Answer to Question 1-1


Answer:
The activation function in a deep architecture needs to be non-linear because linear functions cannot capture complex relationships between inputs and outputs. Non-linear activation functions introduce non-linearity into the model, allowing it to learn and represent more complex patterns and relationships. Linear functions can only model linear relationships, meaning they can only capture a limited range of data and cannot effectively model the non-linear nature of many real-world problems. Non-linear activation functions, such as sigmoid, ReLU, and tanh, introduce non-linearity by introducing a threshold or saturation point, allowing the model to learn and represent more complex relationships between inputs and outputs.

Subquestion 1:
{"Question": "What is the difference between a linear and a non-linear activation function?"}
Answer:
A linear activation function is a function that maps any input directly to an output without any non-linearity. It can be represented by a straight line on a graph. Linear activation functions include the identity function, which simply passes the input through without changing it, and the sigmoid function in its linear region.

A non-linear activation function is a function that introduces non-linearity into the model by introducing a threshold or saturation point. It can be represented by a curve on a graph. Non-linear activation functions include the sigmoid function in its non-linear region, the ReLU function, and the tanh function.

Subquestion 2:
{"Question": "Why is the sigmoid function a popular choice for an activation function in deep learning?"}
Answer:
The sigmoid function is a popular choice for an activation function in deep learning because it introduces non-linearity into the model while also providing some degree of differentiability, which is important for training deep neural networks using backpropagation. The sigmoid function maps any input to an output between 0 and 1, which can be interpreted as a probability. This makes it useful for modeling binary classification problems, where the output is either 0 or 1. However, the sigmoid function can suffer from the vanishing gradient problem, which can make it difficult for the network to learn and optimize deep architectures. To address this issue, other non-linear activation functions, such as ReLU and tanh, have become more popular in deep learning.





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Answer to Question 1-2


Answer:

1. LayerNorm, BatchNorm and InstanceNorm are normalization techniques used in neural networks to help improve the stability and performance of the models.

2. BatchNorm: Batch Normalization (BatchNorm) normalizes the inputs to each layer independently for every batch of data. It calculates the mean and variance for the current batch and normalizes the inputs using these statistics. This helps to reduce the internal covariate shift, which is the change in the distribution of the inputs to a layer due to the variation in the data. This can lead to better convergence and improved generalization of the model.

3. InstanceNorm: Instance Normalization (InstanceNorm) normalizes the inputs to each layer independently for each input instance. It calculates the mean and variance for each channel separately and normalizes the inputs using these statistics. This is useful when dealing with data that has large variations in the input features across different instances, such as images. It can help to improve the stability and robustness of the model by reducing the effect of these variations on the activations of the layers.

4. LayerNorm: Layer Normalization (LayerNorm) normalizes the inputs to each layer across all the features in the layer. It calculates the mean and variance for each layer and normalizes the inputs using these statistics. This is useful when dealing with deep neural networks, where the activations of the layers can have large variations across different features and different layers. It can help to improve the stability and performance of the model by reducing the effect of these variations on the activations of the layers and making the model more robust to input perturbations.

5. Why these normalization layers help neural networks trained more stable: The normalization layers help neural networks trained more stable by reducing the effect of the variations in the inputs to each layer on the activations of the layers. This can help to improve the convergence of the model and reduce the sensitivity of the model to input perturbations. By normalizing the inputs, the normalization layers help to make the activations of the layers more consistent and stable, which can lead to better generalization of the model and improved performance on the test data.





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Answer to Question 1-3


The figure provided is a logistic regression model, not a recurrent network. I cannot determine the function computed by the output unit of a recurrent network based on the provided figure of a logistic regression model.

Therefore, I cannot answer the question as it is currently stated. Please provide the correct figure for the recurrent network you are asking about.





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Answer to Question 1-4


Answer:

a) At the training phase, the inputs to the network are the current time step's word embedding (or one-hot encoded word representation) and the previous hidden state of the network.

b) At the test phase, the inputs to the network are only the current time step's word embedding (or one-hot encoded word representation). The previous hidden state is not provided, as the network is generating the sequence one step at a time without any external guidance.





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Answer to Question 1-5


To answer the question, I will provide the output shapes and the number of parameters for each layer as well as the size of the receptive field.

1. Input: The input shape is (32, 32, 3). The number of parameters is 0 since no weights or biases are learned for the input. The size of the receptive field is 1.

2. CONV3-8: The output shape is (32, 32, 8). The number of parameters is 32 x 32 x 3 x 8 + 8 x 8 = 1,542,720. The size of the receptive field is 3x3 = 9.

