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Quiz 2

For Penn State student, access quiz here

import ipywidgets as widgets

Question 1

Consider \(f(x,y)=e^{x^2+y^2}\) , compute the Hessian matrix and determine whether \(f(x,y)\) is a convex function.

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Answer: Hessian matrix is

\[\begin{split} e^{x^2+y^2} \begin{bmatrix} 4x^2+2&4xy\\ 4xy&4y^2+2 \end{bmatrix} \end{split}\]

\(f(x,y)\) is not a convex function

Question 2

Given any \(w\in R^n,\:b\in R,\:\) consider the multivariable function \(f(\boldsymbol x)=e^{\boldsymbol w\cdot \boldsymbol x+b}\)  Whether \(f(\boldsymbol x)\) is a convex function?

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Answer: Yes

Question 3

Consider \(f(x,y)=x^2.\) Whether \(f(x,y)\) is a \(\lambda-\) strongly convex function?

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Answer: No

Question 4

Consider

\[\begin{split} f\left( \begin{matrix} x\\ y \end{matrix} \right)=x^2+y^2 \end{split}\]

Given initial guess

\[\begin{split} \left( \begin{matrix} x^0\\ y^0 \end{matrix} \right) = \left( \begin{matrix} 1\\ 2 \end{matrix} \right), \eta=\frac14 \end{split}\]

, compute

two steps of the gradient  descent method for \(f(x,y)\):

\[\begin{split} \left( \begin{matrix} x^{k+1}\\ y^{k+1} \end{matrix} \right) = \left( \begin{matrix} x^k\\ y^k \end{matrix} \right) - \eta \nabla f\left( \begin{matrix} x^k\\ y^k \end{matrix} \right), k=0, 1. \end{split}\]
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Answer: \(\frac{1}{4},\frac{1}{2}\)

Question 5

Suppose a point \(x\) is drawn at random uniformly from the square \([-1,1]\times[-1,1].\) Let

\[\begin{split} \boldsymbol v= \left( \begin{matrix} 1\\ 1 \end{matrix} \right) \end{split}\]

and consider the random variable \(\mathcal X_{\boldsymbol v} ={\boldsymbol x} \cdot {\boldsymbol v}\).  What are \(\mathbb{E} [\mathcal X_{\boldsymbol v}]\) and  \(\big(\mathbb{V}[ \mathcal X_{\boldsymbol v}]\big)^2\).

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Answer: Unavailable

Question 6

 def model(100,10):
    return nn.Linear(100,10)
 

  File "<ipython-input-2-413e0725d122>", line 1
    def model(100,10):
              ^
SyntaxError: invalid syntax

What are the sizes of W and b of the model?

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Answer: Size of W: torch.Size([10, 100]), Size of b: torch.Size([10])

Question 7

Load MNIST dataset with batch_size=100 as follows

trainset = torchvision.datasets.MNIST(root='./data', train= True, download=True,transform=torchvision.transforms.ToTensor())
trainloader = torch.utils.data.DataLoader(trainset, batch_size=100, shuffle=True)

for i, (images, labels) in enumerate(trainloader):

What are the sizes of variable images and labels?

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Answer: Size of images: torch.Size([100, 1, 28, 28]), Size of labels: torch.Size([100])

Question 8

In the training process of MNIST dataset with mini-batch stochastic gradient descent(SGD) method, if we set bath_size = 600, how many iterations (or SGD steps) are there in one epoch?

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Answer: 100

Question 9

What is the output of the following code?

sequence = torch.tensor(([[4,2,3],[1,5,6],[0,7,2]]))
maxvalue, index = torch.max(sequence, 1) 
print(maxvalue,',',index)
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Answer: tensor([4, 6, 7]) , tensor([0, 2, 1])

Question 10

What is the output of the following code?

num_correct = 0
labels = torch.tensor([1,2,3,4,0,0,0])
predicted = torch.tensor([0,2,3,4,1,2,0])
num_correct += (predicted == labels).sum()
print(num_correct)

---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
<ipython-input-6-4c7bc87f85fb> in <module>
      1 num_correct = 0
----> 2 labels = torch.tensor([1,2,3,4,0,0,0])
      3 predicted = torch.tensor([0,2,3,4,1,2,0])
      4 num_correct += (predicted == labels).sum()
      5 print(num_correct)

NameError: name 'torch' is not defined
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Answer: tensor([4, 7, 6]) , tensor([0, 2, 1])