Updating mechanism of the perceptron

Objetive:

$$\left\lbrace \begin{array}{ll} \ve{w}_{opt}^t \ve{x} = +1 ... ...\ve{x}_n \quad \textrm{belongs to } C_1\\ \end{array} \right.$$ (3)

Where $\ve{w}_{opt}^t \ve{x}=0$ is one of the infinite set of hyperplanes in $\mathbb{R}^n$ that classifies correctly the input vectors into its respective classes $C_0$ and $C_1$.

$$\left\lbrace \begin{array}{ll} \ve{w}_{n+1} = \ve{w}_n & \... ...textrm{missclassification and } d(n)=-1\\ \end{array} \right.$$ (4)

Where $0 < \eta(n) \leq 1$. Also $\eta(n) = \eta$ can be constant.

Beeing

$$d(n) = \left\lbrace \begin{array}{ll} +1 & if \quad \ve{x}... ...\ve{x}_n \quad \textrm{belongs to } C_1\\ \end{array} \right.$$

Writting (4) compactly:

w_n+1 = w_n + (n)2[ d(n)-y(n) ] x_n

The initial $\ve{w}_0$ would be arbitrary, but it's better that is choosen in a way such as $\sum_{i} w_i = 0$ for the activation function to start in the linear range, so the neuron doesn't begin in saturation.

• $\uparrow \eta \quad \Rightarrow \quad$ fast adaptation
• $\downarrow \eta \quad \Rightarrow \quad$ stable $\ve{w}_n$