Greek letter Δ).

The neural network will employ a similar strategy with a variable called the “learning constant.” We’ll add in the learning constant as follows: NEW WEIGHT = WEIGHT + ERROR * INPUT * LEARNING CONSTANT Notice that a high learning constant means the weight will change more drastically.

This may help us arrive at a solution more quickly, but with such large changes in weight it’s possible we will overshoot the optimal weights.

With a small learning constant, the weights will be adjusted slowly, requiring more training time but allowing the network to make very small adjustments that could improve the network’s overall accuracy.

Assuming the addition of a variable c for the learning constant, we can now write a training function for the perceptron following the above steps.

We could package this up in a class like so: Now the question becomes, how do we pick a point and know whether it is above or below a line?

We are using this scenario, one that we can easily solve without a perceptron, to demonstrate the perceptron’s algorithm as well as easily confirm that it is working properly.

But what *is* a Neural Network? | Deep learning, chapter 1

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