# On the Limitations of the Univariate Marginal Distribution Algorithm to Deception and Where Bivariate EDAs might help

## TL;DR

This paper introduces a new benchmark to evaluate the limitations of the UMDA in deceptive and epistatic problems, showing simple EAs outperform UMDA unless population sizes are very large, and demonstrating the potential of bivariate EDAs like MIMIC.

## Contribution

The paper presents a new benchmark problem and analyzes the limitations of UMDA in deceptive scenarios, highlighting the need for more complex probabilistic models like bivariate EDAs.

## Key findings

- UMDA has exponential runtime on the DLB problem with typical population sizes.
- Simple EAs outperform UMDA unless the population size is extremely high.
- Bivariate MIMIC performs efficiently on the DLB problem.

## Abstract

We introduce a new benchmark problem called Deceptive Leading Blocks (DLB) to rigorously study the runtime of the Univariate Marginal Distribution Algorithm (UMDA) in the presence of epistasis and deception. We show that simple Evolutionary Algorithms (EAs) outperform the UMDA unless the selective pressure $\mu/\lambda$ is extremely high, where $\mu$ and $\lambda$ are the parent and offspring population sizes, respectively. More precisely, we show that the UMDA with a parent population size of $\mu=\Omega(\log n)$ has an expected runtime of $e^{\Omega(\mu)}$ on the DLB problem assuming any selective pressure $\frac{\mu}{\lambda} \geq \frac{14}{1000}$, as opposed to the expected runtime of $\mathcal{O}(n\lambda\log \lambda+n^3)$ for the non-elitist $(\mu,\lambda)~\text{EA}$ with $\mu/\lambda\leq 1/e$. These results illustrate inherent limitations of univariate EDAs against deception and epistasis, which are common characteristics of real-world problems. In contrast, empirical evidence reveals the efficiency of the bi-variate MIMIC algorithm on the DLB problem. Our results suggest that one should consider EDAs with more complex probabilistic models when optimising problems with some degree of epistasis and deception.