General Information

Instructor: Rocco Servedio. Office: 517 CSB. Office hours: Fri 10-12.
Teaching assistants: Felix Sanchez Garcia (OH: Fairchild 607, Tue 2:00-4:00), Li-Yang Tan (OH: CSB 511, Thurs 1:30-3:30), Ilia Vovsha (OH: Mudd 122A, Mon 2:30-4:30)
Classroom: 535 Mudd
Time: Mon/Wed 1:10-2:25.
Course website (this page): http://www.cs.columbia.edu/~cs4252/
Course email (use for all course-related issues): coms4252columbia at gmail dot com.

Introduction

This course will give an introduction to some of the central topics in computational learning theory, a field which studies the above question from a theoretical computer science perspective. We will study well-defined mathematical and computational models of learning in which it is possible to give precise and rigorous analyses of learning problems and learning algorithms. A big focus of the course will be the computational efficiency of learning in these models. We'll develop computationally efficient algorithms for certain learning problems, and will see why efficient algorithms are not likely to exist for other problems.

List of Topics

• Introduction: what is computational learning theory (and why)?
• The online mistake-bound learning model. Online algorithms for simple concept classes.
• General algorithms for online learning. Lower bounds for online learning.
• The Probably Approximately Correct (PAC) learning model: definition and examples. Online to PAC conversions.
• Occam's Razor: learning by finding a consistent hypothesis.
• The VC dimension and uniform convergence.
• Weak versus strong learning: boosting algorithms.
• Computational hardness results for efficient learning based on cryptography.
• PAC learning from noisy data. Malicious noise and random classification noise. Learning from Statistical Queries.
• Exact learning from membership and equivalence queries. Learning monotone DNF and learning finite automata.

Prerequisites

You should be familiar with the following topics:

• Important: Big-O notation and basic analysis of algorithms. Chapter 3 in "Introduction to Algorithms" by Cormen, Leicerson, Rivest and Stein is more than sufficient.
• Important: Basic discrete math and probability. The course Computer Science 3203 and the "Mathematical Background" section (Appendix VIII) of CLRS are good sources here.
• Important: You should be comfortable with reading and writing proofs. You will encounter many proofs in class and will be doing proofs on problem sets. Chapter 0 of Sipser's book "Introduction to the Theory of Computation" is a good source here.
• Less important but helpful: Some basic knowledge of the theory of NP-completeness (e.g. Chapter 7 of Sipser or Chapter 34 of CLRS) and of elementary cryptography (e.g. pages 881-887 of CLRS) would be helpful, but this knowledge is not required -- the course will be self-contained with respect to these topics.

Grading and Requirements

• Biweekly problem sets (70% of grade). Your solutions must be typed and prepared as LaTeX documents (instructions will be given on how to use LaTeX.) The biweekly problem sets will be due on Wednesdays by 1:10pm (start of class); you must bring a hard copy to class and give it to the TA. You are allowed six late days for the semester. Each late day is exactly 24 hours; late days cannot be subdivided -- five minutes late counts as one late day. Note also that problem sets cannot be subdivided with respect to late days (i.e. you can't use 1/6 of a late day on a 6-problem problem set by handing in one problem one day late.) Late days over and beyond the allotted six late days will be penalized at a rate of 10% per late day (with the same system in place; five minutes late counts as one late day).   For an exception, you must have your undergraduate advisor (for undergrads) or your graduate advisor (for graduate students) contact me. Some problems will be challenging; you are advised to start the problem sets early.  Late problem sets should be emailed to the TAs.
• Final project (30% of grade). You may do an experimental project or a theoretical one. An experimental project might involve learning about some specific algorithm and application area and doing an implementation with experiments. A theoretical project might involve reading research papers to get a more in-depth understanding of some topic we touched on in class, or some other CLT topic that you're interested in. For either type of project you will submit a written report clearly explaining what you learned. Final Projects will be due in December (exact date TBA, but it will be around the last day of classes).
• Classroom Participation. Students are expected to come to class and to be prepared to participate. In cases where a student's grade is on the borderline between two grades, coming to class and making a contribution through thoughtgul participation can make the difference.

