Because ANNs were intentionally designed as conceptual models of human brain activity, it is helpful to first understand how biological neurons function. As illustrated in the following figure, incoming signals are received by the cell's dendrites through a biochemical process. The process allows the impulse to be weighted according to its relative importance or frequency. As the cell body begins to accumulate the incoming signals, a threshold is reached at which the cell fires and the output signal is transmitted via an electrochemical process down the axon. At the axon's terminals, the electric signal is again processed as a chemical signal to be passed to the neighboring neurons across a tiny gap known as a synapse.

Figure 7.1: An artistic depiction of a biological neuron

The model of a single artificial neuron can be understood in terms very similar to the biological model. As depicted in the following figure, a directed network diagram defines a relationship between the input signals received by the dendrites (x variables) and the output signal (y variable). Just as with the biological neuron, each dendrite's signal is weighted (w values) according to its importance—ignore, for now, how these weights are determined. The input signals are summed by the cell body and the signal is passed on according to an activation function denoted by f.

Figure 7.2: An artificial neuron is designed to mimic the structure and function of a biological neuron

A typical artificial neuron with n input dendrites can be represented by the formula that follows. The w weights allow each of the n inputs (denoted by x_i) to contribute a greater or lesser amount to the sum of input signals. The net total is used by the activation function f(x), and the resulting signal, y(x), is the output axon:

Neural networks use neurons defined in this way as building blocks to construct complex models of data. Although there are numerous variants of neural networks, each can be defined in terms of the following characteristics:

Let's take a look at some of the variations within each of these categories to see how they can be used to construct typical neural network models.

The activation function is the mechanism by which the artificial neuron processes incoming information and passes it throughout the network. Just as the artificial neuron is modeled after the biological version, so too is the activation function modeled after nature's design.

In the biological case, the activation function could be imagined as a process that involves summing the total input signal and determining whether it meets the firing threshold. If so, the neuron passes on the signal; otherwise, it does nothing. In ANN terms, this is known as a threshold activation function, as it results in an output signal only once a specified input threshold has been attained.

The following figure depicts a typical threshold function; in this case, the neuron fires when the sum of input signals is at least zero. Because its shape resembles a stair, it is sometimes called a unit step activation function.

Figure 7.3: The threshold activation function is "on" only after the input signals meet a threshold

Although the threshold activation function is interesting due to its parallels with biology, it is rarely used in ANNs. Freed from the limitations of biochemistry, ANN activation functions can be chosen based on their ability to demonstrate desirable mathematical characteristics and their ability to accurately model relationships among data.

Perhaps the most commonly used alternative is the sigmoid activation function (more specifically the logistic sigmoid) shown in the following figure. Note that in the formula shown, e is the base of the natural logarithm (approximately 2.72). Although it shares a similar step or "S" shape with the threshold activation function, the output signal is no longer binary; output values can fall anywhere in the range from zero to one.

Additionally, the sigmoid is differentiable, which means that it is possible to calculate the derivative across the entire range of inputs. As you will learn later, this feature is crucial for creating efficient ANN optimization algorithms.

Figure 7.4: The sigmoid activation function mimics the biological activation function with a smooth curve

Although the sigmoid is perhaps the most commonly used activation function and is often used by default, some neural network algorithms allow a choice of alternatives. A selection of such activation functions is shown in the following figure:

Figure 7.5: Several common neural network activation functions

The primary detail that differentiates these activation functions is the output signal range. Typically, this is one of (0, 1), (-1, +1), or (-inf, +inf). The choice of activation function biases the neural network such that it may fit certain types of data more appropriately, allowing the construction of specialized neural networks.

For instance, a linear activation function results in a neural network very similar to a linear regression model, while a Gaussian activation function is the basis of a radial basis function (RBF) network. Each of these has strengths better suited for certain learning tasks and not others.

It's important to recognize that for many of the activation functions, the range of input values that affect the output signal is relatively narrow. For example, in the case of the sigmoid, the output signal is very near zero for an input signal below negative five and very near one for an input signal above positive five. The compression of the signal in this way results in a saturated signal at the high and low ends of very dynamic inputs, just as turning a guitar amplifier up too high results in a distorted sound due to clipping of the peaks of sound waves. Because this essentially squeezes the input values into a smaller range of outputs, activation functions like the sigmoid are sometimes called squashing functions.

One solution to the squashing problem is to transform all neural network inputs such that the feature values fall within a small range around zero. This may involve standardizing or normalizing the features. By restricting the range of input values, the activation function will have action across the entire range. A side benefit is that the model may also be faster to train, since the algorithm can iterate more quickly through the actionable range of input values.

The capacity of a neural network to learn is rooted in its topology, or the patterns and structures of interconnected neurons. Although there are countless forms of network architecture, they can be differentiated by three key characteristics:

The topology determines the complexity of tasks that can be learned by the network. Generally, larger and more complex networks are capable of identifying more subtle patterns and more complex decision boundaries. However, the power of a network is not only a function of the network size, but also the way units are arranged.

