In the preceding section, we looked into a simple neural network without any layers called a perceptron. The perceptron was found to have serious limitations, and in 1969, Marvin Minsky and Seymour Papert worked on research that led to the conclusion that a perceptron is incapable of learning any complex logic.

In fact, they showed that it would be a struggle to learn even logical functions as simple as XOR. That led to a decrease in interest in machine learning in general, and neural networks in particular, and started an era that is now known as the AI winter. Researchers around the world would not take AI seriously, thinking that it was incapable of solving any complex problems.

One of the primary reasons for the so-called AI winter was the limitation of the hardware capabilities available at that time. Either the necessary computing power was not available or it was prohibitively expensive. Toward the end of the 1990s, advances in distributed computing provided easily available and affordable infrastructure, which resulted in the thaw of the AI winter. The thaw reinvigorated research in AI. This eventually resulted in turning the current era into an era that can be called the AI spring, where there is so much interest in AI in general and neural networks in particular. 

For more complex problems, researchers have developed a multilayer neural network called a multilayer perceptron. A multilayer neural network has a few different layers, as shown in the following diagram. These layers are as follows:

• Input layer
• Hidden layer(s)
• Output layer:

An important thing to note is that the neuron is the basic unit of this network, and each neuron of a layer is connected to all neurons of the next layer. For complex networks, the number of these interconnections explodes, and we will explore different ways of reducing these interconnections without sacrificing too much quality.

First, let's try to formulate the problem we are trying to solve.

The input is a feature vector, x, of dimensions n.

We want the neural network to predict values. The predicted values are represented by ý.

Mathematically, we want to determine, given a particular input, the probability that a transaction is fraudulent. In other words, given a particular value of x, what is the probability that y = 1? Mathematically, we can represent this as follows:

Note that x is an n_x-dimensional vector, where n_x is the number of input variables.

This neural network has four layers. The layers between the input and the output are the hidden layers. The number of neurons in the first hidden layer is denoted by . The links between various nodes are multiplied by parameters called weights. Training a neural network is all about finding the right values for the weights. 

Let's see how we can train a neural network.