Artificial neural networks (briefly, nets) represent a class of machine learning models, loosely inspired by studies about the central nervous systems of mammals. Each net is made up of several interconnected neurons, organized in layers, which exchange messages (they fire, in jargon) when certain conditions happen. Initial studies were started in the late 1950s with the introduction of the perceptron (for more information, refer to the article: The Perceptron: A Probabilistic Model for Information Storage and Organization in the Brain, by F. Rosenblatt, Psychological Review, vol. 65, pp. 386 - 408, 1958), a two-layer network used for simple operations, and further expanded in the late 1960s with the introduction of the backpropagation algorithm, used for efficient multilayer networks training (according to the articles: Backpropagation through Time: What It Does and How to Do It, by P. J. Werbos, Proceedings of the IEEE, vol. 78, pp. 1550 - 1560, 1990, and A Fast Learning Algorithm for Deep Belief Nets, by G. E. Hinton, S. Osindero, and Y. W. Teh, Neural Computing, vol. 18, pp. 1527 - 1554, 2006). Some studies argue that these techniques have roots dating further back than normally cited (for more information, refer to the article: Deep Learning in Neural Networks: An Overview, by J. Schmidhuber, vol. 61, pp. 85 - 117, 2015). Neural networks were a topic of intensive academic studies until the 1980s, when other simpler approaches became more relevant. However, there has been a resurrection of interest starting from the mid-2000s, thanks to both a breakthrough fast-learning algorithm proposed by G. Hinton (for more information, refer to the articles: The Roots of Backpropagation: From Ordered Derivatives to Neural Networks and Political Forecasting, Neural Networks, by S. Leven, vol. 9, 1996 and Learning Representations by Backpropagating Errors, by D. E. Rumelhart, G. E. Hinton, and R. J. Williams, vol. 323, 1986) and the introduction of GPUs, roughly in 2011, for massive numeric computation.

These improvements opened the route for modern deep learning, a class of neural networks characterized by a significant number of layers of neurons, which are able to learn rather sophisticated models based on progressive levels of abstraction. People called it deep with 3-5 layers a few years ago, and now it has gone up to 100-200.

This learning via progressive abstraction resembles vision models that have evolved over millions of years in the human brain. The human visual system is indeed organized into different layers. Our eyes are connected to an area of the brain called the visual cortex V1, which is located in the lower posterior part of our brain. This area is common to many mammals and has the role of discriminating basic properties and small changes in visual orientation, spatial frequencies, and colors. It has been estimated that V1 consists of about 140 million neurons, with 10 billion connections between them. V1 is then connected with other areas V2, V3, V4, V5, and V6, doing progressively more complex image processing and recognition of more sophisticated concepts, such as shapes, faces, animals, and many more. This organization in layers is the result of a huge number of attempts tuned over several 100 million years. It has been estimated that there are ~16 billion human cortical neurons, and about 10%-25% of the human cortex is devoted to vision (for more information, refer to the article: The Human Brain in Numbers: A Linearly Scaled-up Primate Brain, by S. Herculano-Houzel, vol. 3, 2009). Deep learning has taken some inspiration from this layer-based organization of the human visual system: early artificial neuron layers learn basic properties of images, while deeper layers learn more sophisticated concepts.

This book covers several major aspects of neural networks by providing working nets coded in Keras, a minimalist and efficient Python library for deep learning computations running on the top of either Google's TensorFlow (for more information, refer to https://www.tensorflow.org/) or University of Montreal's Theano (for more information, refer to http://deeplearning.net/software/theano/) backend. So, let's start.

In this chapter, we will cover the following topics:

• Perceptron
• Multilayer perceptron
• Activation functions
• Gradient descent
• Stochastic gradient descent
• Backpropagation