Course: Math 547, Statistical Learning Theory, Fall 2021
Prerequisite: Working knowledge (graduate or advanced undergraduate level) of: Probability Theory (Math 407/505a, 507a recommended) Real Analysis (Math 425a/b, Math 525a recommended) and Linear Algebra (Math 471).
Course Content: Binary classification, empirical risk minimization, support vector machines, voting algorithms and AdaBoost, Vapnik-Chervonenkis combinatorics, concentration-of-measure inequalities, sparse recovery problems, high-dimensional convex geometry.
Last update: August 16 2021
Instructor: Steven Heilman, stevenmheilman(@-symbol)gmail.com
Office Hours: Tuesdays, 10AM-12PM noon, on zoom [link posted on blackboard]
Lecture Meeting Time/Location: Mondays, Wednesdays, and Fridays, 10AM-1050AM, KAP 140
TA: Jihoon Sohn, jihoon.sohn(@-symbol)usc.edu
TA Office Hours: Held in the Math Center, with schedule provided at that link.
Textbook: There is no official textbook, but the following might be useful references.
Kearns and Vazirani, An Introduction to Computational Learning Theory. I will be following this book the most in the first half of the course.
Shalev-Schwarz and Ben-David, Understanding Machine Learning: From Theory to Algorithms. (A link is available here.)
Gine and Nickl, Mathematical Foundations of Infinite-Dimensional Statistical Models. This encyclopedic book has a lot of nice mathematical tools in it. Chapters 2 and 3 are most relevant.
Vershynin, High-Dimensional Probability: An Introduction with Applications to Data Science. (A link is available here.) Tools and applications.
Ledoux, Concentration of Measure. Another book with useful mathematical tools. Most chapters here are relevant.
Boucheron, Lugosi, Massart, Concentration Inequalities. An encyclopedic book with some overlapping content with Ledoux's book; chapters 12 and 13 are most relevant.
Mossel's Course Webpage on Deep Learning, available here, based on the book Deep Learning, by Goodfellow, Bengio and Courville, available here.
Project Abstract Due: Friday, September 24, 5PM PST (via blackboard)
Progress Report Due: Friday, October 29, 5PM PST (via blackboard)
Final Project Presentations: Last two or three weeks of class, in class, schedule TBD
Final Report Due: Saturday, December 4, 5PM PST (via blackboard)
Exam Procedures: This course has no midterm exams and no final exam. The final project is a replacement for the exams.
Accessibility Services: If you are registered with accessibility services, I would be happy to discuss this at the beginning of the course. Any student requesting accommodations based on a disability is required to register with Accessibility Services (OSAS) each semester. A letter of verification for approved accommodations can be obtained from OSAS. Please be sure the letter is delivered to me as early in the semester as possible. OSAS is located in 301 STU and is open 8:30am-5:00pm, Monday through Friday.
213-740-6948 (TDD only)
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Final Project Guidelines:
The final project is an opportunity to do a bit of research on a subject of your choice related to statistical learning theory. A project could begin with an interesting question or a well-known problem, and perhaps lead to investigating or implementing various algorithms, conducting an empirical analysis, etc.
Along the way, you will review relevant literature, identify appropriate data sources, select appropriate means of evaluation, and either develop novel methodology for your problem or deploy and comprehensively evaluate existing methodology for your new application.
The goal is to say something interesting about a problem in statistical learning theory. You could perhaps develop new methodology for an existing problem or application that has no fully satisfactory solution. You could alternatively tackle a new problem or application with existing methodology; in this case, you should identify one or more questions without satisfactory answers in your chosen domain and explore how the methodology can help you answer those questions. You may draw inspiration from particular datasets, but your focus should rest not on the data itself but rather on the questions about the world that you can answer with that data.
While a substantial theoretical component is not required for this project, (i.e. you do not necessarily have to prove new theorems), you should be able to prove a few things or at very least review or streamline some existing proofs in your investigation.
You may work alone or in a group of two; the standards for a group project will be twice as high. In certain cases I might approve a group of three, but this is unlikely.
We strongly encourage you to come to office hours to discuss your project ideas, progress, and difficulties.
Final Project Milestones: In all cases, please use LaTeX.
I: Project Proposal. By this first milestone, you should have selected a question or problem of interest, identified relevant data sources, begun exploring the literature surrounding the question, and discussed your ideas with the course staff. Your project proposal deliverable is a 1/2 - 1 page report describing the question or problem you intend to tackle, why this question is important or interesting, prior work on this problem, what data you intend to use in your analyses, and the principal challenges that you anticipate.
If you would like to receive feedback about particular aspects of your proposal, please indicate this in your submission.
I can try to help in problem selection. Ideally, the problem should be something you are very interested in. As such, it might be helpful to first tell me about your interests (maybe after class or in office hours), and we can try to think of something to work on. Selecting problems to work on is a difficult skill that takes years to develop, so it would be nice if you find a project idea on your own, but I expect everyone will need at least a little help in their choice. I know some things about some fields but I don't know everything about every field, so I might not be so helpful with certain projects outside my own background, but I can learn a bit myself to help you along if your interests are outside my knowledge.
