First-Hitting Times Under Additive Drift

TL;DR

This paper enhances fundamental drift theorems used in analyzing randomized search heuristics by generalizing their conditions, making them applicable to broader scenarios with fewer restrictions.

Contribution

It generalizes additive, multiplicative, and variable drift theorems, removing unnecessary restrictions and broadening their applicability in analyzing expected run times.

Findings

Improved additive drift theorem for upper bounds with minimal assumptions

Generalized drift theorems applicable to nonnegative processes

Simplified analysis of randomized search heuristics using broader drift conditions

Abstract

For the last ten years, almost every theoretical result concerning the expected run time of a randomized search heuristic used drift theory, making it the arguably most important tool in this domain. Its success is due to its ease of use and its powerful result: drift theory allows the user to derive bounds on the expected first-hitting time of a random process by bounding expected local changes of the process -- the drift. This is usually far easier than bounding the expected first-hitting time directly. Due to the widespread use of drift theory, it is of utmost importance to have the best drift theorems possible. We improve the fundamental additive, multiplicative, and variable drift theorems by stating them in a form as general as possible and providing examples of why the restrictions we keep are still necessary. Our additive drift theorem for upper bounds only requires the process to be nonnegative, that is, we remove unnecessary restrictions like a finite, discrete, or bounded search space. As corollaries, the same is true for our upper bounds in the case of variable and multiplicative drift.