Prof. Michael Kearns

Dr. Jake Abernethy

Mondays 12-3 PM
307 Levine Hall (NOTE ROOM CHANGE)

IMPORTANT NOTE: Since the first Monday of the term (Jan 16) is MLK Day, the first class meeting will be on Monday, Jan 23.

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A previous incarnation of this course:


This course is an introduction to Computational Learning Theory, a field which attempts to provide algorithmic, complexity-theoretic and probabilistic foundations to modern machine learning and related topics.

The first part of the course will closely follow portions of An Introduction to Computational Learning Theory, by M. Kearns and U. Vazirani (MIT Press). We will cover perhaps 6 or 7 of the chapters in K&V over (approximately) the first half of the course, often supplementing with additional readings and materials. Copies of K&V will be available at the Penn bookstore. The second portion of the course will focus on a number of models and topics in learning theory not covered in K&V.

The course will give a broad overview of the kinds of problems and techniques typically studied in Computational Learning Theory, and provide a basic arsenal of powerful mathematical tools for analyzing machine learning problems.

Topics likely to be covered include:

  • Basics of the Probably Approximately Correct (PAC) Learning Model
  • Occam's Razor, Compression and Learning
  • Uniform Convergence and the Vapnik-Chervonenkis Dimension
  • Learning in the Presence of Noise and Statistical Query Learning
  • Learning and Cryptography
  • Query Models
  • Boosting
  • Online and No-Regret Learning
  • Discrete Fourier Methods in Learning
  • Active Learning
  • Agnostic Learning
  • Learning Theory and Algorithmic Mechanism Design
  • Learning Theory and Differential Privacy
  • Learning Theory and Computational Models of Evolution
  • PAC-style Analyses in Reinforcement Learning
  • PAC-style Analyses in Probabilistic Inference


    Much of the course will be in fairly traditional "chalk talk" lecture format, but with ample opportunity for discussion, participation, and critique. The course will meet once a week on Mondays from 12 to 3. Lunch will be served.

    While there are no specific formal prerequisites, background or courses in algorithms, complexity theory, discrete math, combinatorics, probability theory and statistics will prove helpful, as will "mathematical maturity" in general. We will be examining detailed proofs throughout the course. If you have questions about the desired background, please ask me. Auditors and occasional participants are welcome.

    The course requirements for registered students are active in-class participation, a couple of problem sets, possibly co-leading a class discussion, and a final project. The final projects can range from actual research work, to a literature survey, to solving some additional problems.

    The exact timing and set of topics below will depend on our progress and will be updated as we proceed. "[MK]" indicates lectures by Prof. Kearns; "[JA]" by Dr. Abernethy.

    Mon Jan 23
    [MK] In the first meeting, we will go over course mechanics and present a course overview, then immediately begin investigating the Probably Approximately Correct (PAC) model of learning. Detailed topics covered: Learning rectangles in the real plane; definition of the PAC model; PAC-learnability of rectangles in d dimensions; PAC-learnability of conjunctions of boolean variables.

    READING: K&V Chapter 1.

    Mon Jan 30
    [MK] Hardness of PAC-learning 3-term DNF; PAC-learnability of 3-term DNF by 3CNF; relaxing the PAC definition; consistency and PAC learning; Occam's Razor; VC dimension definitions and examples; a polynomial bound on the growth of \Pi_C(m) (Sauer's Lemma).

    READING: K&V Chapters 2 and 3.

    PROBLEM SET #1, DUE AS HARDCOPY IN CLASS FEB 13: Solve the following problems from K&V: 1.1, 1.3, 2.3, 2.4, 3.1, 3.5. You are free to collaborate in groups, but please turn in separate write-ups and clearly acknowledge your collaborations. Please do not attempt to look up solutions in the literature.

    Mon Feb 6
    [JA] Alternate proof of Sauer's Lemma; lower bounds on sample complexity via the VC dimension; beyond VC dimension --- Rademacher complexity, scale-sensitive dimension and friends.

    Mon Feb 13
    [MK] Completion of VC upper bound; introduction to boosting; confidence boosting; a simple majority vote single-stage algorithm for accuracy amplification.

    READING: K&V Chapter 4 and the following papers.

  • The Boosting Apporach to Machine Learning: An Overview. Rob Schapire.
  • A Decision-Theoretic Generalization of On-Line Learning and an Application to Boosting. Yoav Freund and Rob Schapire.
  • Experiments with a New Boosting Algorithm. Yoav Freund and Rob Schapire.

    Mon Feb 20
    [MK] Boosting continued: Adaboost and analysis; introduction to learning with classification noise; a statistics-only algorithm for learning conjunctions; introduction to the statistical query learning model.

    READING: K&V Chapter 5; supplementary reading:

  • Efficient Noise-Tolerant Learning From Statistical Queries. Kearns.

