Multiplicative Drift Analysis

Abstract

In this work, we introduce multiplicative drift analysis as a suitable way to analyze the runtime of randomized search heuristics such as evolutionary algorithms.We give 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 distance to the optimum.To display the strength of this tool, we regard the classical problem how the (1+1) Evolutionary Algorithm optimizes an arbitrary linear pseudo-Boolean function. Here, we first give a relatively simple proof for the fact that any linear function is optimized in expected time O(n log n), where n is the length of the bit string. Afterwards, we show that in fact any such function is optimized in expected time at most (1 + o(1))1.39en ln(n), again using multiplicative drift analysis. We also prove a corresponding lower bound of (1 − o(1))en ln(n) which actually holds for all functions with a unique global optimum.We further demonstrate how our drift theorem immediately gives natural proofs (with better constants) for the best known runtime bounds for the (1+1) Evolutionary Algorithm on combinatorial problems like finding minimum spanning trees, shortest paths, or Euler tours. * Carola Winzen is a recipient of the Google Europe Fellowship in Randomized Algorithms, and this work is supported in part by this Google Fellowship.

Sharp bounds by probability-generating functions and variable drift

“…In addition, we prove a new drift theorem that goes back to Johannsen (2010) and enables us to handle drift that is neither constant from the classical additive point of view nor from the more recent multiplicative point of view (Doerr et al (2010b)). In contrast to the previous so-called variable drift theorem, which was only presented for upper bounds before, our variant allows us to prove lower bounds on the expected runtime.…”

Runtime analysis of the (1+1) evolutionary algorithm on strings over finite alphabets

“…Similar restrictions on the current techniques of analysis have been encounter in [5,6] for the study of the (1+1) EA with mutation rates larger than 1/n.…”

“…The study of the progress of an algorithm with respect to a potential function is called drift analysis (the drift is the expected change in potential) and is briefly introduced in Section 3, see He and Yao [11,13], Doerr, Johannsen, Winzen [5,6] and Hajek [10] for details on this technique.…”

“…For example, in [8] and [6] the progress of the (1+1) EA minimizing linear functions is measured with respect to the potential function g0 : {0, 1} n → R, x → n/2 i=1 2xi + n i= n/2 +1 xi, in [11,12] with respect to the potential function g1 : {0, 1} n → R, x → log 1 + n i=1 xi , and in [14] and [5] with respect to the potential function…”

Superpolynomial Lower Bounds for the $$(1+1)$$ ( 1 + 1 ) EA on Some Easy Combinatorial Problems

“…The use of computationally easy functions as an object of study has also illuminated several fundamental properties of randomized search heuristics and has cultivated the development of new proof techniques [7] and new efficient algorithms [5]. Inspired by these successes, we want to commute this approach into the realm of combinatorial problems with easily understood structure and study the influence of such structure on algorithm behavior.…”