3. Leaky ReLU: This layer does not have an output volume or parameters since it is an activation function applied to the output of the previous layer.

4. Pool-2: The output shape is (16, 16, 8). The number of parameters is 0 since this is a max-pooling layer without any learnable weights. The size of the receptive field is 2x2 = 4.

5. BatchNorm: This layer does not have an output volume or parameters since it is a normalization layer that applies a learned mean and variance to the input.

6. CONV3-16: The output shape is (16, 16, 16). The number of parameters is 16 x 16 x 8 x 16 + 16 x 16 x 16 = 3,276,864. The size of the receptive field is 3x3 = 9.

7. Leaky ReLU: This layer does not have an output volume or parameters since it is an activation function applied to the output of the previous layer.

8. Pool-2: The output shape is (8, 8, 16). The number of parameters is 0 since this is a max-pooling layer without any learnable weights. The size of the receptive field is 2x2 = 4.

9. FLATTEN: The output shape is 1 x (8 x 8 x 16) = 1 x 1024. The number of parameters is 0 since this layer only reshapes the input.

10. FC-10: The output shape is 1 x 10. The number of parameters is 1024 x 10 = 10,240.





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Answer to Question 2-1


Answer:

1. False for the first statement. While it's true that tanh has a smaller gradient than sigmoid near zero, it still suffers from the vanishing gradient problem in deeper layers.
2. True for the second statement. The vanishing gradient problem causes the gradients to become very small in deeper layers, making it harder for them to learn.
3. True for the third statement. Leaky ReLU has a non-zero gradient for negative inputs, which makes it less likely to suffer from the vanishing gradient problem.
4. True for the fifth statement. Xavier initialization helps prevent the vanishing gradient problem by ensuring that the weights are initialized with the correct scale.
5. None of the above is not an option.

Therefore, the correct answer is: [2, 4, 5]





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Answer to Question 2-2


Answer:
1. The size of every convolutional kernel (true)
2. False for the other options

Explanation:
The size of every convolutional kernel can increase the size of the receptive field in a convolutional neural network. This is because each kernel scans the input image with a certain size and stride, and the output feature map has the same dimensions as the input image minus the size of the kernel plus one pixel for each stride. By increasing the size of the kernel, the receptive field of each neuron in the next layer will capture a larger area of the input image.

The number of channels of every convolutional kernel does not directly affect the size of the receptive field. It determines the number of filters that will be applied to the input image, each filter looking for a different feature.

The activation function of each layer does not affect the size of the receptive field. It determines the non-linearity of the neurons in the layer and how they transform the input features.

The size of pooling layer does not increase the size of the receptive field. It reduces the size of the feature maps by downsampling them, which can help to reduce the computational complexity and increase the translation invariance of the network.





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Answer to Question 2-3


Answer: False. Dividing the weight vector W by 2 would change the intercept term, which is a bias term, and thus would affect the test accuracy.





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Answer to Question 2-4


Answer:

1. The function f(x) = min(2,x) can be considered a valid activation function to train a neural net in practice (false). This function is not commonly used as an activation function in neural networks due to its non-differentiability at x = 1, which makes it difficult to optimize the weights using backpropagation.

2. The function f(x) = 3x + 1 can be considered a valid activation function to train a neural net in practice (false). This function is not typically used as an activation function in neural networks because it does not satisfy the requirement of being non-linear and monotonically increasing.

3. The function f(x) = min(x, 0.5x) if x < 0; f(x) = min(x, 0.5x) if x >= 0 can be considered a valid activation function to train a neural net in practice (true). This function is known as the ReLU (Rectified Linear Unit) activation function, which is widely used in deep learning due to its ability to introduce non-linearity and sparsity in the neural network.

4. The function f(x) = min(x, 0.1x) if x < 0; f(x) = max(x, 0.1x) if x >= 0 can be considered a valid activation function to train a neural net in practice (true). This function is known as the Leaky ReLU activation function, which is a variant of the ReLU function that allows for a small negative slope for negative inputs, making it more effective in activating neurons with small negative inputs and preventing the "dying ReLU" problem.





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Answer to Question 2-5


Answer:
[0] True
[1] True
[2] False
[3] False

Explanation:
[0] Data augmentation is a technique that can be used to reduce model overfitting by artificially increasing the size of the training dataset. This is done by applying various transformations to the existing images, such as rotation, flipping, and zooming. By generating new training examples, the model is less likely to learn the noise in the training data and instead focuses on the underlying patterns.

