First-Hitting Times Under Additive Drift

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

Intuitive Analyses via Drift Theory

Humans are bad with probabilities, and the analysis of randomized algorithms offers many pitfalls for the human mind. Drift theory is an intuitive tool for reasoning about random processes. It allows turning expected stepwise changes into expected first-hitting times. While drift theory is used extensively by the community studying randomized search heuristics, it has seen hardly any applications outside of this field, in spite of many research questions which can be formulated as first-hitting times. We state the most useful drift theorems and demonstrate their use for various randomized processes, including approximating vertex cover, the coupon collector process, a random sorting algorithm, and the Moran process. Finally, we consider processes without expected stepwise change and give a lemma based on drift theory applicable in such scenarios without drift. We use this tool for the analysis of the gambler's ruin process, for a coloring algorithm, for an algorithm for 2-SAT, and for a version of the Moran process without bias

Improved Fixed-Budget Results via Drift Analysis

Fixed-budget theory is concerned with computing or bounding the fitness value achievable by randomized search heuristics within a given budget of fitness function evaluations. Despite recent progress in fixed-budget theory, there is a lack of general tools to derive such results. We transfer drift theory, the key tool to derive expected optimization times, to the fixed-budged perspective. A first and easy-to-use statement concerned with iterating drift in so-called greed-admitting scenarios immediately translates into bounds on the expected function value. Afterwards, we consider a more general tool based on the well-known variable drift theorem. Applications of this technique to the LeadingOnes benchmark function yield statements that are more precise than the previous state of the art.Comment: 25 pages. An extended abstract of this paper will be published in the proceedings of PPSN 202

Upper Bounds on the Runtime of the Univariate Marginal Distribution Algorithm on OneMax

A runtime analysis of the Univariate Marginal Distribution Algorithm (UMDA) is presented on the OneMax function for wide ranges of its parameters $\mu$ and $\lambda$. If $\mu\ge c\log n$ for some constant $c>0$ and $\lambda=(1+\Theta(1))\mu$, a general bound $O(\mu n)$ on the expected runtime is obtained. This bound crucially assumes that all marginal probabilities of the algorithm are confined to the interval $[1/n,1-1/n]$. If $\mu\ge c' \sqrt{n}\log n$ for a constant $c'>0$ and $\lambda=(1+\Theta(1))\mu$, the behavior of the algorithm changes and the bound on the expected runtime becomes $O(\mu\sqrt{n})$, which typically even holds if the borders on the marginal probabilities are omitted. The results supplement the recently derived lower bound $\Omega(\mu\sqrt{n}+n\log n)$ by Krejca and Witt (FOGA 2017) and turn out as tight for the two very different values $\mu=c\log n$ and $\mu=c'\sqrt{n}\log n$. They also improve the previously best known upper bound $O(n\log n\log\log n)$ by Dang and Lehre (GECCO 2015).Comment: Version 4: added illustrations and experiments; improved presentation in Section 2.2; to appear in Algorithmica; the final publication is available at Springer via http://dx.doi.org/10.1007/s00453-018-0463-

Surfing on the seascape:adaptation in a changing environment

The environment changes constantly at various time scales and, in order to survive, species need to keep adapting. Whether these species succeed in avoiding extinction is a major evolutionary question. Using a multilocus evolutionary model of a mutation‐limited population adapting under strong selection, we investigate the effects of the frequency of environmental fluctuations on adaptation. Our results rely on an “adaptive‐walk” approximation and use mathematical methods from evolutionary computation theory to investigate the interplay between fluctuation frequency, the similarity of environments, and the number of loci contributing to adaptation. First, we assume a linear additive fitness function, but later generalize our results to include several types of epistasis. We show that frequent environmental changes prevent populations from reaching a fitness peak, but they may also prevent the large fitness loss that occurs after a single environmental change. Thus, the population can survive, although not thrive, in a wide range of conditions. Furthermore, we show that in a frequently changing environment, the similarity of threats that a population faces affects the level of adaptation that it is able to achieve. We check and supplement our analytical results with simulations