Question 9: Consider a sigmoid neuron that takes in an input vector \…

Consider a sigmoid neuron that takes in an input vector $\mathbf{x} = \begin{bmatrix} 1 \\ 0.5 \\ 1 \end{bmatrix}$. The weight vector $\mathbf{w}$ is initialized to $\begin{bmatrix} 0.5 \\ 1 \\ 0.5 \end{bmatrix}$ and $\mathbf{b} = 1$. The output from the sigmoid neuron is

$\hat{\mathrm{y}} = \frac{1}{1 + exp^{-(\mathbf{w^T x} + \mathbf{b})}}$

Suppose we use the following loss function

$\mathcal{L} = \frac{1}{2}(y - \hat{y})^2$

Update the weight vector once by running the vanilla Gradient Descent algorithm with $\eta = 2$. Assume the true label $\mathbf{y} = \mathbf{0}$. What is the new loss value (that is, the loss computed after updating the weight vector)?

(consider upto two digits after the decimal for all the calculations)