  • Weakly Learning DNF and Characterizing Statistical Query Learning Using Fourier Analysis. Blum, Furst, Jackson, Kearns, Mansour, Rudich.

    Mon Feb 27
    [MK] Classification noise and statistical query learning continued: noise-tolerance of all SQ algorithms; almost all PAC algorithms are in SQ; separating PAC and SQ; the SQ dimension and relationship to VC dimension.

    Mon Mar 5
    Spring break; no class meeting.

    Mon Mar 12
    [MK] Learning and cryptography; representation-independence hardness results; learning DFAs from examples and membership queries.

    READING: K&V Chapters 6,7,8.

    PROBLEM SET #2, DUE AS HARDCOPY IN CLASS MAR 26: Solve the following problems from K&V: 5.1, 5.3, 5.4, 7.3, 8.3. You are free to collaborate on the homework assignments, but please turn in separate write-ups and acknowledge your collaborations. Please do not attempt to look up solutions in the literature.

    For those of you registered for credit in the course, your final assignment will be a course project, for which you have three choices:

  • Research Paper. For a research paper, you should either choose an open problem related to the models we have examined in the class and write up your attempts to solve it; or you should write a paper introducing a new model of learning that extends or modifies those we examined in class, and ideally motivate and illustrate your model with some initial results. For those going the "open problem" route, you must both choose your own problem (don't ask me for problems; that's part of doing research), and do enough of a literature search to make sure it hasn't been solved already. Ditto for finding and researching your own new model.
  • Final Problem Set. Here you can choose and solve any 8 problems from K&V that were not already assigned in the homeworks. Try to pick a mix of difficulty levels.
  • Literature Survey/Synthesis. Here you may pick a few related papers from the learning theory literature, and do a write-up surveying their models and results, and relating the papers to each other. Try not to simply regurgitate what's in the papers themselves.

    You are welcome to send email to Prof Kearns proposing what your final project will be in order to get feedback and advice.

    The deadline for final projects will be May 8, via email to Prof Kearns.

    Mon Mar 19
    [JA] Adversarial online learning; halving, weighted majority, and perceptron algorithms; game, minimax via weighted majority, and boosting.

    READING: While the notation and details may differ in class, the following online notes are relevant.

  • Sasha Rahklin notes
  • Rob Schapire notes

    Mon Mar 26
    [MK] PAC analysis of reinforcement learning; efficient exploration and exploitation.


  • "Near-Optimal Reinforcement Learning in Polynomial Time, MK and S. Singh.

    Mon Apr 2
    [JA] Continued: No-regret learning, online convex optimziation, and related topics.


  • Jake's Notes on Weighted Majority, etc.
  • Chapter on No-Regret Learning , Blum and Mansour.

    at least part of class on April 16, be devoted to discussion of your course projects --- Jake and I will have brief individual meetings with each person/team about your proposed project. In preparation for this, you should all send both Jake and I an email sketching your proposed project by FRIDAY APRIL 13. The purpose of the email is not that you have your project entirely figured out, but to get you thinking about what you want to do. The more specific, the better, but it's also fine to say you want to think about papers or problems in some particular area that interests you. That way in the meetings Jake and I can suggest directions, or papers to look at, etc. Please make the subject line of your email "CIS 625 Project" so we can easily sort.

    It is permitted for you to work in small teams on a joint project, but we'll expect a proportionally more ambitious project in such cases.

    Mon Apr 9
    [JA] More fun with no-reget, online learning, convex optimization, etc.

    READING: Papers from last week's discussion:

  • Tracking the Best Expert Herbster and Warmuth 1998

  • A Decision-Theoretic Generalization of On-Line Learning and an Application to Boosting Freund and Schapire 1997

    For this lecture, students should try to read through page 9 of Elad Hazan's review article on online convex optimization:

  • The convex optimization approach to regret minimization Hazan 2009; please read through page 9 of this in preparation for class

    For students that want to know more, and for possible project ideas, here are some links:

  • Exponentiated Gradient versus Gradient Descent for Linear Predictors Kivinen and Warmuth 1997

  • Online Convex Programming and Generalized In finitesimal Gradient Ascent Zinkevich 2003

  • Lecture Notes on Online Learning Rakhlin 2009

  • Universal Portfolios With and Without Transaction Costs Blum and Kalai 1999

    Mon Apr 16
    [JA] Jake finishes up his lectures on online convex optimization.


    Mon Apr 23
    [MK] Some drawbacks of no-regret learning; some topics in ML and finance.


  • Risk-Sensitive Online Learning Even-Dar, K. and Wortman

  • Regret to the Best vs. Regret to the Average Even-Dar, K., Mansour and Wortman

  • Censored Exploration and the Dark Pool Problem Ganchev, K., Nevmyvaka and Wortman

  • Optimal Allocation Strategies for the Dark Pool Problem Agarwal, Bartlett and Dama