# Mathematics for Artificial Intelligence 2

These lectures are offered in the M1 tracks of the Master Mathematics
and Applications of Paris Saclay
### Schedule

Monday 13h30-16h, room 0A3

Wednesday 10h30-12h30, room 0A4

Part I: January 3 - February 21

Part II: March 7 - April 11

### Language

French or English, depending on the audience.

### Prerequisite

Probability theory, linear algebra, convex analysis

### Lecture Notes

Please download the lecture notes.
### Program

This course MathIA-2 can be followed independently of the course MathIA-1 from the
first semester.

The lectures are intended as an introduction to the mathematical tools
involved in the analysis of Machine Learning algorithms.

The course has two parts. A first part is focused on the problem of
sequential learning. The mathematical tools involved are mainly probabilistic, with
some convexity arguments. The second part covers some questions
where matrix analysis and linear algebra play a central role.

Part 1: Sequential learning
- Complement on probability: Sub-Gaussian random vector,
sub-Exponential random variables, concentration inequalities.
- Sequential learning problems, Stochastic Gradient Optimisation,
connection with classical optimisation problems.
- Learning with expert advices, adversarial environnement, mirror
descent.
- Learning in an unknown environnement. Exploration/exploitation
trade-off, Information-Theoretic lower bounds.

Part 2: Matrix analysis for Machine Learning
- Matricial problems in ML: dimension reduction, clustering,
community detection.
- Singular value decomposition, matrix norms, perturbation bounds
- Concentration of the operator norm of random matrices,
application in ML.

### Evaluation

Mid term exam February 21, room 1A7, 14h--17h,

Final exam April 11, room 1A7, 14h--17h.

Exercises for the exam: Sections 1.3.1, 2.3.1, 2.3.2, 2.3.3, 3.4.3 [including Pinsker inequality for arbitrary probability distributions], 4.4., 6.4.2, 7.4.1, 7.4.2, 8.4.1.

### Discovering more on the topic

For a deeper investigation of the first part, I recommend the M2 course
'Apprentissage et optimisation sequentiels'. For the second part, I
recommend the M2 course 'High-dimensional probability and
statistics'. Both courses are given in the master program
'mathématiques de l'aléatoire'.

For exploring more widely on mathematics and machine learning, you may have a look at the nice blogs of Sebastien Bubeck and
Francis Bach.

### Access

The Institut de Mathématiques d'Orsay is located in Building
307, Orsay campus, 5min walk from RER station Orsay-Ville.

Access plan