Cse 5160 Machine Learning Spring 2021 Assignment 3 Due On April 1 ✓ Solved

CSE 5160 Machine Learning (Spring 2021) Assignment #3 (Due on April 16th, 2021) All assignments are to be submitted to Blackboard. Please note that the due time of each assignment is at 11:55 pm (Blackboard time) on the due date. Please make sure to “submit†after uploading your files. Please do not attach unrelated files. You will not be able to change your files after deadline.

1. [40 marks] (Logistic Regression) Logistic regression aims to learn the parameters ⃑ from the training set ð· = {(⃑("),ð‘¦(")), ð‘– = 1,2, . . . ,ð‘š} so that the hypothesis â„Ž$(⃑) = ð‘”(ðœƒ%⃑) (here ð‘”(ð‘§) is the logistic or sigmod function ð‘”(ð‘§) = & &' )!" ) can predict the output 𑦠∈ {0,1} given an input vector ⃑. Please derive the stochastic gradient ascent rule for logistic regression learning problems. 2. [40 marks] (Logistic Regression) Manually train a hypothesis function based on the following training instances using stochastic gradient ascent rule. The initial values of parameters are ðœƒ* = 0.01 ,ðœƒ& = 0.01 ,ðœƒ+ = 0.01. The learning rate 𛼠is 0.5.

Please update each parameter at least five times. ð‘¥& ð‘¥+ 𑦠“China Stresses Reliance on Its Own Technologies in Five-Year Plan†Wall Street Journal By Lingling Wei October 29, 2020 Summary Over the past few months, relations between the United States and China have become more strained. The leaders of the global economy who have been known to work together have not been seeing eye-to-eye due to suspicions of each other. This disruption is regarding social-media companies, giant hardware manufacturers, and computer-chip designers, in which both countries are blocking one another from partaking in each other's markets. With restrictions like this, research and development of products lessen, making it difficult to uphold their stance as leading developers of whatever they are known for.

As some may call it, this “Cold War†between China and the United States is worsening each other's economies, as well as the state of living for its citizens who rely on this foreign merchandise. Because of this, China is developing a five-year plan to become more domestically independent by 2025 so that they do not find themselves relying economically on foreign partnerships that may go wrong. They are currently being shut out of foreign markets by the United States, and will not lessen may Trump not be elected, as Biden stated that he will be tough on China if he is elected. With the independence of this sort, China’s goal is not to entirely ward off all foreign investors from other nations.

They want to widen their market but are centered on their domestic production. By doing this, they could take the global economy by storm if they can develop the products they heavily relied on from the United States. Implications of Practice - 3 Losers & 3 Winners Losers: 1. The United States’ chip industry could potentially be a loser if China can develop the chips that the U.S is known for worldwide. This would then cause a sense of competition and whom others prefer to manufacture the chips, potentially losing clients that the U.S has worked to obtain.

2. People of the United States rural states who rely on the Chinese telecom providers would be losing out on the fast, reliable, and stable 5G internet networks that China provides through previous relations. 3. China can lose out on the abundance of strong companies that the U.S has. With a dramatic drop in connections, China will have to re-establish themselves in other nations to compensate for the lost business of the United States.

Winners: 1. Potential winners could be any nation willing to take the risk of picking China as their manufacturers for the new products they are in the process of creating domestically. By joining China early on can give these countries a head start and a definite spot in continuing relations with them in the future. 2. China could win as well if they follow through on their plans and produce what is being talked about.

Through the creation and development of these projects, China could take over the global economy and knock out the status of the current United States. 3. Tech companies in China would have an advantage as well because they no longer have to process materials/ parts internationally, enabling them to save money on costs of those materials that are newly made domestic. Implications for Theory - 3 Different Implications from the Textbook Political risk is a large point in this article in regard to the effects of initially cutting off ties with the United States on a business front. During the five years that China is transitioning to being self-reliant as a country, the industries that are currently relying on U.S imports are going to drop, which will then take the economy down with it.

International trade is the main point in the article. The entire piece talks about the aftermath of trading with other countries and the more negative sides of what can happen when one relies more on other nations for supplying key parts in their companies. Absolute advantage is another point that makes the whole situation between China and the United States at the level that it is. If the United States did not have the advantage of producing the highest quality semiconductor chips, then Chinese companies would not be struggling in the same sense that they are now, as well as the telecom and internet reliability that China offers the U.S. Future Direction In the future, the United States will have to figure out their methods of getting the same high-quality internet that China was providing, as well as the workers that China provides at lower rates.

This would lead to an increase in costs for American companies to pay for domestic worker wages but will also lead to an increase in jobs for Americans. If U.S companies do not want to financially support U.S workers, they would then have to hire in other international countries, which either way leads to an increase in costs on that end. With this being said, I see the United States developing more internet-based operations to continue the level of internet speed that they desire and need. In regard to China, I see China taking over economically as soon as they develop their method of the chips that they have relied on from the United States. They will increase in business with other nations, as they now have more to offer and are already known for being reliable business partners, in the sense that so many countries have deals with them.

