3990002: Mathematics of Machine Learning (Fall 2023)
Machine learning is the study of algorithms (i.e. gradient descent) that learn functions (i.e. deep networks) from experience (i.e. data). Behind this simple statement, is a lot of mathematical scaffolding: statistics for handling data, optimization for understanding learning algorithms, and linear algebra to create expressive models.
However, the typical computer science degree typical requires only a basic understanding of these mathematical concepts. This means that taking an advanced machine learning course may require taking multiple courses across graduate statistics and mathematics just to get up to speed. It will also get you used to the mathematical lingo of machine learning. If you’ve ever tried reading an ML paper and found it difficult to follow the concepts and equations, this might be the course for you.
To better prepare undergraduates for machine learning coursework and research, this course aims to provide the missing background required to be able understand mathematical concepts commonly used in machine learning. This course will be based on the Mathematics for Machine Learning textbook, which covers the mathematical foundations of machine learning as well as examples of how machine learning algorithms that use these foundations.
Instructor: Eric Wong (exwong@cis)
Class: Monday and Wednesday, 3:30PM4:59PM
Website: https://www.cis.upenn.edu/~exwong/moml/
Registration: To register, you need to sign up both on courses.upenn.edu and also submit the questionaire on the CIS waitlist before I can add you to the course.
Prerequisites: We will assume you’ve taken the minimum mathematics requirements of the Penn CS degree. That is:
 CIS1600 (CS foundations)
 CIS2610 or equivalent (discrete probability)
 MATH1400/MATH1410 or equivalent (vector calculus)
 MATH2400 or equivalent (linear algebra)
If you haven’t yet taken the course CS prerequisites, you may be able to get by going over the review topics in chapters 2, 5, and 6 of the course textbook.
Structure: We will build upon these foundations and cover a more in depth study suited for machine learning problems. Each focus area will be structured in three parts as (1) review of prior material, (2) new ML fundamentals, and (3) an ML example. The review will quickly go over concepts that were already covered in a previous course. The ML fundamentals will introduce the advanced concepts for machine learning. The example will show you how these fundamentals are used in practice. These focus areas are:
 Probability & statistics. Review: probability spaces and discrete probability. Fundamentals: continuous probability. Example: generalization bounds.
 Linear & functional analysis. Review: linear algebra. Fundamentals: function spaces. Example: representer theorems.
 Calculus & optimization. Review: Multivariate calculus. Fundamentals: optimization. Example: convergence rates.
We will accompany these topics with several examples demonstrating how these core techniques are used to prove fundamental results about machine learning algorithms.
Grading: There will be approximately 10 homeworks (estimated weekly) totaling 50% of your grade. There will also be 3 midterms at 15% each, one per focus area. 5% for participation.
A template for your homework solutions can be found here. Homeworks are due a week after they are assigned.
Schedule
Tentative schedule.
Date  Topic  Notes  

August 30  Overview  (1.1, extra notes)  
September 4  Labor day (no class)  
Probability & statistics  (probability lecture notes)  
September 6  Review  Discrete + Continuous Probability Reading: Chapters 6.1, 6.2 Homework: Problems 6.1, 6.4, 6.11 (due September 13) 

September 11  Review  Discrete + Continuous Probability Reading: Chapters 6.3, 6.4 

September 13  Fundamentals  Mean and Variance, Gaussian distribution Reading: Chapters 6.4, 6.5 Homework: Problems 6.5, 6.7, 6.9 (due September 20) 

September 18  Fundamentals  Exponential Distributions and Conjugacy Reading: 6.6 

September 20  Fundamentals  Concentration inequalities (Markov, Chebyshev, WLLN) (concentration lecture notes) Homework: Problems 6.3, 6.12abd, MGF/Chernoff (due September 27) 

September 25  Example  Generalization bounds  
September 27  Example  Generalization bounds  
October 2  Midterm 1  
Linear & functional analysis  
October 4  Review  Linear algebra (2.2,2.4) (linear algebra lecture notes) Homework: 2.1, 2.3, 2.9 (due October 11) 

October 9  Review  Linear algebra (2.5,2.6)  
October 11  Fundamentals  Change of Basis (2.7) Homework: 2.10, 2.16, 2.19 (due October 18) 

October 16  Fundamentals  Inner product spaces and Orthogonality (3.13.8)  
October 18  Fundamentals  Decompositions (4.1, 4.2, 4.4) Homework: 3.1, 3.7, 4.11 (due October 25) 

October 23  No Class  
October 25  Example  Functional analysis, Hilbert spaces, Kernels (12.4) (representer lecture notes) 

October 30  Example  Representer theorems  
November 1  Midterm 2  
Calculus & optimization  
November 6  Review  Multivariate calculus (5.15.4) (calculus notes) 

November 8  Review  Multivariate calculus (5.55.7) Homework: 5.4, 5.5, 5.9 (due November 15) 

November 13  Fundamentals  Multivariate Taylor Series (5.85.9)  
November 15  Fundamentals  Gradient Descent (7.1)) (continuous optimization notes) Homework: 7.4, 7.5, 7.8, 7.9 (due November 29) 

November 20  Fundamentals  Constrained and Convex Optimization (7.27.3)  
November 22  Friday class schedule (no class)  
November 27  Fundamentals  Conjugates & Taylor’s Theorem (SGD convergence notes) 

November 29  Example  Convergence analysis  
December 4  Example  Convergence analysis  
December 6  Midterm 3  
December 11  No class  
December 21  Term ends 