[1] Dropout is a regularization technique that can be used to reduce model overfitting by randomly dropping out some neurons during training. This forces the model to learn more robust features that are not dependent on any single neuron. By randomly dropping out neurons, the model is forced to learn redundant representations, which can help prevent overfitting.

[2] Batch normalization is a technique used to normalize the inputs to each layer of a neural network. It can help improve the training process by reducing internal covariate shift, which is the shift in the distribution of the inputs to a layer as the weights are updated during training. However, it does not directly reduce model overfitting.

[3] Using Adam instead of SGD does not directly reduce model overfitting. Adam is an optimization algorithm that can converge faster than SGD and can handle non-stationary objectives. However, it does not provide any regularization or prevent overfitting.





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Answer to Question 2-6


Answer:
The answer to the question is: The gradient of a sigmoid function during backpropagation decreases in magnitude while maintaining its polarity.

Explanation:
The sigmoid function is defined as y = f(x) = 1 / (1 + e^(-x)). Its derivative is given by y' = f'(x) = y * (1 - y). This derivative shows that the gradient of the sigmoid function decreases in magnitude as y approaches 0 or 1, while maintaining its polarity.

Therefore, the answer to subquestion [0] is false.
The answer to subquestion [1] is false.
The answer to subquestion [2] is false.
The answer to subquestion [3] is true.





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Answer to Question 3-1


Answer:

a) For the attention layer $l$ head $h$ to attend to the previous token position $n$ the most at decoding step $n+1$, the query vector $Q$ of the self-attention mechanism should be closest to the key vector $K$ of the previous token $t_n$ in the sequence. Mathematically, this can be expressed as:

$$
\text{Attend}(T, X^l, W^{l, h})_n = \text{softmax}(\frac{Q \cdot K^T}{\sqrt{d_k}}) \cdot V \cdot O \cdot X^l_{n+1}
$$

where $\text{Attend}(T, X^l, W^{l, h})_n$ is the attention output for the previous token $t_n$, $Q$ is the query matrix, $K$ is the key matrix, $V$ is the value matrix, $O$ is the output matrix, $d_k$ is the dimension of the key vectors, and the softmax function is applied along the second dimension. The dot product between the query vector and the key vector of the previous token $t_n$ should be larger than the dot products between the query vector and the key vectors of other tokens in the sequence for the previous token to be attended to the most.

b) The self-attention mechanism in the Transformer architecture requires the ability to store and access all previous token representations $X^l_{1:n}$ in the input sequence $T$ to fulfill this condition for arbitrary sequences. This is achieved by using a multi-head attention mechanism, which computes attention scores for all possible token pairs in parallel, and a positional encoding scheme, which adds learnable position embeddings to each token representation to capture the position information in the sequence. The query, key, and value matrices are computed based on the input representations and the positional encodings, allowing the attention mechanism to attend to the correct positions in the sequence.





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Answer to Question 3-2


To answer the question, I will provide the answers to each subquestion below, indexed according to the provided JSON.

a) For greedy decoding, the condition for predicting $t_{k+1}$ as the next token is that the probability of $t_{k+1}$ given the previously decoded sequence $T$ is the highest among all possible next tokens. Mathematically, this can be expressed as:

$$
t_{k+1} = \underset{t}{\operatorname{argmax}} \ P(t | T)
$$

b) For the layer $l$ attention head $h$ to attend from position $n+1$ to position $k+1$ the most, the query vectors $Q$ in the self-attention mechanism should be close to the key-value vectors $K$ and $V$ from positions $n+1$ to $k+1$. Mathematically, this can be expressed as:

$$
\operatorname{similarity}(Q, K \oplus V) \geq \operatorname{similarity}(Q, K \oplus V')
$$

where $K \oplus V$ denotes the concatenation of key and value vectors, and $V'$ denotes the key-value vectors from any other position.

c) A Transformer model with only a single attention layer cannot fulfill the condition in (b) for arbitrary sequences and any $k < n$ where $t_k = t_n$. This is because a single attention layer attends to all positions in the sequence at once, and cannot selectively attend to a specific sub-sequence.

d) In a task where multiple attention heads cooperate in solving a task, there is no direct communication channel between them. Instead, they share the same input and output sequences, and each attention head processes the sequence independently. The self-attention mechanism allows each attention head to attend to different positions in the sequence, and the outputs of each attention head can influence the inputs of subsequent attention heads through the residual connections.

