# Brief Overview of a 360-in-525 Minutes Course Set

For more details see Overview of a 360-in-525 Minutes Course Set in Data Sciences, Spring 2018

# 360-in-525-0: Mathematical Statistical Learning Theory Series; An L1 View

This course will introduce a PhD student in mathematics or mathematical statistics to one of the fundamental problems at the very core of various probabilistic theories of decision-making.

We will mainly focus on the relation between the combinatorial geometric complexity of the (sigma) algebras of a simple measurable space and the rates of convergence of empirical measures over them in one of the simplest posable decision problems – *nonparametric density estimation* of an unknown density f in L1 based on finitely many observations drawn independently from it, but without making any mathematical compromise whatsoever, and thereby giving the so-called *universal performance guarantee* .

This course was given in another form at CMAP, Ecole Polytechnique, Palaiseau, France for PhD students in mathematics there. Students in Geometry and Combinatorial probability as well as analysis may find this course insightful for their own research, as one of the basic theorems involves the combined use of several unique inequalities in a specific partial order of implications.

The emphasis will involve constructive mathematics and perhaps delve into tree arithmetics towards such decision with universal performance guarantees along with their combinatorial, algebraic and analytic properties if time permits. Unfortunately such guarantees are not available for big data sets and may be necessary for being able to impose legal requirements and standards on automated decision-making systems.

# Course Content

The lectures are based on Combinatorial Methods in Density Estimation by Luc Devroye and Gábor Lugosi, with an emphasis on the problem of selecting from k density estimates with *Universal Performance Guarantees* via minimum distance estimates that bound the Vapnik-Chervonenkis shatter coefficients over the Yatracos class of all k(k-1) Scheffe sets.

**YouTube Archive of black-board lectures:**

- https://youtu.be/wVmN0awhRf4
- https://youtu.be/LBgP3CcDHGU
- https://youtu.be/hNXBW1siyN4
- https://youtu.be/-Y6_cHCEQio
- https://youtu.be/TZLJSqh9iKQ

*These lectures are dedicated to Luc Devroye and the uncompromising principle of universal performance gurantee.*