About

I am a PhD student in the theory group at UCSD where I am thankful to be advised by Daniel Kane and Shachar Lovett. I did my undergraduate BA in Mathematics at Harvard University with a minor in Computer Science. In my time there, I was lucky enough to work under Michael Mitzenmacher and Madhu Sudan. My research is generously supported by NSF GRFP, ARCS, and JSOE fellowships.

I am grateful to have spent '22-'23 as a visiting student at the Weizmann institute with Irit Dinur, and will spend '24-'27 as a postdoc at Princeton and the Institute for Advanced Study with Pravesh Kothari and Avi Wigderson respectively.

If you are looking for UCSD Theory Lunch, it has been passed to the next generation.

I like singing, boardgames, squash, and sushi (in no particular order).

Interests

I am broadly interested in areas such as boolean function analysis, hardness of approximation, learning theory, privacy, and computational geometry. Currently, I mostly think about:

1. High Dimensional Expanders

High Dimensional Expansion (HDX) is an emerging area with applications across topics such as sampling, error correction, approximation algorithms, and more. I study analysis of boolean functions and random walks on HDX, especially in the context of hardness of approximation. In Spring 2021, Shachar Lovett and I co-taught an introductory course on expanders and HDX. Detailed course notes can be found here, and corresponding lectures here.

2. Learning beyond Uniform Convergence

Many classical results in learning theory rely on the idea of uniform convergence, that empirical risk should converge to true risk across all hypotheses simultaneously, but the paradigm fails to capture rates and other phenomena observed in practice. I study statistical learning beyond uniform convergence, including basic settings such as learning with data-dependent assumptions which remain largely enigmatic at a theoretical level.

3. Active Learning through Enriched Queries

When does adaptivity help us learn? In standard models, it is well known that adaptivity often provides no asymptotic benefit. I study how asking better questions allows us to build adaptive algorithms that drastically speed up learning. If you're interested in this area, check out this blog post for a basic primer.

Papers

Preprints

• Replicability in High Dimensional Statistics
  Max Hopkins, Russell Impagliazzo, Daniel Kane, Sihan Liu, Chris Ye
  Coming soon!

• Chernoff Bounds and Reverse Hypercontractivity on HDX
  Yotam Dikstein, Max Hopkins
  

• Do PAC-Learners Learn the Marginal Distribution?
  Max Hopkins, Daniel Kane, Shachar Lovett, Gaurav Mahajan
  

Publications

• Stability is Stable: Connections between Replicability, Privacy, and Adaptive Generalization
  Mark Bun, Marco Gaboardi, Max Hopkins, Russell Impagliazzo, Rex Lei, Toniann Pitassi, Satchit Sivakumar, Jess Sorrell
  STOC 2023

• Robust Empirical Risk Minimization with Tolerance
  Robi Bhattacharjee, Max Hopkins, Akash Kumar, Hantao Yu, Kamalika Chaudhuri
  ALT 2023

• Sampling Equilibria: Fast No-Regret Learning in Structured Games
  Daniel Beaglehole, Max Hopkins, Daniel Kane, Sihan Liu, Shachar Lovett
  SODA 2023

• Active Learning Polynomial Threshold Functions
  Omri Ben-Eliezer, Max Hopkins, Chutong Yang, Hantao Yu
  NeurIPS 2022

• Explicit Lower Bounds Against Ω(n)-Rounds of Sum-of-Squares
  Max Hopkins, Ting-Chun Lin
  FOCS 2022

• Eigenstripping, Spectral Decay, and Edge-Expansion on Posets
  Jason Gaitonde, Max Hopkins, Tali Kaufman, Shachar Lovett, Ruizhe Zhang
  RANDOM 2022 (Talk)

• Realizable Learning is All You Need
  Max Hopkins, Daniel Kane, Shachar Lovett, Gaurav Mahajan
  COLT 2022 (Long Talk, Short Talk)

• Hypercontractivity on High Dimensional Expanders: A Local-to-Global Method for Higher Moments
  Mitali Bafna, Max Hopkins, Tali Kaufman, Shachar Lovett
  STOC 2022 (Talk)

