CS 294-92: Analysis of Boolean Functions (Spring 2023)

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.

Undergraduate students who wish to take this class in Spring 2023 should fill out the following Google Form.

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:

Semester: Spring 2023

Time and Place: Tuesday, Thursday 12:30-2:00 PM -- 306 Soda Hall (lecture will not be recorded)

Instructor: Avishay Tal, Soda 635, atal "at" berkeley.edu

Office Hours: Thursday 2-3 PM (start at Berkeley time) - 306 Soda Hall (or fix an appointment by email).

TA: Xin Lyu, Soda 634, xinlyu "at" berkeley.edu

Office Hours: TBD.

Grading: Homework - 40% (4 assignments), Lecture Scribe - 10%, Final Project & Presentation - 50%.

Problem Sets:

  • Homework 1 - Due Monday, Feb 6.

  • Homework 2

  • Homework 3

  • Homework 4

Discussions: Ed

HW submissions: Gradescope

Lectures Schedule:

For each lecture - see the relevant chapters in O'Donnell's book + additional resources + lecture notes.

  1. Jan 17 - The Fourier expansion, orthogonality of characters - Chapters 1.1-1.4

  2. Jan 19 - BLR linearity testing - Chapters 1.5-1.6

  3. Jan 24 - Social Choice, Influences, Effects, Derivatives - Chapters 2.1, 2.2

  4. Jan 26 - Influences, Effects, Isoperimetric Inequalities - Chapters 2.2, 2.3

  5. Jan 31 - Noise Stability, Arrow's Theorem - Chapters 2.4, 2.5

  6. Feb 2 - Spectral Concentration and Low-Degree Learning - Chapters 3.1-3.4

  7. Feb 7 - Goldreich-Levin Algorithm - Chapter 3.5

  8. Feb 9 - Learning Juntas - [Mossel-O'Donnell-Servedio'04], [Valiant'12]

  9. Feb 14 - DNFs, Random Restrictions - Chapters 3.3, 4.1, 4.3

  10. Feb 16 - Fourier Concentration of DNFs - Chapter 4.4

  11. Feb 21 - Proof of the Switching Lemma - Thapen's Notes

  12. Feb 23 - AC0 Circuits + Intro to Pseudorandomness - Chapter 4.5 + Chapters 6.1

  13. Feb 28 - Pseudorandomness - small-biased distributions - Chapters 6.3 + 6.4

  14. Mar 2 - Pseudorandomness - fooling low-degree polynomials, Fractional PRGs - Chapter 6.5 + CHHL'18

  15. Mar 7 - LTFs and Central Limit Theorems - Chapters 5.1, 5.2

  16. Mar 9 - Noise Stability of the Majority Function and LTFs - Chapters 5.2, 5.5

  17. Mar 14 - Hypercontractivity - Bonami's Lemma - Chapter 9

  18. Mar 16 - Hypercontractivity - KKL Thm, Friedgut's Junta Lemma - Chapter 9

  19. Mar 21 - Dictator Testing & PCPPs - Chapter 7

  20. Mar 23- Hardness of Approximation from UGC - Chapter 7

Spring Break

  1. Apr 4 - The Invariance Principle - Chapter 11

  2. Apr 6 - Majority is Stablest & Hardness of Max-CUT - Chapter 11

  3. Apr 11 - Query Complexity - [Buhrman, de Wolf'00]

  4. Apr 13 - The Sensitivity Theorem - [Huang'19]

  5. Apr 18 - Extremal Combinatorics, The Sunflower Lemma - [Alweiss-Lovett-Wu-Zhang'19]

  6. Apr 20 - Threshold Phenomena, Proof of the Kahn-Kalai Conjecture - [Park-Pham'22]

  7. Apr 28 - Presentations # 1

  8. Apr 30 - Presentations # 2

RRR Week: More Presentations

Resources and Other Courses:

Analysis and TCS: new frontiers -- Simons Institute program -- Summer 2023
Open Problems by Ryan O'Donnell (2012)
Open Problems - Simons Institute program on Real Analysis in Computer Science (2014)
Theory of Computing - Special Issue on Analysis of Boolean Functions

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

Videos about Fourier concentration and random restriction-based PRGs:

PRGs for Small Space via Fourier Analysis

Pseudorandom Generators from Polarizing Random Walks

Better Pseudorandom Generators from Milder Pseudorandom Restrictions

Fourier tails for Boolean functions and their applications

Other Courses:

Dor Minzer - MIT (2021)
Yuval Filmus - Technion (2021)
Li-Yang Tan - Stanford (2018)
Shachar Lovett - UCSD (2017)
Hamad Hatami - McGill (2014)
Oded Regev - NYU (2012)
Ryan O'Donnell - CMU (2012)
Guy Kindler - Weizmann (2008)
Ryan O'Donnell - CMU (2007)
Irit Dinur and Ehud Friedgut - HUJI (2005)
Ryan O'Donnell - AOBF - Mini-Course (2012) Scribe Notes by Li-Yang Tan