1. The output of a single neuron is given by y = f(w₁x₁ +w₂x₂ +w₀x₀), where x₁ and x₂ are inputs, x₀ has the constant value 1, w₀, w₁ and w₂ are parameters (weights), and f : R ! R is a function. Suggest a function f typically used in this context. Explain how this neuron could be used to classify inputs (x₁; x₂) into one of two classes. The input data consists of 8 points:

(1:4; 5) (2:3; 9) (9:6; 9) (0:3; 1) (a₀; a₁) (a₂; a₃) (a₄; a₅) (a₆; a₇)

where a₀ a₁ a₂ a₃ a₄ a₅ a₆ a₇ is your student enrolment number. Assign these data points to classes, in such a way that the classification could be learnt by a single neuron. Suggest values for w₀, w₁ and w₂ that would achieve a correct classification, for your choice of function f. Draw a plot of the data points and the boundary line between the classes.

Explain in full detail how the weights of the single neuron in question 1 could be automatically adjusted using gradient descent i.e. how it could be trained to correctly distinguish the classes. Assume f to be the logistic function, and demonstrate why this choice simplifies the numerical calculation of the value of f'.