Stochastic Black-Box Optimization and Benchmarking in Large Dimensions

Abstract : Because of the generally high computational costs that come with large-scale problems, more so on real world problems, the use of benchmarks is a common practice in algorithm design, algorithm tuning or algorithm choice/evaluation. The question is then the forms in which these real-world problems come. Answering this question is generally hard due to the variety of these problems and the tediousness of describing each of them. Instead, one can investigate the commonly encountered difficulties when solving continuous optimization problems. Once the difficulties identified, one can construct relevant benchmark functions that reproduce these difficulties and allow assessing the ability of algorithms to solve them. In the case of large-scale benchmarking, it would be natural and convenient to build on the work that was already done on smaller dimensions, and be able to extend it to larger ones. When doing so, we must take into account the added constraints that come with a large-scale scenario. We need to be able to reproduce, as much as possible, the effects and properties of any part of the benchmark that needs to be replaced or adapted for large-scales. This is done in order for the new benchmarks to remain relevant. It is common to classify the problems, and thus the benchmarks, according to the difficulties they present and properties they possess. It is true that in a black-box scenario, such information (difficulties, properties...) is supposed unknown to the algorithm. However, in a benchmarking setting, this classification becomes important and allows to better identify and understand the shortcomings of a method, and thus make it easier to improve it or alternatively to switch to a more efficient one (one needs to make sure the algorithms are exploiting this knowledge when solving the problems). Thus the importance of identifying the difficulties and properties of the problems of a benchmarking suite and, in our case, preserving them. One other question that rises particularly when dealing with large-scale problems is the relevance of the decision variables. In a small dimension problem, it is common to have all variable contribute a fair amount to the fitness value of the solution or, at least, to be in a scenario where all variables need to be optimized in order to reach high quality solutions. This is however not always the case in large-scales; with the increasing number of variables, some of them become redundant or groups of variables can be replaced with smaller groups since it is then increasingly difficult to find a minimalistic representation of a problem. This minimalistic representation is sometimes not even desired, for example when it makes the resulting problem more complex and the trade-off with the increase in number of variables is not favorable, or larger numbers of variables and different representations of the same features within a same problem allow a better exploration. This encourages the design of both algorithms and benchmarks for this class of problems, especially if such algorithms can take advantage of the low effective dimensionality of the problems, or, in a complete black-box scenario, cost little to test for it (low effective dimension) and optimize assuming a small effective dimension. In this thesis, we address three questions that generally arise in stochastic continuous black-box optimization and benchmarking in high dimensions: 1. How to design cheap and yet efficient step-size adaptation mechanism for evolution strategies? 2. How to construct and generalize low effective dimension problems? 3. How to extend a low/medium dimension benchmark to large dimensions while remaining computationally reasonable, non-trivial and preserving the properties of the original problem?
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Ouassim Ait Elhara. Stochastic Black-Box Optimization and Benchmarking in Large Dimensions. Artificial Intelligence [cs.AI]. Université Paris-Saclay, 2017. English. ⟨NNT : 2017SACLS211⟩. ⟨tel-01615829⟩

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