Theory of Stochastic Drift

TL;DR

This paper develops the theory of stochastic drift, explaining how expected biases in stochastic processes influence the time to reach certain values, with applications in randomized search heuristics.

Contribution

It introduces the theory of stochastic drift, reviews main drift theorems, and demonstrates their application in diverse contexts, including counterintuitive cases.

Findings

Provides a comprehensive review of drift theorems

Shows how to apply drift methods in various scenarios

Includes examples illustrating stochastic drift concepts

Abstract

In studying randomized search heuristics, a frequent quantity of interest is the first time a (real-valued) stochastic process obtains (or passes) a certain value. The processes under investigation commonly show a bias towards this goal, the \emph{stochastic drift}. Turning an iteration-wise expected bias into a first time of obtaining a value is the main result of \emph{drift theorems}. This thesis introduces the theory of stochastic drift, providing examples and reviewing the main drift theorems available. Furthermore, the thesis explains how these methods can be applied in various contexts, including those where drift theorems seem a counterintuitive choice. Later sections examine related methods and approaches.