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Tuning a neural network by hand

The essence of neural networks is just simple function fitting. A neural network is a universal function approximator and given enough data and network parameters it can approximate any (continuous) function to any desired accuracy[1]. This is immensely powerful given that pretty much any behaviour (identifying faces, playing chess, holding a conversation) can be defined as a function, and using the right encoding any function can be described by a mathematical function.

Above is the simplest case of a feed-forward neural network, a single hidden layer with a single neuron. Using the activation function ReLU this is simply a linear function that has been cut off at zero:

A simple line segment that has been cutoff at zero is of course not that interesting but by adding more nodes one gets more line segments which, by adjusting the parameters, can be combined to approximate any function. Here follows a schematic of a neural network with three nodes in its hidden layer.

This corresponds to three line segments

Node 1

Node 2

Node 3

Output node

The output node is the weighted sum of the hidden nodes and its bias. For simplicity the weights of the final edges have been set to 1. It is now fairly easy to hand-tune the network parameters to approximate a target function of:||[0.012 \cdot x^{2} + 1||]

Show output value

Show target function