Concentration Inequalities: Theory and Applications

**Prerequisites:** Probability Theory and Linear Algebra.

- Introduction
- variance and concentration
- isoperimetry and Lipschitz concentration
- martingale method
- entropy method
- transportation method

- Basics
- Cramer-Chernoff method
- Hoeffding's, Bennet's, Bernstein's
- Efron-Stein-Steele, Tensorization
- Bounded differences, self-bounding functions
- Poincaré inequalities

- Application: JL and Random projections
- Talagrand's convex distance inequality
- Application: TSP
- Application: longest increasing subsequence
- Suprema of empirical processes
- variance and concentration
- Nemirovski's inequality
- Symmetrization and contraction
- Talagrand's inequality, Bousquet's inequality
- chaining
- uniform laws of large numbers

- Application: analysis of regression for misspecified models
- Application: linear inverse problems, sparse recovery, restricted isometry, Gaussian widths
- Application: model selection
- Markov semi-group proofs
- Extensions to martingales, sequential complexities
- In-depth analysis:
- Log-Sobolev inequalities
- The entropy method
- Isoperimetry
- The transportation method
- Information Inequalities

- Stein's method
- Superconcentration
- Boolean functions, Fourier analysis
- Concentration of multivariate polynomials (Kim and Vu)
- Matrix concentration
- Application: Learning without concentration, small ball property, offset Rademacher averages

#### books/notes:

- S. Boucheron, G. Lugosi, P. Massart. Concentration Inequalities: A Nonasymptotic Theory of Independence.
- R. van Handel. Probability in High Dimension.
- M. Ledoux. The Concentration of Measure Phenomenon.
- J.M. Steele. Probability Theory and Combinatorial Optimization.
- R. Vershynin. Introduction to the non-asymptotic analysis of random matrices.
- M. Raginsky and I. Sason. Concentration of measure inequalities in information theory, communications and coding.
- S. Chatterjee. Superconcentration and Related Topics.
- M. Ledoux and M. Talagrand. Probability in Banach Spaces.
- A. van der Vaart and J. Wellner. Weak Convergence and Empirical Processes: With Applications to Statistics.

#### Articles: