CS 294-92: Analysis of Boolean Functions (Spring 2020)

Boolean functions are central objects of study in theoretical computer science and combinatorics. Analysis of Boolean functions, and in particular Fourier analysis, has been a successful tool in the areas of circuit lower bounds, hardness of approximation, social choice, threshold phenomena, pseudo-randomness, property testing, learning theory, cryptography, quantum computing, query complexity, and others. 

These applications are derived by understanding fundamental beautiful concepts in the study of Boolean functions, such as influence, noise-sensitivity, approximation by polynomials, hyper-contractivity, and the invariance-principle (connecting the discrete Boolean domain with the continuous Gaussian domain).

We will study these foundational concepts of Boolean function as well as their applications to diverse areas in TCS and combinatorics.

Textbook:
The course will be mainly based on the wonderful book by Ryan O'Donnell. The book is available for free download via this link, or available for purchase on Amazon.
In addition, we will highlight some recent exciting results that are not covered in the book.

General Information: 

Time and Place: Tuesday, Thursday 5:00-6:30 PM -- 310 Soda Hall

Instructor: Avishay Tal, Soda 635, atal "at" berkeley.edu
Office Hours: Monday 1:30-3:30 PM
TA: Orr Paradise, Soda 626, orrp "at" eecs.berkeley.edu
Office Hours: Wednesday 2:30-3:30 PM (or fix an appointment by email). Please send questions/topics in advance.

Grading: Homeworks - 40% (4 assignments), Participation in class - 10%, Final Project & Presentation - 50%. Scribe - an exempt from one problem set and 10%.

Problem Sets:
Discussions: Piazza

HW submissions: Gradescope

Topics (not necessarily in that order):

Property Testing:
- Linearity Testing [Blum-Luby-Rubinfeld]

Influence and Hypercontractivity:
- Arrow's Theorem (Kalai's Proof)
- Bonami-Beckner Theorem
- The Influence of Variables on Boolean Functions [Kahn-Kalai-Linial]
- Friedgut's Junta Theorem

Learning Theory:
- The Goldreich-Levin Algorithm (stemming from cryptography)
- Learning Shallow Circuits [Linial-Mansour-Nisan]
- Learning DNFs and Mansour's conjecture
- Learning Juntas [Mossel-O'Donnell-Servedio]

Circuit Complexity:
- Random Restrictions
- Parity not in AC0: the Switching Lemma  [Håstad]
- Shrinkage of De Morgan Formulae  [Håstad]

Pseudo-randomness:
- Small-biased distributions  [Naor-Naor]
- Pseudo-randomness for low-degree functions  [Bogdanov-Viola, Lovett, Viola].
- Pseudo-randomness based on random restrictions  [Ajtai-Wigderson, Gopalan-Meka-Reingold-Trevisan-Vadhan, Steinke-Reingold-Vadhan]
- Fractional pseudo-random generators and polarizing random walks  [Chattopadhyay-Hatami-Hosseini-Lovett]

Hardness-of-Approximation:
- The unique-games conjecture and its implications.

The Invariance Principle:
- Central limit theorems [Berry–Esseen]
- "Majority is Stablest" [Mossel-O'Donnell-Oleszkiewicz]

Query Complexity:
- The decision tree model, and its randomized, quantum, and non-deterministic analogs
- The sensitivity theorem [Huang]
- The polynomial method for quantum lower bounds [Beals-Buhrman-Cleve-Mosca-de Wolf]

Lectures:
For each lecture - see the relevant chapters in O'Donnell's book.
  1. Jan 21 - The Fourier expansion, orthogonality of characters -  Chapters 1.1-1.4
  2. Jan 23 - BLR linearity testing -  Chapters 1.5-1.6 
  3. Jan 28 - Social Choice, Influences, Effects, Derivatives -  Chapters 2.1, 2.2 
  4. Jan 30 - Influences, Effects, Isoperimetric Inequalities -  Chapters 2.2, 2.3
  5. Feb 4 - Noise Stability, Arrow's Theorem -  Chapters 2.4, 2.5 
  6. Feb 6 - Spectral Concentration and Low-Degree Learning -  Chapters 3.1-3.4
  7. Feb 11 - Goldreich-Levin Algorithm -  Chapter 3.5
  8. Feb 13 - Learning Juntas - [Mossel-O'Donnell-Servedio'04], [Valiant'12]
  9. Feb 18 - DNFs, Random Restrictions - Chapters 3.3, 4.1, 4.3
  10. Feb 20 - Fourier Concentration of DNFs - Chapter 4.4
  11. Feb 25 - Proof of the Switching Lemma -  Thapen's Notes
  12. Feb 27 - AC0 Circuits + Intro to Pseudorandomness - Chapter 4.5 + Chapters 6.1
  13. Mar 3 - Pseudorandomness - small-biased distributions - Chapters 6.3 + 6.4
  14. Mar 5 - Pseudorandomness - fooling low-degree polynomials, Fractional PRGs - Chapter 6.5 + CHHL'18 
  15. Mar 10 - LTFs and Central Limit Theorems - Chapters 5.1, 5.2
  16. Mar 12 - Noise Stability of the Majority Function and LTFs - Chapters 5.2, 5.5
  17. Mar 17 - Hypercontractivity - Bonami's Lemma - Chapter 9
  18. Mar 19 - Hypercontractivity - KKL Thm, Friedgut's Junta Lemma - Chapter 9, continued

Lecture notes: currently for lectures 3-5. 

Google Drive folder for other lecture notes.

Resources and Other Courses:


Ryan O'Donnell - AOBF - Mini-Course (2012) Scribe Notes by Li-Yang Tan


Videos from Real Analysis in Computer Science `Boot Camp' at the Simons Institute (2013):
Inapproximability of Constraint Satisfaction Problems - Johan Håstad - 5 talks
Analytic Methods for Supervised Learning​ - Adam Klivans - 4 talks
Introduction to Analysis on the Discrete Cube - Krzysztof Oleszkiewicz - 4 talks