# Multiplicative Up-Drift

## TL;DR

This paper introduces a new drift theorem for analyzing processes with multiplicative growth, improving run time guarantees for evolutionary algorithms by providing near-linear dependence on key parameters.

## Contribution

It presents a novel drift theorem for multiplicative growth, offering the first near-linear dependence on 1/δ and population size, enhancing analysis of evolutionary algorithms.

## Key findings

- Proves a drift theorem for multiplicative growth in expectation.
- Provides stronger run time guarantees for evolutionary algorithms.
- Simplifies the proof of the level-based theorem.

## Abstract

Drift analysis aims at translating the expected progress of an evolutionary algorithm (or more generally, a random process) into a probabilistic guarantee on its run time (hitting time). So far, drift arguments have been successfully employed in the rigorous analysis of evolutionary algorithms, however, only for the situation that the progress is constant or becomes weaker when approaching the target. Motivated by questions like how fast fit individuals take over a population, we analyze random processes exhibiting a $(1+\delta)$-multiplicative growth in expectation. We prove a drift theorem translating this expected progress into a hitting time. This drift theorem gives a simple and insightful proof of the level-based theorem first proposed by Lehre (2011). Our version of this theorem has, for the first time, the best-possible near-linear dependence on $1/\delta$ (the previous results had an at least near-quadratic dependence), and it only requires a population size near-linear in $\delta$ (this was super-quadratic in previous results). These improvements immediately lead to stronger run time guarantees for a number of applications. We also discuss the case of large $\delta$ and show stronger results for this setting.