II: Progress Report. By this second milestone, you should have some initial results to share; for example, you may have implemented and evaluated the performance of existing algorithms on your dataset and task of interest, or you may have conducted an initial study with simulated data to better understand the properties of certain methods, or you were able to prove some preliminary result about some question of interest, etc.
Your progress report deliverable is a write-up of no more than 2 pages (not including references) describing what you have accomplished so far and, briefly, what you intend to do in the remainder of the term. You should be able to reuse at least part of the text of this milestone in your final report.
III: In class presentation. You will present your work in class using your choice of (a combination of) the blackboard or a computer presentation. Paper handouts are optional. The lecture itself should be 15 minutes. Since the talk will be short, you should consider practicing (and timing) your talk in advance. You can practice part of it with me in office hours if you want. Expect around 2 minutes of questions from myself and your fellow students. Unlike other times in the class, attendance is mandatory during the presentations, and I will be taking attendance during them. Once I set the time limits they will be strict. I will indicate when you go over time. Going over time will result in penalties.
You will be graded on your presentation skills, e.g. voice volume, board usage, pacing of material, choice of material, etc. Minor technical problems will not be penalized, but major technical problems will be penalized. If you use a computer, make sure to have a backup plan in case of a technical problem to avoid such a penalty. If you want to give a short version of your presentation in office hours before the actual presentation, and then have me give feedback that might be a good idea.
IV: Final Report. Your final project report (not including acknowledgements and references) should be around 5-8 pages in length (using at most 12 point font and maximum 1 inch margins) and should follow a typical scientific style (with abstract, introduction, etc.). The write-up should clearly define your problem or question of interest, review relevant past work, and introduce and detail your approach. A comprehensive empirical evaluation could follow, or some proofs of some results, along with an interpretation of your results. Any elucidation of the theoretical properties of an empirical method under consideration is also welcome.
If this work was done in collaboration with someone outside of the class (e.g., a professor), please describe their contributions in an acknowledgements section.
The final report PDF file should be emailed. No hardcopy is needed.
Some Project Ideas: (to browse your own ideas, you could e.g. look through the proceedings of recent COLT conference such as COLT 2021, COLT 2020, etc.)
Streaming Algorithms and Online Learning
Adversarially Robust Learning
Robust Algorithmic Statistics
Embeddings and the "kernel trick"
Tentative Schedule: (This schedule may change slightly during the course.)
|1||Aug 23: Introduction||Aug 24||Aug 25: Introduction, Vertex Cover, SVD, PCA||Aug 26||Aug 27: k-means, power method|
|2||Aug 30: Learning linear threshold functions||Aug 31||Sep 1: Perceptron Algorithm||Sep 2||Sep 3: Perceptron Algorithm|
|3||Sep 6: No class||Sep 7||Sep 8: Embeddings and the "kernel trick"||Sep 9||Sep 10: Homework 1 due. Mercer's Theorem|
|4||Sep 13: Concentration of Measure, Hoeffding's Inequality||Sep 14||Sep 15: Gaussian Concentration, Johnson-Lindenstrauss||Sep 16||Sep 17: Bennett's Inequality|
|5||Sep 20: Empirical Risk Minimization||Sep 21||Sep 22: Gaussian Processes||Sep 23||Sep 24: Project Proposal (Abstract) due. Gaussian Processes|
|6||Sep 27: Sub-Gaussian Processes||Sep 28||Sep 29: General Empirical Processes||Sep 30||Oct 1: General Empirical Processes|
|7||Oct 4: Probably Approximately Correct (PAC) Learning||Oct 5||Oct 6: PAC Learning||Oct 7||Oct 8: Homework 2 due. Vapnis-Chernovenkis (VC) Theory|
|8||Oct 11: VC Theory||Oct 12||Oct 13: VC Theory||Oct 14||Oct 15: No class|
|9||Oct 18: VC Theory||Oct 19||Oct 20: Sparse recovery||Oct 21||Oct 22: high-dimensional convex geometry|
|10||Oct 25: Compressed sensing and LASSO||Oct 26||Oct 27: Low rank matrix recovery||Oct 28||Oct 29: Progress report due. Low rank matrix recovery|
|11||Nov 1: Boosting||Nov 2||Nov 3: Boosting||Nov 4||Nov 5: Homework 3 due. Deep Learning|
|12||Nov 8: Deep Learning||Nov 9||Nov 10: Deep Learning||Nov 11||Nov 12: Deep Learning|
|13||Nov 15: Review of Course||Nov 16||Nov 17: Leeway||Nov 18||Nov 19: Leeway|
|14||Nov 22: Leeway||Nov 23||Nov 24: No class||Nov 25||Nov 26: No class|
|15||Nov 29: Student Presentations||Nov 30||Dec 1: Student Presentations||Dec 2||Dec 3: Student Presentations [Dec 4: Final Report Due]|
Advice on succeeding in a math class:
Homework .tex files