How to find bias for perceptron algorithm?

Question

My question is very basic. I am starting with ML and am working on the perceptron algorithm. I successfully computed the weights for this input data:

X = [[0.8, 0.1], [0.7, 0.2], [0.9, 0.3], [0.3, 0.8], [0.1, 0.7], [0.1, 0.9]]
Y = [-1, -1, -1, 1, 1, 1]

Output_weights = [-0.7, 0.5]

But I didn't take bias into account, i.e. I assumed the discriminator line goes through the origin. Now, let's say I add another point into my training set:

new_X = [4,4]
new_Y = [-1]

How do I proceed if I want to compute the bias as well? In the first iteration for example, I'd set default weights to $[0,0]$, so I find the first point that is incorrectly classified.

Without bias, it is easy. I compute the dot product

0.8*0 + 0.1*0 = 0

should be $-1$, so it is incorrectly classified. I update the weights to:

[-0.8,-0.1]

However, taking bias into account, I get:

0.8*0 + 0.1*0 + bias 

Now, how do I update the weights and the bias? What is the procedure?

I have searched several tutorials like this or this but didn't find an answer. A link to some resource would help, too.

Answer 1

Rather than viewing your data as

X = [[0.8, 0.1], [0.7, 0.2], [0.9, 0.3], [0.3, 0.8], [0.1, 0.7], [0.1, 0.9]]
Y = [-1, -1, -1, 1, 1, 1]

You could have treat the problem as

X = [[1, 0.8, 0.1], [1, 0.7, 0.2], [1, 0.9, 0.3], [1, 0.3, 0.8], [1, 0.1, 0.7], [1, 0.1, 0.9]]
Y = [-1, -1, -1, 1, 1, 1]

That is to append a $1$ in every single entry of $X$. The weight corresponding to $1$ would be the bias.

That is $Y=Ax+b$ can be written as $Y=[A, e]\begin{bmatrix}x \\b\end{bmatrix}$ where $e$ is the all one vector.