# 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

### 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 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