Ex-2: Computational Derivation and Matrix Multiplication

Overview

In this exercise, you will practice computational derivation and matrix multiplication.

Objectives

1. Calculate the derivative of a function at a given point.
2. Perform matrix multiplication with tensors.
3. Implement a simple RMSE function.

The Assignment

Computational Derivation (1 point)

1. Given a function $f()$, calculate the derivative of the function $f$ at point $x$ using the definition of the derivative and evaluating the slope between points $x$ and $x+h$.
2. So, $f'()$ at point $x$ is ${{f(x+h) - f(x)} \over h}$ when $h$ is very small.
3. You can also use the definition ${{f(x+h) - f(x-h)} \over 2h}$. For the purpose of this exercise, you can take $h$ to a small value of your choosing (e.g., $h=0.001$).

RMSE Implementation (1 points)

1. Follow the instructions provided in the rmse.py file.
2. RMSE (Root Mean Square Error) = $\sqrt{\frac{1}{n}\sum_{i=1}^{n}(y_i - \hat{y_i})^2}$, where $y_i$ are the true values, $\hat{y_i}$ are the predicted values, and $n$ is the total number of samples.

Matrix Multiplication (3 points)

1. Follow the instructions provided in the tensor_multiplication.py file.
2. Your solution should be concise, typically a one-liner.
3. We introduce the transpose operator W.T, which switches the dimensions around. This is to conform with conventions of how the weight matrix is typically written in PyTorch and to ensure that when we perform matrix multiplication, the inner dimensions match.

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Validating and Evaluating Your Results

Online

1. After committing and pushing your code, check the mark on the top line (near the commit ID).
2. If some tests are failing, click on the ❌ to open up a popup, which will show details about the errors.
3. You can click the Details link to see what went wrong. Pay special attention to lines with the words "Failed" or "error".

4. Near the bottom of the Details page, you can see your score. Here are examples of 0/5 and 5/5:

5. When you achieve a perfect score, you will see a green checkmark near the commit ID.

Locally

1. You can test your code locally by installing and running pytest (pip install pytest or conda install pytest).
2. Run the tests using the command pytest in your terminal. This will show the status of each test and any errors that occurred.

Good luck!