Welcome to your first CS221 assignment! The goal of this assignment is to sharpen your math and programming skills needed for this class. If you meet the prerequisites, you should find these problems relatively innocuous. Some of these problems will occur again as subproblems of later homeworks, so make sure you know how to do them. If you're unsure about them or need a refresher, we recommend going through our prerequisites module or other resources on the internet, or coming to study halls or office hours.

Before you get started, please read the Assignments section on the course website thoroughly.

We also recommend you check the Piazza release post for this homework here for any corrections/clarifications made post-release.

In this class, we will cast a lot of AI problems as optimization problems, that is, finding the best solution in a rigorous mathematical sense. At the same time, we must be adroit at coping with uncertainty in the world, and for that, we appeal to tools from probability.

1. Let $x_1, \dots, x_n$ be real numbers representing positions on a number line. Let $w_1, \dots, w_n$ be positive real numbers representing the importance of each of these positions. Consider the quadratic function: $f(\theta) = \frac{1}{2} \sum_{i=1}^n w_i (\theta - x_i)^2$. Note that $\theta$ here is a scalar. What value of $\theta$ minimizes $f(\theta)$? Show that the optimum you find is indeed a minimum. What problematic issues could arise if some of the $w_i$'s are negative?
   
   Note: You can think about this problem as trying to find the point $\theta$ that's not too far away from the $x_i$'s. Over time, hopefully you'll appreciate how nice quadratic functions are to minimize.
   An expression for the value of $\theta$ that minimizes $f(\theta)$ and how you got it. A short calculation/argument to show that it is a minimum. 1-2 sentences describing a problem that could arise if some of the $w_i$'s are negative.
2. In this class, there will be a lot of sums and maxes. Let's see what happens if we switch the order. Let $f(\mathbf x) = \sum_{i=1}^d \max_{s \in \{1,-1\}} s x_i$ and $g(\mathbf x) = \max_{s \in \{1,-1\}} \sum_{i=1}^d s x_i$, where $\mathbf x = (x_1, \dots, x_d) \in \mathbb{R}^d$ is a real vector. Does $f(\mathbf x) \le g(\mathbf x)$, $f(\mathbf x) = g(\mathbf x)$, or $f(\mathbf x) \ge g(\mathbf x)$ hold for all $\mathbf x$? Prove it.
   
   Hint: You may find it helpful to refactor the expressions so that they are maximizing the same quantity over different sized sets.
   A short (3-5) line/sentence proof. You should use mathematical notation in your proof, but can also make your argument in words.
3. Suppose you repeatedly roll a fair six-sided die until you roll a $1$ (and then you stop). Every time you roll a $2$, you lose $a$ points, and every time you roll a 6, you win $b$ points. You do not win or lose any points if you roll a 3, 4, or a 5. What is the expected number of points (as a function of $a$ and $b$) you will have when you stop?
   
   Hint: You will find it helpful to define a recurrence. If you define $V$ as the expected number of points you get from playing the game, what happens if you roll a 2? You lose $a$ points and then get to play again. What about the other cases? Can you write this as a recurrence?
   A recurrence to represent the problem and the resulting expression from solving the recurrence (No more than 1-2 lines).
4. Suppose the probability of a coin turning up heads is $0 \lt p \lt 1$, and that we flip it 7 times and get $\{ \text{H}, \text{H}, \text{T}, \text{H}, \text{T} , \text{T}, \text{H} \}$. We know the probability (likelihood) of obtaining this sequence is $L(p) = p p (1-p) p (1-p) (1-p) p = p^4(1-p)^3$. What value of $p$ maximizes $L(p)$? What is an intuitive interpretation of this value of $p$?
   
   Hint: Consider taking the derivative of $\log L(p)$. You can also directly take the derivative of $L(p)$, but it is cleaner and more natural to differentiate $\log L(p)$. You can verify for yourself that the value of $p$ which maximizes $\log L(p)$ must also maximize $L(p)$ (you are not required to prove this in your solution).
   The value of $p$ that maximizes $L(p)$ and the work/calculation used to solve for it. Note that you must prove/show that it is a maximum. A 1 sentence intuitive interpretation of the value of $p$.
5. Now for a little bit of practice manipulating conditional probabilities. Suppose that $A$ and $B$ are two events such that $P(A|B) = P(B|A)$. We also know that and $P(A \cup B) = 1$ and $P(A \cap B) > 0$. Prove that $P(A) > \frac{1}{2}$.
   
