Answer to Question 1-1
The activation function in a deep architecture needs to be non-linear for several reasons:

1. **Non-linearity**: If the activation function in a deep architecture was linear, then no matter how many layers we add, the overall network would still behave like a single-layer perceptron, as the output would be a linear function of the input. This limits the model's ability to learn complex patterns and relationships in the data.

2. **Capturing Complex Patterns**: Non-linear activation functions such as ReLU, Sigmoid, and Tanh introduce non-linearities to the network, allowing it to capture complex patterns and relationships in the data that cannot be represented by a linear model.

3. **Gradient Descent**: Non-linear activation functions introduce non-linearities in the network, which helps in mitigating the vanishing gradient problem during backpropagation. This allows for more effective training of deep neural networks with multiple layers.

4. **Model Expressiveness**: Non-linear activation functions increase the expressiveness of the model, enabling it to learn and represent non-linear relationships between input features and target variables, which is crucial for tasks like image recognition, natural language processing, and other complex problems.

In conclusion, using a non-linear activation function in a deep architecture allows the neural network to learn complex patterns, avoid the vanishing gradient problem, and increase its overall expressiveness, making it more capable of tackling challenging tasks.

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Figure: N/A





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Answer to Question 1-2
Layer normalization (LayerNorm), Batch normalization (BatchNorm), and Instance normalization (InstanceNorm) are all methods used to normalize the inputs to each layer of a neural network. Here are the differences between them:

1. **Batch normalization (BatchNorm)**:
   - BatchNorm computes the mean and variance of each input channel over a mini-batch of samples.
   - It normalizes the input by subtracting the mean and dividing by the standard deviation calculated for each batch.
   - BatchNorm helps in reducing internal covariate shift, enables higher learning rates, and acts as a regularizer.
   - It is commonly used in feedforward neural networks and CNNs.

2. **Instance normalization (InstanceNorm)**:
   - InstanceNorm normalizes the mean and variance of each individual sample.
   - It calculates mean and variance for each sample across all its channels.
   - InstanceNorm is often used in style transfer networks or generative models where the mean and variance of features need to be adjusted independently for each image.

3. **Layer normalization (LayerNorm)**:
   - LayerNorm normalizes the mean and variance of each feature map separately.
   - It computes the mean and variance independently for each sample in a batch across all channels and spatial locations.
   - LayerNorm is frequently used in recurrent neural networks (RNNs) where batch sizes can vary.

The normalization layers help neural networks train more stably by addressing the internal covariate shift problem. Internal covariate shift occurs when the distribution of inputs to a layer changes during training, leading to slower convergence and making training more difficult. By normalizing the inputs to each layer, these techniques help in reducing this shift and ensuring that the network learns more effectively by providing a more stable gradient flow during training.





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Answer to Question 1-3
To determine the function computed by the output unit at the final time step in the recurrent network shown in the provided figure, we need to understand the structure and operations of the network.

1. The figure shows a simple recurrent neural network with 3 nodes (input, hidden, output) and connections between them.
2. The input sequence enter the input node and propagate through the network iteratively.
3. The values in the nodes are updated for each time step based on the input, weights, and the sigmoid activation function.
4. At the final time step, the output of the output unit will provide the final result computed by the network.

Since the figure is not provided, I cannot analyze the specific weights and connections in the network to determine the exact function computed. To find the output function, the weights in the connections and the specific input sequence would need to be known.

If the figure could be displayed, I could provide a more detailed analysis of the network structure and operation to determine the final output function.





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Answer to Question 1-4
a) At the training phase of an RNN language model using teacher forcing, the inputs to the network are the actual ground truth words from the training data. This means that at each time step, the input to the network is the word that should have been predicted at the previous time step.

b) At the test phase of an RNN language model, the inputs to the network are the words that were predicted by the model at the previous time step. In other words, the model uses its own predictions as inputs during the test phase.

Figures: N/A





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Answer to Question 1-5
Layer 1:
- Output volume dimension: (32, 32, 8)
- Number of Parameters: 224 (9 parameters per filter x 8 filters + 8 bias terms)
- Size of receptive field: 9

Layer 2 (Leaky Relu does not change the volume dimension, parameters, or receptive field size):
- Output volume dimension: (32, 32, 8)
- Number of Parameters: 0
- Size of receptive field: 9

Layer 3:
- Output volume dimension: (16, 16, 8)
- Number of Parameters: 0
- Size of receptive field: 9

Layer 4 (Batch Normalization does not change the volume dimension, parameters, or receptive field size):
- Output volume dimension: (16, 16, 8)
- Number of Parameters: 0
- Size of receptive field: 9

Layer 5:
- Output volume dimension: (16, 16, 16)
- Number of Parameters: 1168 (9 parameters per filter x 16 filters + 16 bias terms)
- Size of receptive field: 9

Layer 6:
- Output volume dimension: (8, 8, 16)
- Number of Parameters: 0
- Size of receptive field: 9

Layer 7:
- Output volume dimension: (1, 1, 128)
- Number of Parameters: 0
- Size of receptive field: 9

Layer 8:
- Output volume dimension: (1, 1, 10)
- Number of Parameters: 1290 (128*10 weights + 10 bias terms)
- Size of receptive field: 9





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Answer to Question 2-1
1. false
2. true
3. true
4. true
5. false





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Answer to Question 2-2
The factors that can increase the size of the receptive field when using a convolutional neural network are:

1. [true] The size of every convolutional kernel
3. [true] The activation function of each layer
4. [true] The size of the pooling layer





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Answer to Question 2-3
True







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Answer to Question 2-4
To train a neural network in practice, the valid activation functions need to be able to introduce non-linearity to the model. Therefore, the activation functions need to be differentiable and continuous. 

