Algorithms for Heavy-Tailed Statistics - Regression, Covariance Estimation, and Beyond

Explore advanced algorithms for heavy-tailed statistics in this 20-minute conference talk from the Association for Computing Machinery (ACM). Delve into high probability estimation techniques, Gaussian covariance estimation, and linear regression methods. Learn about covariance estimation and linear regression rates under weak assumptions, and examine key SOS (Sum of Squares) assumptions. Investigate the Median of Means framework, including one-dimensional cases and high-dimensional tournament estimators. Analyze SOS relaxation techniques, covering optimization problems, concentration steps, and expectation steps. Discuss Matrix Bernstein inequalities and their limitations, as well as evidence of hardness for covariance estimation. Conclude by exploring low-degree tests for detection in heavy-tailed statistical scenarios.

Intro
High Probability Estimation
Gaussian Covariance Estimation
Gaussian Linear Regression
Covariance Estimation under Weak Assumptions
Linear Regression Rates under Weak Assumptions
Key SOS Assumptions
Towards Statistical Optimality for Covariance Estimation
Towards Statistical Optimality for Linear Regression
Outline
Median of Means Framework
Median of Means - One Dimensional Case
Tournament Estimator - High Dimensional Version
Testing a Candidate Matrix - Optimization Problem
Sos Relaxation - Analysis
Sos Relaxation - Concentration Step
Sos Relaxation - Expectation Step
Matrix Bernstein?
Getting Around Matrix Bernstein
Evidence of Hardness for Covariance Estimation
Low degree Tests for Detection