# Martin Krejca

## Theoretical Analyses of Univariate Estimation-of-Distribution Algorithms

Optimization is a core part of technological advancement and is usually heavily aided by computers. However, since many optimization problems are hard, it is unrealistic to expect an optimal solution within reasonable time. Hence, heuristics are employed, that is, computer programs that try to produce solutions of high quality quickly. One special class are estimation-of-distribution algorithms (EDAs), which are characterized by maintaining a probabilistic model over the problem domain, which they evolve over time. In an iterative fashion, an EDA uses its model in order to generate a set of solutions, which it then uses to reÿne the model such that the probability of producing good solutions is increased.

In this thesis, we theoretically analyze the class of univariate EDAs over the Boolean domain, that is, over the space of all length-n bit strings. In this setting, the probabilistic model of a univariate EDA consists of an n-dimensional probability vector where each component denotes the probability to sample a 1 for that position in order to generate a bit string.

My contribution follows two main directions: ÿrst, we analyze general in-herent properties of univariate EDAs. Second, we determine the expected run times of speciÿc EDAs on benchmark functions from theory. In the ÿrst part, we characterize when EDAs are unbiased with respect to the problem encoding. We then consider a setting where all solutions look equally good to an EDA, and we show that the probabilistic model of an EDA quickly evolves into an incorrect model if it is always updated such that it does not change in expectation.

In the second part, we ÿrst show that the algorithms cGA and MMAS-fp are able to e˝ciently optimize a noisy version of the classical benchmark function OneMax. We perturb the function by adding Gaussian noise with a variance of ˙ 2, and we prove that the algorithms are able to generate the true optimum in a time polynomial in ˙ 2 and the problem size n. For the MMAS-fp, we generalizethis result to linear functions. Further, we prove a run time of ????n log(n) for the
algorithm UMDA on (unnoisy) OneMax. Last, we introduce a new algorithm that is able???? to optimize the benchmark functions OneMax and LeadingOnesboth in O n log(n) , which is a novelty for heuristics in the domain we consider.