The valid activation functions among the given options are:
- f(x) = min(x, 0.5x) if x < 0; f(x) = min(x, 0.5x) if x >= 0
- f(x) = min(x, 0.1x) if x < 0; f(x) = max(x, 0.1x) if x >= 0

Therefore, the correct answer is:
- [] true

Subquestions:
a. Which activation functions are considered invalid and why?
b. If you were to plot the valid activation functions on a graph, how would they look?

a. The activation functions considered invalid are:
- f(x) = min(2,x) because it is not differentiable at x = 2.
- f(x) = 3x + 1 because it is a linear function and does not introduce non-linearity.
These functions do not fulfill the requirements of being differentiable and introducing non-linearity for training a neural network effectively.

b. To plot the valid activation functions:
- For f(x) = min(x, 0.5x) if x < 0; f(x) = min(x, 0.5x) if x >= 0, the graph would consist of two linear segments intersecting at (0,0).
- For f(x) = min(x, 0.1x) if x < 0; f(x) = max(x, 0.1x) if x >= 0, the graph would have a V-shape, with the minimum and maximum functions meeting at (0,0).

Figure paths:
- No figures provided for subquestion (b).





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Answer to Question 2-5
The following methods can reduce model overfitting:
1. Dropout
2. Batch normalization

These methods help in preventing the neural network from fitting too closely to the training data and thus improve the generalization and performance of the model.





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Answer to Question 2-6
During backpropagation, as the gradient flows backward through a sigmoid, the gradient will always:
- [ ] Increase in magnitude, maintain polarity

Explanation: When the gradient flows backward through a sigmoid function, the gradient will decrease in magnitude but maintain its polarity due to the derivative of the sigmoid function being y * (1-y).





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Answer to Question 3-1
a) In order for the attention layer $l$ head $h$ to attend to the previous token position $n$ the most at decoding step $n+1, the condition that must be met can be formulated as:
\[ \text{Softmax}\left( (W^{l,h}_Q x^l_{n+1})^T(W^{l,h}_K x^l_n) \right) = 1\]

b) The self-attention mechanism in the Transformer architecture requires the positional encoding to be able to fulfill this condition for arbitrary sequences.

Figure path: images/transformer_architecture.png





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Answer to Question 3-2
a. In order for $t_{k+1}$ to be predicted as the next token using greedy decoding, the condition that must be met at decoding step $n+1$ is: the probability $p(t_{k+1} | T)$ given by the decoder model's softmax output for the next token prediction must be the highest among all possible tokens in the vocabulary.

b. The condition that must be met for the layer $l$ attention head $h$ to attend from position $n+1$ to position $k+1$ the most is that the dot product attention mechanism must assign the highest attention weight (after softmax normalization) to the position embedding of $k+1$ when calculating the attention scores between the query at position $n+1$ and the keys/values at all positions in the sequence.

c. No, a Transformer model with only a single attention layer cannot fulfill the condition for arbitrary sequences and any $k < n$ where $t_k = t_n$. This is because for self-attention to work effectively in identifying the relevant context tokens, multiple attention heads and layers are necessary to capture different aspects of the input sequence.

d. 
   - For attention heads of the same layer in a Transformer model, there is no direct communication channel between them. Each attention head attends independently to the input tokens and contributes to the final output through their weighted sums.
   - For attention heads of successive layers, the communication channel is the hidden states passed from the encoder layers to the decoder layers. The Write operation of the self-attention decides what information to communicate to the next layer, and the Read operation decides what information to read from the previous layer's outputs.

e. In a two-layer Transformer model with attention layers only, we can design a sequence of self-attention operations where the layer $l > 1$ attention head $h$ attends to position $k+1$ by setting the query to be the position embedding of $n+1$, keys and values to be the embeddings of all positions in the sequence, and applying a linear mapping with appropriate weights to attend more on $k+1$ based on the similarity between query and key vectors. This way, the attention head can focus on the relevant historical context provided by position $k+1$ for the next token prediction.





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Answer to Question 4-1
**Answer:**

1. The input vector for the token $s_k$ being the $i$-th vocabulary word is the $i$-th row of the word embedding matrix $W_E$, denoted as $W_E[i]$.
   - As each column $i$ in the word embedding matrix $W_E$ represents a vectorial representation for the $i$-th vocabulary word, the $i$-th row contains the vector representation for the token $s_k$.

2. The input vector $x_k$ is multiplied with the $i$-th row of the word embedding matrix $W_E$, $W_E[i]$, in a matrix-vector multiplication to obtain the vectorial representation for the token $s_k$. 

Path to Figures: None





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Answer to Question 4-2
a. The function g must be differentiable if we want to perform training via gradient descent.

b. The gradient ∇l with respect to WE is ∂l/∂WE = ∂l/∂f(w) * ∂f(w)/∂WE. The unresolved term is ∂g/∂E(w). 

c. The gradient ∂l/∂wij for i ≠ k is 0, since the error does not directly depend on the word embeddings for words other than the actual input word w.

d. The insight from part c signifies that during the forward pass, the computational complexity of the embedding layer remains constant regardless of the vocabulary size since only the embedding for the input word is used. However, during the backward pass, the memory complexity scales with the vocabulary size as gradients for all words need to be stored for updating the embeddings.





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