   Hint: Note that $A$ and $B$ are not necessarily mutually exclusive. Consider how we can relate $P(A \cup B)$ and $P(A \cap B)$.
   A short (~5 line) proof/derivation.
6. Let's practice taking gradients, which is a key operation for being able to optimize continuous functions. For $\mathbf w \in \mathbb R^d$ (represented as a column vector) and constants $\mathbf a_i, \mathbf b_j \in \mathbb R^d$ (also represented as column vectors) and $\lambda \in \mathbb R$, define the scalar-valued function $$f(\mathbf w) = \Bigg( \sum_{i=1}^n \sum_{j=1}^n (\mathbf a_i^\top \mathbf w - \mathbf b_j^\top \mathbf w)^2 \Bigg) + \lambda \|\mathbf w\|_2^2,$$ where the vector is $\mathbf w = (w_1, \dots, w_d)^\top$ and $\|\mathbf w\|_2 = \sqrt{\sum_{k=1}^d w_k^2} = \sqrt{{\mathbf w}^T {\mathbf w}}$ is known as the $L_2$ norm. Compute the gradient $\nabla f(\mathbf w)$.
   
   Recall: the gradient is a $d$-dimensional vector of the partial derivatives with respect to each $w_i$: $$\nabla f(\mathbf w) = \left(\frac{\partial f(\mathbf w)}{\partial w_1}, \dots \frac{\partial f(\mathbf w)}{\partial w_d}\right)^\top.$$ If you're not comfortable with vector calculus, first warm up by working out this problem using scalars in place of vectors and derivatives in place of gradients. Not everything for scalars goes through for vectors, but the two should at least be consistent with each other (when $d=1$). Do not write out summation over dimensions, because that gets tedious.
   An expression for the gradient and the work used to derive it. (~5 lines). No need to expand out terms unnecessarily, try to write the final answer compactly.

When designing algorithms, it's useful to be able to do quick back of the envelope calculations to see how much time or space an algorithm needs. Hopefully, you'll start to get more intuition for this by being exposed to different types of problems.

1. Suppose we have an image of a human face consisting of $n \times n$ pixels. In our simplified setting, a face consists of two eyes, two ears, one nose, and one mouth, each represented as an arbitrary axis-aligned rectangle (i.e. the axes of the rectangle are aligned with the axes of the image). As we'd like to handle Picasso portraits too, there are no constraints on the location or size of the rectangles. For example, it is possible for all four corners of a single rectangle to be the same point (which is a rectangle of size 0) or for all 6 rectangles to be on top of each other. How many possible faces (choice of its 6 component rectangles) are there? In general, we only care about asymptotic complexity, so give your answer in the form of $O(n^c)$ or $O(c^n)$ for some integer $c$.
   A big-O bound for the number of possible faces and some simple explanation/reasoning for the answer (~2 sentences).
2. Suppose we have an $n\times n$ grid. We start in the upper-left corner (position $(1,1)$), and we would like to reach the lower-right corner (position $(n,n)$) by taking single steps down or to the right. Suppose we are provided with a function $c(i, j)$ that outputs the cost of touching position $(i, j)$, and assume it takes constant time to compute for each position. Note that $c(i, j)$ can be negative. Give an algorithm for computing the minimum cost in the most efficient way. What is the runtime (just give the big-O)?
   A description of the algorithm for computing the minimum cost as efficiently as possible (~5 sentences). The big-O runtime and a short explanation of how it arises from the algorithm.

In this problem, you will implement a bunch of short functions. The main purpose of this exercise is to familiarize yourself with Python, but as a bonus, the functions that you will implement will come in handy in subsequent homeworks.

Do not import any outside libraries (e.g. numpy). Only standard python libraries and/or the libraries imported in the starter code are allowed.

If you're new to Python, the following provide pointers to various tutorials and examples for the language:

• Python for Programmers
• Example programs of increasing complexity

1. Implement find_alphabetically_last_word in submission.py.
2. Implement euclidean_distance in submission.py.
3. Implement mutate_sentences in submission.py.
4. Implement sparse_vector_dot_product in submission.py.
5. Implement increment_sparse_vector in submission.py.
6. Implement find_singleton_words in submission.py.
7. Implement compute_longest_palindrome_length in submission.py.

As a big class and the first AI/ML course taken by many students, CS221 has a wide range of people coming from different backgrounds. On top of that, the transition to remote learning has created new challenges for all of us. We know that these challenges affect students unevenly and to different degrees depending on their individual circumstances. As a result, we've put together a quick survey to get a little more information about the background of everyone in the class and figure out what we can do to best support each one of you this quarter.

Please fill out the Google form here.

This survey is optional and will not be graded, so your answers will not affect your grade whatsoever. However, it will be a huge help to us in making CS221 a great experience for everyone!

Submission is done on Gradescope.

Written: When submitting the written parts, make sure to select all the pages that contain part of your answer for that problem, or else you will not get credit. To double check after submission, you can click on each problem link on the right side and it should show the pages that are selected for that problem.

Programming: After you submit, the autograder will take a few minutes to run. Check back after it runs to make sure that your submission succeeded. If your autograder crashes, you will receive a 0 on the programming part of the assignment.

More details can be found in the Submission section on the course website.