Learning Theory from First Principles

Francis Bach

Mastere MASH 2020/2021

The class will be taught in French or 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 9 three-hour sessions, each with a precise topic except the last one dedicated to recent learning theory results. See tentative schedule below.

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

Given the sanitary situation, all classes will be online, with the following tentative plan. Detailed class notes will be made available 2 days before each class, while connection details sent to the registered participants the night before (I will use GotoMeeting).

Each student is expected to read the class notes before the class. During class, I will go over them, provide additional details and answer questions. Classes will be held on Friday between 8.30am and 11.30am.

Date	Topics	Class notes
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)	lecture1.pdf
September 25	Liner 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	lecture2.pdf
October 2	Empirical risk minimization -Convexification of the risk -Risk decomposition -Estimation error: finite number of hypotheses and covering numbers -Rademacher complexity -Penalized problems	lecture3.pdf
October 16	Optimization for machine learning -Gradient descent -Stochastic gradient descent -Generalization bounds through stochastic gradient descent	lecture4.pdf
October 23	Local averaging techniques -Partition estimators -Nadaraya-Watson estimators -K-nearest-neighbors -Universal consistency	lecture5.pdf
October 30	Kernel methods -Kernels and representer theorems -Algorithms -Analysis of well-specified models -Sharp analysis of ridge regression -Universal consistency	lecture6.pdf
November 6	Model selection -L0 penalty -L1 penalty -High-dimensional estimation	lecture7.pdf
November 13	Neural networks -Single hidden layer neural networks - Estimation error - Approximation properties and universality	lecture8.pdf
November 20	Special topics -Generalization/optimization properties of infinitely wide neural networks -Double descent	lecture9.pdf
December 4	EXAM

Evaluation

One written in-class exam, and (very) simple coding assignments (to illustrate convergence results, to be sent to learning.theory.first.principles@gmail.com). For all classes, the coding assignment is to reproduce the experiments shown in the lecture notes and send only the figures to the address above.