e) To design a sequence of self-attention operations in a two-layer Transformer model such that the layer $l > 1$ attention head $h$ attends to position $k+1$ for arbitrary sequences (given $t_k = t_n$), we can use position-wise feed-forward networks (FFNs) to add or subtract position embeddings before feeding them into the self-attention layers. Specifically, we can add or subtract a constant vector to the position embeddings of positions $k$ and $k+1$ to make them closer or farther apart in the embedding space, so that the layer $l > 1$ attention head $h$ attends to position $k+1$ instead of position $k$. This can be expressed as:

$$
X^{l-1}_{k+1} \leftarrow X^{l-1}_{k+1} + \Delta
$$

$$
X^{l-1}_{k} \leftarrow X^{l-1}_{k} - \Delta
$$

where $\Delta$ is a constant vector. The self-attention layers then attend to the modified input sequences $X^{l-1}$ and output the attended vectors $A^{l-1}$, which are then fed into the FFNs and added to the output of the previous layer to obtain the final output $Y^{l-1}$. The same process is repeated for the layer $l$ attention head $h$. This way, the layer $l > 1$ attention head $h$ attends to position $k+1$ for arbitrary sequences, given $t_k = t_n$.





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Answer to Question 4-1


Answer:

Given a neural network with a word embedding layer $E$ as its first layer, we have a vocabulary $V$ with $|V|$ words, and each word $w_i \in V$ is represented by a vector $e_i \in \mathbb{R}^d$. The word embedding matrix $W_E \in \mathbb{R}^{|V| \times d}$ is formed by stacking all these vectors column-wise.

When the input sequence $S = (s_1, \dots, s_n)$ is processed, the embedding layer independently maps each input token $s_k$ to its vectorial representation $x_k$.

To describe the layer $E$ as a matrix-vector multiplication for the $i$-th vocabulary word $w_i$, we first identify the corresponding column vector $e_i$ in the word embedding matrix $W_E$.

So, the input vector $x_i$ when $s_k = w_i$ is simply the $i$-th column vector $e_i$ of the word embedding matrix $W_E$.

Then, the matrix-vector multiplication is performed by multiplying the input vector $x_i = e_i$ with the word embedding matrix $W_E$. The result is the vectorial representation $x_k$ of the input token $s_k = w_i$.

Therefore, the matrix-vector multiplication can be written as:

$$x_k = E(s_k) = W_E \cdot e_i$$

where $x_k \in \mathbb{R}^d$ is the output vector, $e_i \in \mathbb{R}^d$ is the input vector (column vector from $W_E$), and $W_E \in \mathbb{R}^{|V| \times d}$ is the word embedding matrix.





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Answer to Question 4-2


Answer:

a) The function g must be differentiable for us to perform training via gradient descent. This is because gradient descent relies on the gradient of the loss function to update the parameters.

b) To find the gradient of the loss l with respect to the word embedding matrix W_E, we first compute the forward pass through the network to obtain the output f(w) = g(E(w)). Then, we compute the gradient of the loss with respect to the network output:

∇l = ∇l[f(w)] ∇f(w)[g(E(w))] ∇g(E(w))[E(w)]

The term ∇l[f(w)] is the derivative of the loss with respect to the network output, ∇f(w)[g(E(w))] is the derivative of the function g with respect to its input (the network output), and ∇g(E(w))[E(w)] is the derivative of g with respect to its input (the word embedding). The only unresolved term is ∇g(E(w))[E(w)], which represents the gradient of g with respect to its input embedding.

c) For i ≠ k, the gradient ∂l/∂w_ij is given by:

∂l/∂w_ij = ∂l/∂f_j ∂f_j/∂E_k ∂E_k/∂w_ij

where f_j is the j-th component of the network output f(w), E_k is the embedding vector for the input word w (which is the i-th row of W_E), and w_ij is the j-th element of the word embedding matrix W_E. Since the loss only depends on the difference between the predicted and target outputs, and the input word w does not appear in the target, the gradient ∂l/∂E_k is zero. Therefore, ∂l/∂w_ij = 0 for i ≠ k.

d) The insight from part (b) signifies that during the forward pass, the computational complexity of the embedding layer is O(|V|d), as we need to compute the product of the word embedding matrix and the input sequence. The memory complexity is also O(|V|d), as we need to store the word embedding matrix. During the backward pass, the computational complexity is O(|V|d^2), as we need to compute the transpose of the word embedding matrix to compute the gradient ∂l/∂W_E. The memory complexity is O(d|V|), as we need to store the intermediate representations of the network output and the gradient.





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