The number of layers

To define topology, we need terminology that distinguishes artificial neurons based on their position in the network. The figure that follows illustrates the topology of a very simple network. A set of neurons called input nodes receives unprocessed signals directly from the input data. Each input node is responsible for processing a single feature in the dataset; the feature's value will be transformed by the corresponding node's activation function. The signals sent by the input nodes are received by the output node, which uses its own activation function to generate a final prediction (denoted here as p).

The input and output nodes are arranged in groups known as layers. Because the input nodes process the incoming data exactly as received, the network has only one set of connection weights (labeled here as w₁, w₂, and w₃). It is therefore termed a single-layer network. Single-layer networks can be used for basic pattern classification, particularly for patterns that are linearly separable, but more sophisticated networks are required for most learning tasks.

Figure 7.6: A simple single-layer ANN with three input nodes

As you might expect, an obvious way to create more complex networks is by adding additional layers. As depicted here, a multilayer network adds one or more hidden layers that process the signals from the input nodes prior to reaching the output node. Most multilayer networks are fully connected, which means that every node in one layer is connected to every node in the next layer, but this is not required.

Figure 7.7: A multilayer network with a single two-node hidden layer

The direction of information travel

You may have noticed that in the prior examples, arrowheads were used to indicate signals traveling in only one direction. Networks in which the input signal is fed continuously in one direction from the input layer to the output layer are called feedforward networks.

In spite of the restriction on information flow, feedforward networks offer a surprising amount of flexibility. For instance, the number of levels and nodes at each level can be varied, multiple outcomes can be modeled simultaneously, or multiple hidden layers can be applied. A neural network with multiple hidden layers is called a deep neural network (DNN), and the practice of training such networks is referred to as deep learning. Deep neural networks trained on large datasets are capable of human-like performance on complex tasks like image recognition and text processing.

Figure 7.8: Complex ANNs can have multiple output nodes or multiple hidden layers

In contrast to feedforward networks, a recurrent network (or feedback network) allows signals to travel backward using loops. This property, which more closely mirrors how a biological neural network works, allows extremely complex patterns to be learned. The addition of a short-term memory, or delay, increases the power of recurrent networks immensely. Notably, this includes the capability to understand sequences of events over a period of time. This could be used for stock market prediction, speech comprehension, or weather forecasting. A simple recurrent network is depicted as follows:

Figure 7.9: Allowing information to travel backward in the network can model a time delay

DNNs and recurrent networks are increasingly being used for a variety of high-profile applications and consequently have become highly popular. However, building such networks uses techniques and software outside the scope of this book, and often requires access to specialized computing hardware or cloud servers. On the other hand, simpler feedforward networks are also very capable of modeling many real-world tasks. In fact, the multilayer feedforward network, also known as the multilayer perceptron (MLP), is the de facto standard ANN topology. If you are interested in deep learning, understanding the MLP topology provides a strong theoretical basis for building more complex DNN models later on.

The network topology is a blank slate that by itself has not learned anything. Like a newborn child, it must be trained with experience. As the neural network processes the input data, connections between the neurons are strengthened or weakened, similar to how a baby's brain develops as he or she experiences the environment. The network's connection weights are adjusted to reflect the patterns observed over time.

Training a neural network by adjusting connection weights is very computationally intensive. Consequently, though they had been studied for decades prior, ANNs were rarely applied to real-world learning tasks until the mid-to-late 1980s, when an efficient method of training an ANN was discovered. The algorithm, which used a strategy of back-propagating errors, is now known simply as backpropagation.

Although still somewhat computationally expensive relative to many other machine learning algorithms, the backpropagation method led to a resurgence of interest in ANNs. As a result, multilayer feedforward networks that use the backpropagation algorithm are now common in the field of data mining. Such models offer the following strengths and weaknesses:

In its most general form, the backpropagation algorithm iterates through many cycles of two processes. Each cycle is known as an epoch. Because the network contains no a priori (existing) knowledge, the starting weights are typically set at random. Then, the algorithm iterates through the processes until a stopping criterion is reached. Each epoch in the backpropagation algorithm includes:

Over time, the algorithm uses the information sent backward to reduce the total error of the network. Yet one question remains: because the relationship between each neuron's inputs and outputs is complex, how does the algorithm determine how much a weight should be changed? The answer to this question involves a technique called gradient descent. Conceptually, this works similarly to how an explorer trapped in the jungle might find a path to water. By examining the terrain and continually walking in the direction with the greatest downward slope, the explorer will eventually reach the lowest valley, which is likely to be a riverbed.

In a similar process, the backpropagation algorithm uses the derivative of each neuron's activation function to identify the gradient in the direction of each of the incoming weights—hence the importance of having a differentiable activation function. The gradient suggests how steeply the error will be reduced or increased for a change in the weight. The algorithm will attempt to change the weights that result in the greatest reduction in error by an amount known as the learning rate. The greater the learning rate, the faster the algorithm will attempt to descend down the gradients, which could reduce training time at the risk of overshooting the valley.

Figure 7.10: The gradient decent algorithm seeks the minimum error but may also find a local minimum

Although this process seems complex, it is easy to apply in practice. Let's apply our understanding of multilayer feedforward networks to a real-world problem.