Learning Theory from First Principles

Francis Bach

Masters M2 IASD 2025/2026

The class will be taught in French or (most probably) English, depending on attendance (all slides and class notes are in English).

Summary

The goal of this class is to present old and recent results in learning theory, for the most widely-used learning architectures. This class is geared towards theory-oriented students as well as students who want to acquire a basic mathematical understanding of algorithms used throughout the masters program.

A particular effort will be made to prove many results from first principles, while keeping the exposition as simple as possible. This will naturally lead to a choice of key results that show-case in simple but relevant instances the important concepts in learning theory. Some general results will also be presented without proofs.

The class will be organized in nine three-hour sessions, each with a precise topic (a chapter from the book in preparation "Learning theory from first principles"). See tentative schedule below. Credit: 4 ECTS.

Prerequisites: We will prove results in class so a good knowledge of undergraduate mathematics is important, as well as basic notions in probability. Having followed an introductory class on machine learning is beneficial.

Dates

All classes will be "in real life" at the auditorium of PSL, 16 bis rue de l'Estrapade, 75005 Paris, on Thursdays between 9am and 12.15pm.

The class will follow the book (final draft available here)

Each student will benefit more from the class if the corresponding sections are read before class.

Date	Topics	Book chapters Exercises / Boards
September 18	Learning with infinite data (population setting) -Decision theory (loss, risk, optimal predictors) -Decomposition of excess risk into approximation and estimation errors -No free lunch theorems -Basic notions of concentration inequalities (MacDiarmid, Hoeffding, Bernstein)	Chapter 2 Exercises 2.5, 2.6 Board
September 25	Linear Least-squares regression -Guarantees in the fixed design settings (simple in closed-form) -Ridge regression: dimension independent bounds -Guarantees in the random design settings -Lower bound of performance	Chapter 3 Exercises 3.3, 3.5
October 2	Empirical risk minimization -Convexification of the risk -Risk decomposition -Estimation error: finite number of hypotheses and covering numbers -Rademacher complexity -Penalized problems	Chapter 4 Exercises 4.12, 4.14 Board
October 9	Optimization for machine learning -Gradient descent -Stochastic gradient descent -Generalization bounds through stochastic gradient descent	Chapter 5 Exercises 5.20, 5.25 Board
October 16	Local averaging techniques -Partition estimators -Nadaraya-Watson estimators -K-nearest-neighbors	Chapter 6
October 23	Kernel methods -Kernels and representer theorems -Algorithms -Universal consistency	Chapter 7
October 30	Model selection -L0 penalty -L1 penalty -High-dimensional estimation	Chapter 8
November 6	Neural networks -Single hidden layer neural networks -Estimation error -Approximation properties and universality	Chapter 9
December 11	Exam

Evaluation

Take-home exercises, one per class to be sent before the next class (25%). One written in-class exam (75%).
Extra points for writing in latex solutions to exercises from the textbook that are not already written.

Procedure for sending exercise solutions: Send your PDF file to fbachorsay2017@gmail.com, before the end of the next class. Do not forget to add your name to the pdf.

Solutions to the exercises will be posted here one week after they are due.