Multiplicative drift analysis

Abstract

Drift analysis is one of the strongest tools in the analysis of evolutionary algorithms. Its main weakness is that it is often very hard to find a good drift function.In this paper, we make progress in this direction. We prove a multiplicative version of the classical drift theorem. This allows easier analyses in those settings, where the optimization progress is roughly proportional to the current objective value.Our drift theorem immediately gives natural proofs for the best known run-time bounds for the (1+1) Evolutionary Algorithm computing minimum spanning trees and shortest paths, since here we may simply take the objective function as drift function.As a more challenging example, we give a relatively simple proof for the fact that any linear function is optimized in time O(n log n). In the multiplicative setting, a simple linear function can be used as drift function (without taking any logarithms).However, we also show that, both in the classical and the multiplicative setting, drift functions yielding good results for all linear functions exist only if the mutation probability is at most c/n for a small constant c.

A Runtime Analysis of Parallel Evolutionary Algorithms in Dynamic Optimization

“…It is taken from [6], except for the final tail bound "P(T > · · · ) < · · · ", which stems from [5].…”

Lyapunov Design of a Simple Step-Size Adaptation Strategy Based on Success

“…The use of drift analysis combined with Markov chain theory in order to obtain lower and upper bounds on the expected hitting time on discrete search spaces is presented, for example, in [6][7][8][9][10]. Proofs of convergence of the (1 + 1)-EA applied to pseudo-Boolean linear functions can be found in [6,9,10]. In [6], the author combines drift analysis and Markov-chains for the first time to state bounds on the expected optimization time.…”

A Parameterized Runtime Analysis of Simple Evolutionary Algorithms for Makespan Scheduling

“…Such a move occurs with probability at least 1/(en). By the multiplicative drift theorem [2], the expected number of steps until the d value has reached zero conditioned on the event that no bits corresponding to C(z) are flipped is at most en (1 + ln d(x )) ≤ en (ln n + ln p 1 + 1) since d(x ) ≤ np 1 . Consider a run of the (1+1) EA of length t = cen(ln n + ln p 1 + 1).…”

“…The expected time until the d value has reduced to zero conditioned on the event that no bits of index at most k are flipped follows from the multiplicative drift theorem of Doerr et al [2] and is at most t = en (1 + ln d(x )). By the Markov inequality, the probability that this occurs after 2t steps (again, conditioned on the event that no bits with index at most k are flipped) is at least 1/2 = Ω(1).…”