General Information

Instructor: Rocco Servedio. Office: CSB 517. Office hours: TBA.
Instructional assistants: TBA.
Classroom: TBA
Time: Tues/Thurs 8:40-9:55.
Course website (this page): http://www.cs.columbia.edu/~cs4252/
Course email (use for all course-related issues): coms4252columbia2018 at gmail dot com.

Introduction

This course will give an introduction to some of the central topics in computational learning theory, a field which approaches 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)? Basic notions (learning models, concept classes).
• The online mistake-bound learning model. Online algorithms for simple learning problems (elimination, Perceptron, Winnow)
• General algorithms and lower bounds for online learning (halving algorithm, Weighted Majority algorithm, VC dimension)
• 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 sufficient background.
• 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 Thursdays by 8:40am (start of class); you must bring a hard copy to class and give it to the TA, and also turn in an electronic copy to the course account coms4252columbia2018. You are allowed a total of 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 cannot 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% of the initial grade per late day (with the same system in place; five minutes late for one problem counts as one late day for the entire set), thus a problem set which is four days late would receive 60% of the points that it would have received had it been turned in on time. No problem sets will be accepted after two weeks have elapsed from the initial due date. For any exceptions, 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.
• Midterm evaluation (10% of grade). Details TBA.
• In-class final evaluation (20% of grade). There will be an in-class evaluation on the last day of class, Thursday, December 6. It will be closed-book and closed-note, and will be cumulative -- anything that was covered this semester is fair game. The best way to prepare for the evaluation is to go over course notes, readings, and homework problems and solution sets.

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 must 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 any portion 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 Tuesday, your regrade request would have to be submitted before the start of class on the next Tuesday). 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. It's also available on reserve in the science and engineering library, and is electronically available through the Columbia library here (you will need to be signed in to access this). 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.

• There are countless online sources where you can find expositions of the Perceptron algorithm and proofs of convergence. The language tends to vary from presentation to presentation; one which hews fairly closely to the presentation we'll give in class can be found here.

A handy overview of Chernoff and Hoeffding bounds can be found here (thanks to Clement Canonne for preparing this).

For the unit on boosting, a good reference is Rob Schapire's survey paper "The Boosting approach to machine learning: An overview". The first three sections give a (very condensed) treatment of the AdaBoost algorithm we'll cover in class, and later sections have good overviews of more advanced material (this is a good starting point for a project on boosting). Of course, the original paper on Adaboost, "A decision-theoretic generalization of on-line learning and an application to boosting", is a good sources as well. Click here for a handy copy of the AdaBoost algorithm.

Advice

"Realizing the gravity of the situation, we immediately set about taking notes." -- Witold Gombrowicz, Philidor's Child Within

Problem Sets

Schedule of Topics

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

Link to Courseworks/Piazza

Schedule of Deadlines