Homework policy: You are encouraged to discuss the course material and the homework problems with each other in small groups (2-3 people), but you must list all discussion partners on your problem set. Discussion of homework problems may include brainstorming and verbally discussing possible solution approaches, but should not go as far as one person telling the others how to solve the problem. In addition, each person must write up her/his solutions independently; you may not look at another student's written solutions. You may consult outside materials, but all materials used must be appropriately acknowledged, and you must always write up your solutions in your own words. Violation of this policy will result in a penalty to be assessed at the instructor's discretion. This may include receiving a zero grade for the assignment in question AND a failing grade for the whole course, even for the first infraction.

Students are expected to adhere to the Academic Honesty policy of the Computer Science Department; this policy can be found in full here. Please contact the instructor with any questions.

Re-grade policy: If you dispute the grade received for an assignment, you must submit, in writing, a detailed and clearly stated argument for what you believe is incorrect and why. This must be submitted no later than the beginning of class 1 week after the assignment was returned. (For example, if the assignment were returned to the class on Wednesday, your regrade request would have to be submitted before the start of class on the next Wednesday). Requests for a re-grade after this time will not be accepted. A written response will be provided within one week indicating your final score. Requests of re-grade of a specific problem may result in a regrade of the entire assignment. This re-grade and written response is final. Keep in mind that a re-grade request may result in the overall score for an assignment being lowered.

Readings

"I therefore perused this Sheet with wonderful Application, and in about a Day's time discovered that I could not understand it." -- Henry Fielding, A Journey from This World to the Next

The textbook for this course is:

• M. Kearns and U. Vazirani.  An Introduction to Computational Learning Theory. 

This book is available for purchase on-line and at the Columbia Bookstore; it's also available for reserve in the engineering library, but you will likely want to have your own copy. Note that several topics which we'll cover (particularly early in the semester) are not in the book. A good source for material which we will cover in the first few weeks of class are the following two papers:

• Avrim Blum's survey paper "Online Algorithms in Machine Learning" This has an overview of the online mistake-bound learning model, the Weighted Majority algorithm, the online decision list learning algorithm, and much more.

• Nick Littlestone's original paper introducing the Winnow algorithm: "Learning Quickly When Irrelevant Attributes Abound: A New Linear-threshold Algorithm". This has lots of details on the Winnow algorithm.

Problem Sets

Final Projects

Schedule of Topics

• Lecture 1 Introduction to machine learning theory. Learning models and learning problems.
• Lecture 2 Online mistake bound learning. Algorithms for simple concept classes. Attribute efficient learning with the Winnow algorithm.
• Lecture 3 Winnow algorithm continued. Perceptron algorithm. General bounds for online mistake bound learning. The Halving algorithm, the Standard Optimal Algorithm.
• Lecture 4 Best Experts and Weighted Majority.
• Lecture 5 Weighted Majority algorithm continued.
• Lecture 6 The Probably Approximately Correct(PAC) Learning model. PAC learning algorithms for simple concept classes.
• Lecture 7 More PAC learning algorithms. Conversions from online learning to PAC learning. (KV chapter 1)
• Lecture 8 Occam's Razor: learning by finding consistent hypotheses. Applications (KV chapter 2).
• Lecture 9 Computational hardness results for finding consistent hypotheses.
• Lecture 10 Vapnik-Chervonenkis dimension. Upper and lower bounds on sample complexity (KV chapter 3).
• Lecture 11 VC dimension and sample complexity continued.
• Lecture 12 VC dimension and sample complexity continued.
• Lecture 13 Weak learning, strong learning, and Boosting (KV chapter 4).
• Lecture 14 Boosting continued.
• Lecture 15 Boosting continued.
• Lecture 16 Cryptographic limitations on efficient learning (KV chapter 6).
• Lecture 17 Cryptographic limitations on learning continued.
• Lecture 18 Cryptographic limitations and reductions in PAC learning (KV chapters 6,7).
• Lecture 19 Learning in the presence of noise. Classification noise, malicious noice,
• Lecture 20 Statistical Query learning (KV chapter 5).
• Lecture 21 Learning with noise continued.
• Lecture 22 Learning with noise continued.
• Lecture 23 The model of exact learning from membership and equivalence queries. Learning monotone DNF formulas.
• Lecture 24 Learning deterministic finite automata.
• Lecture 25 Learning deterministic finite automata continued.
• Lecture 26 Learning deterministic finite automata continued.

Lecture Notes