• High Dimensional Expanders: Eigenstripping, Pseudorandomness, and Unique Games
  Mitali Bafna, Max Hopkins, Tali Kaufman, Shachar Lovett
  SODA 2022

• Active Learning with Bounded Memory through Enriched Queries
  Max Hopkins, Daniel Kane, Shachar Lovett, Michal Moshkovitz
  COLT 2021 (Talk)

• The Power of Comparisons for Actively Learning Linear Separators
  Max Hopkins, Daniel Kane, Shachar Lovett
  NeurIPS 2020

• Point Location and Active Learning: Learning Halfspaces Almost Optimally
  Max Hopkins, Daniel Kane, Shachar Lovett, Gaurav Mahajan
  FOCS 2020 (Long Talk, Short Talk)

• Noise Tolerant, Reliable Active Classification with Comparison Queries
  Max Hopkins, Daniel Kane, Shachar Lovett, Gaurav Mahajan
  COLT 2020 (Talk)

• A Novel CMB Component Separation Method: Hierarchical Generalized Morphological Component Analysis
  Sebastian Wagner-Carena, Max Hopkins, Ana Rivero, Cora Dvorkin
  MNRAS, May 2020
  Code

• Doppelgangers: the Ur-Operation and Posets of Bounded Height
  Thomas Browning, Max Hopkins, Zander Kelley
  FPSAC 2018 (Extended Abstract)

• Simulated Annealing for JPEG Quantization
  Max Hopkins, Michael Mitzenmacher, Sebastian Wagner-Carena
  DCC 2018 (poster)
  Code

Blog Posts

• How to Actively Learn in Bounded Memory
  A basic exposition of new work towards characterizing bounded memory active learning with a few concrete examples.
  

• The Power of Comparison in Active Learning
  An introduction to active learning with comparison queries covering seminal work of KLMZ and recent extensions.
  

Miscellaneous Writing

• Notes on Expanders and High Dimensional Expansion
  Lecture notes from my course with Shachar Lovett on Expanders and HDX
  

• A Few Tips for Getting into CS Theory Research as an Undergraduate
  A couple slides with tricks for entering theory research and links to Summer program opportunities
  

• Representation-Theoretic Techniques for Independence Bounds of Cayley Graphs
  My undergraduate thesis under Madhu Sudan
  

• Bordism Homology
  A survey of un-oriented bordism homology

Talks

• Ben-Gurion University (March 27, April 16)
  

• On High Dimensional Expanders and Hardness of Approximation (IISC Bangalore)
  

• Hypercontractivity on High Dimensional Expanders (STOC '22, TTIC, UMich Theory Seminar, MIT A&C Seminar, Weizmann Guest Lecture)
  

• Realizable Learning is All You Need (COLT '22, UWaterloo, GaTech Student Seminars, Stanford Theory Lunch)
  

• High Dimensional Expanders and Unique Games (SODA '22, UW Seminar, SoS+TCS Seminar, Cornell Theory Tea, Chicago/TTIC (Reading Group))
  

• Active Learning in Bounded Memory (COLT '21)
  

• Point Location and Active Learning (FOCS '20, Harvard CMSA)
  

• Noise Tolerant Active Learning with Comparisons (COLT '20)
  

• On the Equivalence of Realizable and Agnostic Learning
  

• Active Learning with Bounded Memory
  

• Point Location and Active Learning
  

• An Open Problem in Noisy Sorting
  

• Small Set Expansion in the Johnson Graph through High Dimensional Expansion
  

• High Dimensional Expanders: The Basics
  

• Reasoning in the Presence of Noise
  

• The Power of Comparisons for Active Learning
  

• On the Cohomology of Dihedral Groups
  

• Understanding Doppelgangers and the Ur-Operation
  

Teaching

1. Co-taught CSE 291, Expander graphs and high-dimensional expanders, Spring 2021.
   

2. TA for CSE 200, Computability and Complexity, Winter 2020.
   

1. TA for APMTH 106, Applied Algebra, Fall 2017.