Random matrix theory of high-dimensional optimization - Lecture 1

Speaker: Elliot Paquette

Date: Tue, Jul 2, 2024

Location: CRM, Montreal

Conference: 2024 CRM-PIMS Summer School in Probability

Subject: Mathematics, Probability

Class: Scientific


Optimization theory seeks to show the performance of algorithms to find the (or a) minimizer x∈ℝd of an objective function. The dimension of the parameter space d has long been known to be a source of difficulty in designing good algorithms and in analyzing the objective function landscape. With the rise of machine learning in recent years, this has been proven that this is a manageable problem, but why? One explanation is that this high dimensionality is simultaneously mollified by three essential types of randomness: the data are random, the optimization algorithms are stochastic gradient methods, and the model parameters are randomly initialized (and much of this randomness remains). The resulting loss surfaces defy low-dimensional intuitions, especially in nonconvex settings.
Random matrix theory and spin glass theory provides a toolkit for theanalysis of these landscapes when the dimension $d$ becomes large. In this course, we will show

how random matrices can be used to describe high-dimensional inference
nonconvex landscape properties
high-dimensional limits of stochastic gradient methods.