# Neural Network (Function Representation)
📗 Neural networks are usually used for classification or regression in supervised learning. They can be used to represent functions for example \(Q\left(s, a\right)\) or \(\pi\left(s\right)\) for large or continuous state spaces.
📗 Given a state represented by a vector \(s\), a perceptron (for example, logistic regression), approximates \(Q\left(s, a\right) \approx g\left(w^\top s + b\right)\) for every action \(a\).
➭ If the identity function \(g\left(z\right) = z\) is used, then it is a linear regression.
➭ If the sigmoid function \(g\left(z\right) = \dfrac{1}{1 - e^{-z}}\) is used as the activation function, then it is a logistic regression.
➭ Another popular activation function is Rectified Linear Unit (ReLU) \(g\left(z\right) = \displaystyle\max\left\{0, z\right\}\).
📗 The outputs of the perceptrons can be used as inputs of another layer of perceptrons. The resulting network is called a multi-layer perceptron or a neural network, i.e. \(Q\left(s, a\right) \approx g\left(w_{l^\top} h_{l} + b_{l}\right)\), where \(h_{l} = g\left(w_{l-1^\top} h_{l-1} + b_{l-1}\right)\) are called hidden units in hidden layer \(l\) and can be defined recursively with \(h_{0} = s\).
📗 Similarly, \(\pi\left(s\right)\) can be approximated by a neural network.
📗 Universal Approximation Theorem says neural networks with one hidden layer and large number of units can approximate any continuous function arbitrarily closely; and networks with two hidden layers and large number of units can approximate any function arbitrarily closely.