To address each question in the document, here’s a step-by-step breakdown:

1. Compute $\hat{y} = g(x)$ for $x = (0, 0)^T$ and Loss $L(g(x), y)$:

   • Input vector $x = [0, 0]^T$.
   • First dense layer matrix $A = \begin{bmatrix} 6 & -2 \\ 1 & 1 \end{bmatrix}$.
   • Applying the function $g(x) = B(f \circ (A \cdot x))$ where $f$ is the sigmoid.
   • Compute the output of each layer, apply $f$, and then compute the final output $\hat{y}$.
   • Compare with $y = 1$ and compute the loss $L = (\hat{y} - y)^2$.

2. Compute $\hat{y} = g(x)$ for $x = (1, 1)^T$ and Loss $L(g(x), y)$:

   • Use the new input vector $x = [1, 1]^T$.
   • Repeat the process to compute $\hat{y}$, then calculate the loss $L$ with $y = 2.12$.

3. Show that $g(x) = 0$ for $x = (1, 3)^T$ and compute $L(g(x), y)$:

   • Substitute $x = [1, 3]^T$.
   • Show that after applying $g(x)$, the output $\hat{y} = 0$.
   • Compute the loss $L$ for $y = 2$.

4. Compute Gradients $\frac{\partial L(g(x), y)}{\partial b_{12}}$ and $\frac{\partial L(g(x), y)}{\partial a_{22}}$:

   • Using the values from Exercise 3, calculate the partial derivatives of the loss with respect to $b_{12}$ and $a_{22}$.

5. Update $b_{12}$ and $a_{22}$ using Stochastic Gradient Descent with learning rate $\alpha = 2$:

   • Apply the SGD update rule: $\theta := \theta - \alpha \frac{\partial L}{\partial \theta}$, using the gradients from Exercise 4.

6. Assume $f = 0$ (constant 0 function), and update $a_{11}, a_{12}, a_{21}, a_{22}$ using SGD with $\alpha = 3.14$:

   • With $f = 0$, recalculate the output values and gradients, then apply SGD updates for each parameter $a_{11}, a_{12}, a_{21}, a_{22}$.

7. Cross-Entropy Loss in Classification:

   • Discuss the implications of minimizing or maximizing cross-entropy loss in classification. Explain with best and worst case predictions.

Let me know if you’d like detailed calculations for any of these steps, or have any questions.

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Extended Questions:

1. What happens to the loss if we increase the learning rate significantly in each update?
2. How does the choice of activation function, like ReLU vs. sigmoid, affect the backpropagation process?
3. Could a different optimizer (e.g., Adam) alter the results for updates in steps 5 and 6? If so, how?
4. What impact does the loss function choice (MSE vs. Cross-Entropy) have in different neural network tasks?
5. How would the network behavior change if an additional hidden layer were added?

Tip: When computing gradients, keep track of each partial derivative separately to simplify the backpropagation steps, especially in multi-layer networks.