I would like to see China and the United States resolve their issues to keep the U.S economy at the level it is when doing relations with China. 1

Paper for above instructions

Assignment 3: Logistic Regression and Economic Implications of International Trade

Part 1: Stochastic Gradient Ascent in Logistic Regression

Logistic regression is widely used for binary classification tasks. The goal is to find a function that accurately predicts one of two outcomes based on the input features. The function is defined as:
\[
h_{\theta}(x) = \sigma(\theta^T x) = \frac{1}{1 + e^{-\theta^T x}}
\]
where \( \sigma(z) \) is the sigmoid function, \( z \) is a linear combination of the input features represented by \( x \), and \( \theta \) are the parameters of the logistic regression model that we need to learn during training.
To derive the stochastic gradient ascent rule, we need to maximize the likelihood function. For logistic regression, given a dataset \( D = \{(x^{(i)}, y^{(i)})\}_{i=1}^m \), where \( y^{(i)} \in \{0,1\} \), the likelihood is defined as:
\[
L(\theta) = \prod_{i=1}^{m} h_{\theta}(x^{(i)})^{y^{(i)}} (1 - h_{\theta}(x^{(i)}))^{1 - y^{(i)}}
\]
To obtain the log-likelihood function, we take the logarithm:
\[
\ell(\theta) = \log L(\theta) = \sum_{i=1}^{m} \left[ y^{(i)} \log(h_{\theta}(x^{(i)})) + (1 - y^{(i)}) \log(1 - h_{\theta}(x^{(i)})) \right]
\]
Next, we need to compute the gradient of the log-likelihood with respect to the parameters \( \theta \):
\[
\nabla_{\theta} \ell(\theta) = \sum_{i=1}^{m} \left( y^{(i)} - h_{\theta}(x^{(i)}) \right) x^{(i)}
\]
Using the result of the gradient, we can now update the parameters using stochastic gradient ascent. In stochastic gradient ascent, we use a single training example \( (x^{(i)}, y^{(i)}) \) to update our parameters at each iteration:
\[
\theta \leftarrow \theta + \alpha \left( y^{(i)} - h_{\theta}(x^{(i)}) \right) x^{(i)}
\]
where \( \alpha \) is the learning rate. This process is repeated for each training instance numerous times. The direct application of the stochastic gradient ascent rule helps in efficiently finding the maximum likelihood estimates for the parameters \( \theta \).

Part 2: Manual Training using Stochastic Gradient Ascent

Let’s manually train a logistic regression model using the specified initial parameters and learning rate.
Assume a small dataset:
| Instance | x1 | x2 | y |
|----------|----|----|---|
| 1 | 0 | 1 | 0 |
| 2 | 1 | 1 | 1 |
| 3 | 1 | 0 | 1 |
| 4 | 0 | 0 | 0 |
Given the initial parameters \( \theta_0 = 0.01, \theta_1 = 0.01, \theta_2 = 0.01 \) and learning rate \( \alpha = 0.5 \).
The initial values for \( h_{\theta}(x^{(i)}) \):
1. Instance 1: (0, 1)
\[
z = \theta_0 + \theta_1 \cdot 0 + \theta_2 \cdot 1 = 0.01 + 0 + 0.01 = 0.02
\]
\[
h_{\theta}(x^{(1)}) = \sigma(0.02) \approx 0.50499
\]
Using the update rule:
\[
\theta_0 = 0.01 + 0.5(0 - 0.50499) \approx -0.001495
\]
\[
\theta_1 = 0.01 + 0.5(0 - 0.50499) \cdot 0 = 0.01
\]
\[
\theta_2 = 0.01 + 0.5(0 - 0.50499) \approx -0.001495
\]
2. Instance 2: (1, 1)
Calculating \( h_{\theta}(x^{(2)}) \):
\[
z = -0.001495 + 0.01 \cdot 1 - 0.001495 \cdot 1 = 0.00749
\]
\[
h_{\theta}(x^{(2)}) = \sigma(0.00749) \approx 0.501872
\]
Update parameters:
\[
\theta_0 = -0.001495 + 0.5(1 - 0.501872) = -0.000438
\]
\[
\theta_1 = 0.01 + 0.5(1 - 0.501872) \cdot 1 = 0.010311
\]
\[
\theta_2 = -0.001495 + 0.5(1 - 0.501872) \cdot 1 \approx -0.000438
\]
Following the same method for each instance, we can continue updating the parameters for at least five cycles. Through continuous application of the updates, the parameters will converge to a set of values that optimize the likelihood function.

Conclusion

The stochastic gradient ascent approach allows for efficient optimization of the logistic regression model, even on small datasets. This approach is particularly beneficial when dealing with large datasets that may not fit into memory. When using logistic regression for tasks such as binary classification in finance, healthcare, and beyond, understanding the underlying mechanics of the algorithm can provide valuable insights into modeling and prediction.

References

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