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📗 [4 points] Given the following neural network that classifies all the training instances correctly. What are the labels (0 or 1) of the training data? The activation functions are LTU for all units: \(1_{\left\{z \geq 0\right\}}\). The first layer weight matrix is , with bias vector , and the second layer weight vector is , with bias
\(x_{i1}\)
\(x_{i2}\)
\(y_{i}\) or \(o_{1}\)
0
0
?
0
1
?
1
0
?
1
1
?
Note: if the weights are not shown clearly, you could move the nodes around with mouse or touch.
📗 Answer (comma separated vector): .
📗 [4 points] Given the following neural network that classifies all the training instances correctly. What are the labels (0 or 1) of the training data? The activation functions are LTU for all units: \(1_{\left\{z \geq 0\right\}}\). The first layer weight matrix is , with bias vector , and the second layer weight vector is , with bias
\(x_{i1}\)
\(x_{i2}\)
\(y_{i}\) or \(o_{1}\)
0
0
?
0
1
?
1
0
?
1
1
?
Note: if the weights are not shown clearly, you could move the nodes around with mouse or touch.
📗 Answer (comma separated vector): .
📗 [4 points] Given the following neural network that classifies all the training instances correctly. What are the labels (0 or 1) of the training data? The activation functions are LTU for all units: \(1_{\left\{z \geq 0\right\}}\). The first layer weight matrix is , with bias vector , and the second layer weight vector is , with bias
\(x_{i1}\)
\(x_{i2}\)
\(y_{i}\) or \(o_{1}\)
0
0
?
0
1
?
1
0
?
1
1
?
Note: if the weights are not shown clearly, you could move the nodes around with mouse or touch.
📗 Answer (comma separated vector): .
📗 [3 points] Suppose you are given a neural network with hidden layers, input units, output units, and hidden units. In one backpropogation step when computing the gradient of the cost (for example, squared loss) with respect to \(w^{\left(1\right)}_{11}\), the weight in layer \(1\) connecting input \(1\) and hidden unit \(1\), how many weights (including \(w^{\left(1\right)}_{11}\) itself, and including biases) are used in the backpropogation step of \(\dfrac{\partial C}{\partial w^{\left(1\right)}_{11}}\)?
📗 Note: the backpropogation step assumes the activations in all layers are already known so do not count the weights and biases in the forward step computing the activations.
📗 Answer: .
📗 [3 points] Suppose you are given a neural network with hidden layers, input units, output units, and hidden units. In one backpropogation step when computing the gradient of the cost (for example, squared loss) with respect to \(w^{\left(1\right)}_{11}\), the weight in layer \(1\) connecting input \(1\) and hidden unit \(1\), how many weights (including \(w^{\left(1\right)}_{11}\) itself, and including biases) are used in the backpropogation step of \(\dfrac{\partial C}{\partial w^{\left(1\right)}_{11}}\)?
📗 The above is a diagram of the network, the nodes labelled "1" are the bias units. You can highlight the edges representing the weights in the diagram, but they are not graded. Note: the backpropogation step assumes the activations in all layers are already known so do not count the weights and biases in the forward step computing the activations.
📗 Answer: .
📗 [2 points] In a three-layer (fully connected) neural network, the first layer contains sigmoid units, the second layer contains units, and the output layer contains units. The input is dimensional. How many weights plus biases does this neural network have? Enter one number.
📗 Answer: .
📗 [2 points] In a three-layer (fully connected) neural network, the first layer contains sigmoid units, the second layer contains units, and the output layer contains units. The input is dimensional. How many weights plus biases does this neural network have? Enter one number.
📗 The above is a diagram of the network, the nodes labelled "1" are the bias units.
📗 Answer: .
📗 [4 points] Fill in the missing weight below so that it computes the following function. All inputs takes value 0 or 1, and the perceptrons are linear threshold units. The first layer weight matrix is , with bias vector , and the second layer weight vector is , with bias .
\(x_{1}\)
\(x_{2}\)
\(y\) or \(o_{1}\)
0
0
0
1
1
0
1
1
📗 Note: if the weights are not shown clearly, you could move the nodes around with mouse or touch.
📗 Answer: .
📗 [4 points] Fill in the missing weight below so that it computes the following function. All inputs takes value 0 or 1, and the perceptrons are linear threshold units. The first layer weight matrix is , with bias vector , and the second layer weight vector is , with bias .
\(x_{1}\)
\(x_{2}\)
\(y\) or \(o_{1}\)
0
0
0
1
1
0
1
1
📗 Note: if the weights are not shown clearly, you could move the nodes around with mouse or touch.
📗 Answer: .
📗 [4 points] Fill in the missing weight below so that it computes the following function. All inputs takes value 0 or 1, and the perceptrons are linear threshold units. The first layer weight matrix is , with bias vector , and the second layer weight vector is , with bias .
\(x_{1}\)
\(x_{2}\)
\(y\) or \(o_{1}\)
0
0
0
1
1
0
1
1
📗 Note: if the weights are not shown clearly, you could move the nodes around with mouse or touch.
📗 Answer: .
📗 [2 points] Let the input \(x \in \mathbb{R}\). Thus the input layer has a single \(x\) input. The network has 5 hidden layers. Each hidden layer has 10 units. The output layer has a single unit and outputs \(y \in \mathbb{R}\). Between layers, the network is fully connected. All units in the network have a bias input. All units are linear units, namely the activation function is the identity function \(a = g\left(z\right) = z\), while \(z = w^\top x + b\) is a linear combination of all inputs to that unit (including the bias). Which functions can this network compute?
📗 Choices:
None of the above
📗 [1 points] You want to design a neural network with sigmoid units to predict the academic role from his webpage. Possible roles are "professor" (label 0), "student" (label 1), "staff" (label 2). Suppose each person can take on only one of these roles at the same time. The neural network uses one-hot encoding, label 0 is encoded by \(\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}\), label 1 is encoded by \(\begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}\), and label 2 is encoded by \(\begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}\). What is the role (enter a label, not a string) if the output is ?
📗 Answer: .
📗 [3 points] The sigmoid function in a neural network is defined as \(g\left(x\right) = \dfrac{1}{1 + e^{-x}}\). There is an another activation function defined as \(h\left(x\right)\) = . If \(h\left(x\right) = a \cdot g\left(b \cdot x\right) + c\), write down the values of \(a, b, c\) (constants, they cannot be functions of \(x\)). In the diagram, the green line is \(h\left(x\right)\) and the red line is \(a \cdot g\left(b \cdot x\right) + c\) with the \(a, b, c\) you selected.
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