Brief course description: The course will cover the main topics in high-dimensional probability theory. High-dimensional probability is an area of probability theory that studies random objects in Rn where the dimension n can be very large. The applications in data science of the introduced theoretical tools will be discussed.

Course topics:

• Basic tail and concentration bounds: from Markov to Chernoff, sub-Gaussian vari- ables and Hoeffding bounds, sub-exponential variables and Bernstein bounds, some one- sided results, uniform laws of large numbers.

• Sparse linear models in high dimensions: problem formulation and applications and penalized estimators, recovery in the noiseless setting, estimation in noisy settings and LASSO estimator, bounds on prediction error, variable or subset selection.

• Nonparametric least squares: problem set-up, oracle inequalities, regularized esti- mators.

  Textbooks:

• R. Vershynin (2018) High-dimensional probability. An introduction with applications in Data Science. Cambridge University Press.

• M. J. Wainwright (2019) High-dimensional statistics: A non-asymptotic viewpoint. Cam- bridge University Press.