Putting neurons together: networks

A single neuron would be way too simple to make decisions and prediction reliably in most real-life applications. To unleash the full potential of neural networks, we can use the output of one neuron as the input of other neurons, whose outputs can be the input to yet other neurons, and so on. The output of the whole network is obtained as the output of a certain subset of the neurons, which are called the output layer.

Layers

Often the network architecture is composed of layers. The input layer consists of neurons that get their inputs directly from the data. So for example, in an image recognition task, the input layer would use the pixel values of the input image as the inputs of the input layer. The network typically also has hidden layers that use the other neurons´ outputs as their input, and whose output is used as the input to other layers of neurons. Finally, the output layer produces the output of the whole network. All the neurons on a given layer get inputs from neurons on the previous layer and feed their output to the next.

A classical example of a multilayer network is the so-called multilayer perceptron. As we discussed above, Rosenblatt’s Perceptron algorithm can be used to learn the weights of a perceptron. For multilayer perceptron, the corresponding learning problem is way harder and it took a long time before a working solution was discovered. But eventually, one was invented: the backpropagation algorithm lead to a revival of neural networks in the late 1980s. It is still at the heart of many of the most advanced deep learning solutions.

The path(s) leading to the backpropagation algorithm are rather long and winding. An interesting part of the history is related to the computer science department of the University of Helsinki. About three years after the founding of the department in 1967, a Master’s thesis was written by a student called Seppo Linnainmaa. The topic of the thesis was “Cumulative rounding error of algorithms as a Taylor approximation of individual rounding errors” (the thesis was written in Finnish, so this is a translation of the actual title “Algoritmin kumulatiivinen pyöristysvirhe yksittäisten pyöristysvirheiden Taylor-kehitelmänä”).

The automatic differentiation method developed in the thesis was later applied by other researchers to quantify the sensitivity of the output of a multilayer neural network with respect to the individual weights, which is the key idea in backpropagation.