CS 294-92: Analysis of Boolean Functions (Spring 2025)

Description:

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 and their applications to diverse areas in TCS and combinatorics.

Undergraduate students who wish to take this class in Spring 2025 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 2025

Time and Place: Tuesday, Thursday 11:00A-12:30P -- 310 Soda Hall

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

Office Hours: Tuesday 1:30-2:30 pm @ 635 Soda.

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

Problem Sets:

HW1 -- Due Monday, Feb 10.
HW2 - Due Monday, Feb 24.
HW3 - Due Monday, Mar 17.
HW4 - Due Monday, Apr 7.

Discussions on Ed

HW submissions on Gradescope

Lectures Schedule (tentative):

For each lecture - please take a look at the relevant chapters in O'Donnell's book & additional resources & lecture notes.

Jan 21 - The Fourier expansion, orthogonality of characters - Chapters 1.1-1.4 -- notes (Joyce Lu)
Jan 23 - BLR linearity testing - Chapters 1.5-1.6 -- notes (Austin Pechan)
Jan 28 - Social Choice, Influences, Effects, Derivatives - Chapters 2.1, 2.2 -- notes (Vivien Goyal)
Jan 30 - Influences, Effects- Chapters 2.2, 2.3 -- notes (Patrick Bales)
Feb 4- Isoperimetric Inequalities, Noise Stability, Arrow's Theorem - Chapters 2.4, 2.5 -- notes (Prastik Mohanraj)
Feb 6 - Spectral Concentration and Low-Degree Learning - Chapters 3.1-3.4
Feb 11 - Goldreich-Levin Algorithm - Chapter 3.5 -- notes (Thomas Culhane)
Feb 13 - Learning Juntas - [Mossel-O'Donnell-Servedio'04], [Valiant'12] -- notes (Timothe Kasriel)
Feb 18 - DNFs, Random Restrictions - Chapters 3.3, 4.1, 4.3 -- notes (Qinggao Hong)
Feb 20 - Fourier Concentration of DNFs - Chapter 4.4 -- notes (Lucas Ericsson)
Feb 25 - Proof of the Switching Lemma - Thapen's Notes
Feb 27 - Finish Mansour's Theorem + AC0 Circuits - Chapter 4.4, 4.5
Mar 4 - Pseudorandomness - small-biased distributions - Chapters 6.1, 6.3, 6.4
Mar 6 - Pseudorandomness - fooling low-degree polynomials, Fractional PRGs - Chapter 6.5 + CHHL'18
Mar 11 - LTFs and Central Limit Theorems - Chapters 5.1, 5.2.
Mar 13 - Noise Stability of the Majority Function and LTFs - Chapters 5.2, 5.5
Mar 18 - Hypercontractivity - Bonami's Lemma - Chapter 9
Mar 20 - Hypercontractivity - KKL Thm, Friedgut's Junta Lemma - Chapter 9

-- Spring Break

Apr 1 - Dictator Testing & PCPPs - Chapter 7
Apr 3- Hardness of Approximation from UGC - Chapter 7
Apr 8 - The Invariance Principle - Chapter 11
Apr 10 - Majority is Stablest & Hardness of Max-CUT - Chapter 11
Apr 15 - Query Complexity - [Buhrman, de Wolf'00]
Apr 17 - The Sensitivity Theorem - [Huang'19]
Apr 22 - Extremal Combinatorics, The Sunflower Lemma - [Alweiss-Lovett-Wu-Zhang'19] [Rao'19]
Apr 24 - Threshold Phenomena, Proof of the Kahn-Kalai Conjecture - [Park-Pham'22] [Frankston-Kahn-Narayanan-Park'19]
Apr 29 - Presentations # 1
May 1 - Presentations # 2