## 这是一篇来自美国的关于为了增强对目标函数、回归模型和优化的梯度下降算法理解的CS代写。它包括一个编程分配（具有可选的奖励积分扩展）和一个报告。这个项目是个人的工作，没有代码共享请，但你可以发布bug问题到广场寻求帮助。

**Topic **

Design and implement a gradient descent algorithm or algorithms for regression.

**Programming work **

**A) Data pre-processing **

Pre-process the attribute values of your data by normalizing or standardizing each variable. Make sure to keep a copy that was not pre-processed, so you can analyze the effect that pre-processing the data has on the optimization.

**B) Univariate linear regression **

In lecture, we discussed univariate linear regression y = f(x) = mx+b, where there is only a single independent variable x, using MSE as the loss function.

Your program must specify the objective function of mean squared error and be able to apply the gradient descent algorithm for optimizing a univariate linear regression model.

**C) Multivariate linear regression **

In practice, we typically have multi-dimensional (or multi-variate) data, i.e., the input **x **is a vector of features with length p. Assigning a parameter to each of these features, plus the b parameter, results in p+1 model parameters. Multi-variate linear models can be succinctly represented as:

y = f(**x**) = (**m **· **x**)

(i.e., dot product between **m **and **x**),

where **m **= (m0, m1, …, mp) T and **x **= (1, x1, …, xp) T, with m0 in place of b in the model.

Your program must be able to apply the gradient descent algorithm for optimizing a multivariate linear regression model using the mean squared error objective function.

**D) Optional extension 1 – Mean Absolute Error as the loss function **

For bonus points, include the option of optimizing for the MAE instead of MSE. Calculating MAE as your error is insufficient! You must define a new gradient calculation to be used for the gradient descent optimization process.

**E) Optional extension 2 – Ridge Regression **

For bonus points, include the option of optimizing an l2 penalty as part of your loss function.

Calculating MSE + l2 as your error is insufficient! You must define a new gradient calculation to be used for the gradient descent optimization process. You must tune the ? hyperparameter value for minimizing test error.**IMPORTANT**: Regression is basic, so there are many implementations available, but you MUST implement your method yourself. This means that you cannot use an embedded function for regression or gradient descent from a software package. You may use other basic functions like matrix math, but the gradient descent and regression algorithm must be implemented by yourself.

**Data to be used **

We will use the Concrete Compressive Strength dataset in the UCI repository at UCI Machine Learning Repository: Concrete Compressive Strength Data Set

(https://archive.ics.uci.edu/ml/datasets/Concrete+Compressive+Strength)

Note that the last column of the dataset is the response variable (i.e., y).

There are 1030 instances in this dataset.

Use 900 instances for training and 130 instances for testing, randomly selected. This means that you should learn parameter values for your regression models using the training data, and then use the trained models to predict the testing data’s response values without ever training on the testing dataset.

**What to submit – follow the instructions here to earn full points **

- (80 pts total + 17 bonus points) The report as a pdf

o Introduction (15 pts + 5 bonus points)

- (4 pts) Your description/formulation of the problem (what’s the data and what practical application could there be for your work with it, beyond just “this is my homework” or “I want to optimize this equation”),

- (3 pts) a description of how you normalized or standardized your data. Include some figures that illustrate how the distribution of feature values changed because of your pre-processing

- (5 pts) the details of your algorithm (e.g., stopping criterion, is this stochastic gradient descent or not, how you chose your learning rate, etc),

- (3 pts) pseudo-code of your algorithm (see Canvas for an example)
- (+2 bonus pts) if you include a description of how you implemented MAE
- (+3 bonus pts) if you include a description of how you implemented Ridge Regression

o Results (52 pts + 8 bonus points)

- To report the performance of your models, calculate the variance explained (eg.

R-squared) for the response variable, which is:

1-*MSE/Variance(observed)*

In other words, calculate the average squared error between predicted responses and actual responses (MSE). Then calculate the average squared difference between actual responses and mean actual response (Variance). Divide the former by the latter, then subtract from 1.§ (26 pts) Variance explained of your models on the training dataset when using only one of the predictor variables (univariate regression) and when using all eight (multivariate regression).

You should have a total of nine values from optimizing on the raw data, and nine values from optimizing on the pre-processed data.

At least two of your models optimized on raw data must achieve a positive variance explained on the training data.

At least two of your models optimized on pre-processed data must achieve a positive variance explained on the training data

- (10 pts) Variance explained of your models on the testing data.

You should have a total of nine values from optimizing on the raw data, and nine values from optimizing on the pre-processed data.

- (16 pts) Plots of your univariate models on top of scatterplots of the training data used. Please plot the data using the x-axis for the predictor variable and the y-axis for the response variable.

- (+4 bonus points) if you include results from using the MAE loss function on the pre-processed data

- (+4 bonus points) if you include results from using Ridge Regression on the preprocessed data

o Discussion (13 pts + 4 bonus points)

- (8 pts) Compare and contrast your models.
- Did the same models that accurately predicted the training data also accurately predict the testing data?

- Did different models take longer to train or require different hyperparameter values?

- How did pre-processing change your results or optimization approach?
- (5 pts) Draw some conclusions about what factors predict concrete compressive strength. What would you recommend for making the hardest possible concrete?

- (+2 bonus points) if you include comparisons from using MAE
- (+2 bonus points) if you include comparisons from using Ridge Regression

Cement (component 1)(kg in a m^3 mixture)

0.0805381**Note: **We won’t be grading for good writing practices, but you may have points taken off if you don’t write in full sentences and paragraphs, or if you fail to correct spelling and grammar that a simple spell-check tool would alert you of. Results may be presented as a table, but you must label the rows/columns with enough detail for a reader to interpret it without searching your text, and the figures must be labeled as well.

- (20 pts total + 8 bonus points) Your program (in a language you choose) including:

o (15 pts) The code itself

o (5 pts) Sufficient instructions/documentation on how to run your program (input/output plus execution environment and compilation if needed)

o (+4 bonus points) if you include code for using MAE as the loss function

o (+4 bonus points) if you include code for Ridge Regression

**Note: **We won’t grade your program’s code for good coding practices or documentation.

However, if we find your code difficult to understand or run, we may ask you to run your program to show it works on a new dataset.

**Due date **

Monday, March 6 (midnight, STL time). Submission to Gradescope via course Canvas.

A one-week late extension is available in exchange for a 20% penalty on your final score.

**About the extra credit: **

The bonus point values listed are the upper limit of extra credit you can earn for each extension. How many points you get will depend on how well you completed each task. Feel free to include partially completed extensions for partial extra credit!

In total, you can earn up to 25 bonus points on this assignment, which means you can get a 125% as your score if you submit it on time, or you can submit this assignment late with the 20% penalty and still get a 100% as your score. It’s up to you how you would prefer to manage your time and effort.

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