Stora avvikelser och stokastiska approximationer

Stochastic approximation is the general the theory for finding fixed points with stochastic algorithms, originally developed by Robbins and Monro in the 1950´s.

Classical applications include statistical inference, pattern classification, and control theory, whereas modern applications include areas of high-dimensional statistics, large-scale optimization, machine learning and statistical physics. Large deviations theory is not well developed for stochastic approximation methods.

The existing results were developed in the mid 1980´s under restrictive assumptions.

The aim of this project is to develop a general large deviations theory for stochastic approximation algorithms by extending the existing theory, using the weak convergence approach to large deviations, to general conditions, providing alternative and more accesible representations of the rate function and establishing connections to high-dimensional partial differential equations of Hamilton-Jacobi type.

Presently large deviations theory provides useful information on the performance and design of efficient computational methods in related areas such as rare-event simulation, statistical physics and molecular dynamics.

This project aims for similar success for stochastic approximation algorithms, leading to potential breakthroughs in the analysis of efficient algorithms in stochastic gradient methods, reinforcement learning, adapted Markov chain Monte Carlo, and extended ensemble methods in statistical physics. .