Lecture 1-1: Intro and First Day Stuff

Lecture 1-1: Intro and First Day Stuff#

Download the original slides: CMSE382-Lec1_1.pdf

Warning

This is an AI-generated transcript of the lecture slides and may contain errors or inaccuracies. Please refer to the original course materials for authoritative content.

Course Information#

People in this Lecture#

Prof. Elizabeth Munch (she/her)
Depts of CMSE and Math

Omeiza Olumoye
Graduate Student, CMSE, MSU

What is This Course About?#

Machine Learning Pipeline
Finding the Optimum

There will be calculus! And matrices!

Course Outline#

See: msu-cmse-courses.github.io/cmse382-spring2026/

Course outline diagram showing the topics covered in CMSE 382 across the semester, organized by chapter.

Course Website#

Course slides and Jupyter notebooks:
https://elizabethmunch.com/CMSE382
— or —
https://msu-cmse-courses.github.io/cmse382-spring2026/

Note the syllabus link above!

D2L and Where to Find Grades#

https://d2l.msu.edu/d2l/home/2387926

Crowdmark and Where to Submit Assignments#

You should already have an automated email from the system. If not, talk to the instructor.

Office Hours#

Zoom link and schedule: msu-cmse-courses.github.io/cmse382-spring2026/help/#office-hours

Dr. Munch: Time TBD (Starting next week). Zoom & EGR 1511.

Omeiza Olumoye: Time TBD. Zoom & EGR (Room TBD).

Textbook#

Introduction to Nonlinear Optimization by Amir Beck.

Free download from MSU Library Website.
More info: msu-cmse-courses.github.io/cmse382-spring2026/textbook/

Class Structure#

Most classes have videos to watch before class. These will be posted on the course website and D2L.
Class is a brief review of the video content, and group work/coding time.
- Bring computer every day
- Jupyter notebooks
- Python
- In-class worksheets graded on completion. No credit unless present in class. 5 drops.
Approximately every two weeks there is a 15 minute quiz at the end of class.
- Drop one lowest grade
- Can bring a cheat sheet

Class Structure, Continued#

Homeworks due approximately every 2 weeks.
- Drop two lowest grades
- Sliding scale:
  - 24 hours late: 5% penalty
  - 48 hours late: 10% penalty
  - More than 48 hours: No late work accepted
Three Midterms
- See schedule for dates
- Not cumulative
One Project
- Analyze dataset using tools in class, submit written report
- 100 points
- Due at the end of the semester

Collaboration and Generative AI Policy#

You may use generative AI tools (e.g., ChatGPT, Bard, DALL-E) for brainstorming, drafting, and coding assistance on homework assignments.
You must clearly indicate any use of generative AI tools in your submissions, specifying the tool used and the nature of its contribution.
You may discuss homework problems with classmates, but all submitted work must be your own.
No collaboration or use of generative AI tools is allowed on quizzes and exams.

Approximate Schedule#

msu-cmse-courses.github.io/cmse382-spring2026/schedule/

Grade Distribution#

	Number	Number of drops	Percentage
Homeworks	6	2	40%
In-class worksheets	~30	5	10%
Quizzes	6	1	10%
Midterms	3	0	40%

Math Preliminaries - Part 1#

The Space \(\mathbb{R}^n\)#

Definition: The space \(\mathbb{R}^n\) is the set of \(n\)-dimensional column vectors with real components.

Addition:

\[\begin{split}\begin{pmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{pmatrix} + \begin{pmatrix} y_1 \\ y_2 \\ \vdots \\ y_n \end{pmatrix} = \begin{pmatrix} x_1 + y_1 \\ x_2 + y_2 \\ \vdots \\ x_n + y_n \end{pmatrix}\end{split}\]

Scalar multiplication:

\[\begin{split}\lambda \cdot \begin{pmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{pmatrix} = \begin{pmatrix} \lambda x_1 \\ \lambda x_2 \\ \vdots \\ \lambda x_n \end{pmatrix}\end{split}\]

Standard Basis#

Definition: The standard basis vectors in \(\mathbb{R}^n\) are the vectors

\[\begin{split}e_1 = \begin{pmatrix} 1 \\ 0 \\ \vdots \\ 0 \end{pmatrix}, \quad e_2 = \begin{pmatrix} 0 \\ 1 \\ \vdots \\ 0 \end{pmatrix}, \quad \ldots, \quad e_n = \begin{pmatrix} 0 \\ 0 \\ \vdots \\ 1 \end{pmatrix}.\end{split}\]

The Space \(\mathbb{R}^{m \times n}\)#

Definition: The space \(\mathbb{R}^{m \times n}\) is the set of \(m\) by \(n\) matrices with real components.

Special matrices:

Identity matrix: \(\mathrm{I}_n\)
Zeros matrix: \(\mathrm{0}_{m \times n}\)

Inner Product#

Definition: An inner product on \(\mathbb{R}^n\) is a function \(\langle \cdot, \cdot \rangle : \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}\) that satisfies the following properties for all \(x,y,z \in \mathbb{R}^n\) and \(\alpha \in \mathbb{R}\):

Symmetry: \(\langle x, y \rangle = \langle y, x \rangle\)
Additivity: \(\langle x, y + z \rangle = \langle x, y \rangle + \langle x, z \rangle\)
Homogeneity: \(\langle x, \lambda y \rangle = \lambda \langle x, y \rangle\)
Positive-definiteness: \(\langle x, x \rangle \geq 0\) with equality if and only if \(x = 0\).

Example: dot product \(\langle x, y \rangle = x^\top y = \sum_{i=1}^n x_i y_i.\)

Vector Norms#

Definition: A norm on \(\mathbb{R}^n\) is a function \(\|\cdot\| : \mathbb{R}^n \to \mathbb{R}\) that satisfies the following properties for all \(x,y \in \mathbb{R}^n\) and \(\lambda \in \mathbb{R}\):

Nonnegativity: \(\|x\| \geq 0\) with equality if and only if \(x = 0\)
Positive homogeneity: \(\|\lambda x\| = |\lambda| \|x\|\)
Triangle inequality: \(\|x + y\| \leq \|x\| + \|y\|\)

Example: \(\ell_p\) norm for \(p\geq 1\): \(\|x\|_p = \left( \sum_{i=1}^n |x_i|^p \right)^{1/p}.\)

Norms from Inner Products#

Given an inner product \(\langle \cdot, \cdot \rangle\) on \(\mathbb{R}^n\), we can define a norm by \(\|x\| = \sqrt{\langle x, x \rangle}.\)

Example: Given the dot product, we have the Euclidean norm, also called the \(\ell_2\) norm:

\[\|x\|_2 = \sqrt{x^\top x} = \sqrt{\sum_{i=1}^n x_i^2}.\]

Cauchy-Schwarz Inequality#

Lemma (Cauchy-Schwarz Inequality): For any \(x, y \in \mathbb{R}^n\),

\[| x^T \cdot y | \leq \|x\|_2 \cdot\|y\|_2.\]

Written another way and for more general inner products,

\[|\langle x,y \rangle| \leq \|x\| \cdot \|y\|.\]