Analysis of Different Types of Regret in Continuous Noisy Optimization

The performance measure of an algorithm is a crucial part of its analysis. The performance can be determined by the study on the convergence rate of the algorithm in question. It is necessary to study some (hopefully convergent) sequence that will measure how "good" is the approximated optimum compared to the real optimum. The concept of Regret is widely used in the bandit literature for assessing the performance of an algorithm. The same concept is also used in the framework of optimization algorithms, sometimes under other names or without a specific name. And the numerical evaluation of convergence rate of noisy algorithms often involves approximations of regrets. We discuss here two types of approximations of Simple Regret used in practice for the evaluation of algorithms for noisy optimization. We use specific algorithms of different nature and the noisy sphere function to show the following results. The approximation of Simple Regret, termed here Approximate Simple Regret, used in some optimization testbeds, fails to estimate the Simple Regret convergence rate. We also discuss a recent new approximation of Simple Regret, that we term Robust Simple Regret, and show its advantages and disadvantages.Comment: Genetic and Evolutionary Computation Conference 2016, Jul 2016, Denver, United States. 201

Sorting by Swaps with Noisy Comparisons

We study sorting of permutations by random swaps if each comparison gives the wrong result with some fixed probability $p<1/2$. We use this process as prototype for the behaviour of randomized, comparison-based optimization heuristics in the presence of noisy comparisons. As quality measure, we compute the expected fitness of the stationary distribution. To measure the runtime, we compute the minimal number of steps after which the average fitness approximates the expected fitness of the stationary distribution. We study the process where in each round a random pair of elements at distance at most $r$ are compared. We give theoretical results for the extreme cases $r=1$ and $r=n$, and experimental results for the intermediate cases. We find a trade-off between faster convergence (for large $r$) and better quality of the solution after convergence (for small $r$).Comment: An extended abstract of this paper has been presented at Genetic and Evolutionary Computation Conference (GECCO 2017

Analysis of Evolutionary Algorithms in Dynamic and Stochastic Environments

Many real-world optimization problems occur in environments that change dynamically or involve stochastic components. Evolutionary algorithms and other bio-inspired algorithms have been widely applied to dynamic and stochastic problems. This survey gives an overview of major theoretical developments in the area of runtime analysis for these problems. We review recent theoretical studies of evolutionary algorithms and ant colony optimization for problems where the objective functions or the constraints change over time. Furthermore, we consider stochastic problems under various noise models and point out some directions for future research.Comment: This book chapter is to appear in the book "Theory of Randomized Search Heuristics in Discrete Search Spaces", which is edited by Benjamin Doerr and Frank Neumann and is scheduled to be published by Springer in 201