The document provided is a bachelor project on "Deep Learning for Audio Processing," focusing on exercises related to backpropagation in a neural network. Here’s a breakdown of each exercise in the document:

1. Compute the Network Output and Loss for $x = (0, 0)^T$:

   • Calculate the output $\hat{y} = g(x)$ given the input vector $x = (0, 0)^T$.
   • Assume the true label $y = 1$ and compute the loss $L(g(x), y)$ using the squared error loss function.

2. Compute the Network Output and Loss for $x = (1, 1)^T$:

   • Calculate $\hat{y} = g(x)$ for $x = (1, 1)^T$.
   • With a true value $y = 2.12$, compute the loss $L(g(x), y)$.

3. Demonstrate that $g(x) = 0$ for $x = (1, 3)^T$:

   • Show that the output $g(x)$ is zero when the input is $x = (1, 3)^T$.
   • Compute the loss assuming $y = 2$.

4. Compute the Gradients for Loss with respect to $b_{12}$ and $a_{22}$:

   • Using the values from Exercise 3, calculate the partial derivatives $\frac{\partial L(g(x), y)}{\partial b_{12}}$ and $\frac{\partial L(g(x), y)}{\partial a_{22}}$.

5. Parameter Update Using Stochastic Gradient Descent (SGD):

   • Update the values of $b_{12}$ and $a_{22}$ using the gradients from Exercise 4 and a learning rate $\alpha = 2$.

6. Parameter Update with $f = 0$:

   • Assuming $f = 0$ (constant zero function), update the parameters $a_{11}, a_{12}, a_{21},$ and $a_{22}$ using SGD with a learning rate $\alpha = 3.14$.

7. Cross-Entropy Loss in Classification:

   • Discuss whether it’s logical to minimize or maximize the cross-entropy loss, explaining the behavior in cases of best and worst predictions.

Would you like me to walk you through the calculations for each exercise? Here are some related questions that can further your understanding:

1. How does backpropagation adjust weights in a neural network using the chain rule?
2. Why is the sigmoid function often used in neural networks, and what are its limitations?
3. What are the advantages and disadvantages of using squared error as a loss function in neural networks?
4. How does the choice of learning rate $\alpha$ affect the training process and convergence?
5. What are the key differences between cross-entropy loss and squared error loss in terms of their applications?

Tip: When computing gradients, remember that each layer's gradient depends on the output of the previous layer, which is a fundamental principle of backpropagation.