Sampling in CMA-ES: Low Numbers of Low Discrepancy Points

The Covariance Matrix Adaptation Evolution Strategy (CMA-ES) is one of the most successful examples of a derandomized evolution strategy. However, it still relies on randomly sampling offspring, which can be done via a uniform distribution and subsequently transforming into the required Gaussian. Previous work has shown that replacing this uniform sampling with a low-discrepancy sampler, such as Halton or Sobol sequences, can improve performance over a wide set of problems. We show that iterating through small, fixed sets of low-discrepancy points can still perform better than the default uniform distribution. Moreover, using only 128 points throughout the search is sufficient to closely approximate the empirical performance of using the complete pseudorandom sequence up to dimensionality 40 on the BBOB benchmark. For lower dimensionalities (below 10), we find that using as little as 32 unique low discrepancy points performs similar or better than uniform sampling. In 2D, for which we have highly optimized low discrepancy samples available, we demonstrate that using these points yields the highest empirical performance and requires only 16 samples to improve over uniform sampling. Overall, we establish a clear relation between the $L_2$ discrepancy of the used point set and the empirical performance of the CMA-ES.

Quantifying Individual and Joint Module Impact in Modular Optimization Frameworks

This study explores the influence of modules on the performance of modular optimization frameworks for continuous single-objective black-box optimization. There is an extensive variety of modules to choose from when designing algorithm variants, however, there is a rather limited understanding of how each module individually influences the algorithm performance and how the modules interact with each other when combined. We use the functional ANOVA (f-ANOVA) framework to quantify the influence of individual modules and module combinations for two algorithms, the modular Covariance Matrix Adaptation (modCMA) and the modular Differential Evolution (modDE). We analyze the performance data from 324 modCMA and 576 modDE variants on the BBOB benchmark collection, for two problem dimensions, and three computational budgets. Noteworthy findings include the identification of important modules that strongly influence the performance of modCMA, such as the~\textit{weights\ option} and~\textit{mirrored} modules for low dimensional problems, and the~\textit{base\ sampler} for high dimensional problems. The large individual influence of the~\textit{lpsr} module makes it very important for the performance of modDE, regardless of the problem dimensionality and the computational budget. When comparing modCMA and modDE, modDE undergoes a shift from individual modules being more influential, to module combinations being more influential, while modCMA follows the opposite pattern, with an increase in problem dimensionality and computational budget.

Avoiding Redundant Restarts in Multimodal Global Optimization

Na\"ive restarts of global optimization solvers when operating on multimodal search landscapes may resemble the Coupon's Collector Problem, with a potential to waste significant function evaluations budget on revisiting the same basins of attractions. In this paper, we assess the degree to which such ``duplicate restarts'' occur on standard multimodal benchmark functions, which defines the \textit{redundancy potential} of each particular landscape. We then propose a repelling mechanism to avoid such wasted restarts with the CMA-ES and investigate its efficacy on test cases with high redundancy potential compared to the standard restart mechanism.

Empirical Analysis of the Dynamic Binary Value Problem with IOHprofiler

Optimization problems in dynamic environments have recently been the source of several theoretical studies. One of these problems is the monotonic Dynamic Binary Value problem, which theoretically has high discriminatory power between different Genetic Algorithms. Given this theoretical foundation, we integrate several versions of this problem into the IOHprofiler benchmarking framework. Using this integration, we perform several large-scale benchmarking experiments to both recreate theoretical results on moderate dimensional problems and investigate aspects of GA's performance which have not yet been studied theoretically. Our results highlight some of the many synergies between theory and benchmarking and offer a platform through which further research into dynamic optimization problems can be performed.

Impact of Training Instance Selection on Automated Algorithm Selection Models for Numerical Black-box Optimization

The recently proposed MA-BBOB function generator provides a way to create numerical black-box benchmark problems based on the well-established BBOB suite. Initial studies on this generator highlighted its ability to smoothly transition between the component functions, both from a low-level landscape feature perspective, as well as with regard to algorithm performance. This suggests that MA-BBOB-generated functions can be an ideal testbed for automated machine learning methods, such as automated algorithm selection (AAS). In this paper, we generate 11800 functions in dimensions $d=2$ and $d=5$, respectively, and analyze the potential gains from AAS by studying performance complementarity within a set of eight algorithms. We combine this performance data with exploratory landscape features to create an AAS pipeline that we use to investigate how to efficiently select training sets within this space. We show that simply using the BBOB component functions for training yields poor test performance, while the ranking between uniformly chosen and diversity-based training sets strongly depends on the distribution of the test set.

Using the Empirical Attainment Function for Analyzing Single-objective Black-box Optimization Algorithms

A widely accepted way to assess the performance of iterative black-box optimizers is to analyze their empirical cumulative distribution function (ECDF) of pre-defined quality targets achieved not later than a given runtime. In this work, we consider an alternative approach, based on the empirical attainment function (EAF) and we show that the target-based ECDF is an approximation of the EAF. We argue that the EAF has several advantages over the target-based ECDF. In particular, it does not require defining a priori quality targets per function, captures performance differences more precisely, and enables the use of additional summary statistics that enrich the analysis. We also show that the average area over the convergence curves is a simpler-to-calculate, but equivalent, measure of anytime performance. To facilitate the accessibility of the EAF, we integrate a module to compute it into the IOHanalyzer platform. Finally, we illustrate the use of the EAF via synthetic examples and via the data available for the BBOB suite.

Large-scale Benchmarking of Metaphor-based Optimization Heuristics

The number of proposed iterative optimization heuristics is growing steadily, and with this growth, there have been many points of discussion within the wider community. One particular criticism that is raised towards many new algorithms is their focus on metaphors used to present the method, rather than emphasizing their potential algorithmic contributions. Several studies into popular metaphor-based algorithms have highlighted these problems, even showcasing algorithms that are functionally equivalent to older existing methods. Unfortunately, this detailed approach is not scalable to the whole set of metaphor-based algorithms. Because of this, we investigate ways in which benchmarking can shed light on these algorithms. To this end, we run a set of 294 algorithm implementations on the BBOB function suite. We investigate how the choice of the budget, the performance measure, or other aspects of experimental design impact the comparison of these algorithms. Our results emphasize why benchmarking is a key step in expanding our understanding of the algorithm space, and what challenges still need to be overcome to fully gauge the potential improvements to the state-of-the-art hiding behind the metaphors.

Impact of spatial transformations on landscape features of CEC2022 basic benchmark problems

When benchmarking optimization heuristics, we need to take care to avoid an algorithm exploiting biases in the construction of the used problems. One way in which this might be done is by providing different versions of each problem but with transformations applied to ensure the algorithms are equipped with mechanisms for successfully tackling a range of problems. In this paper, we investigate several of these problem transformations and show how they influence the low-level landscape features of a set of 5 problems from the CEC2022 benchmark suite. Our results highlight that even relatively small transformations can significantly alter the measured landscape features. This poses a wider question of what properties we want to preserve when creating problem transformations, and how to fairly measure them.

Explainable Benchmarking for Iterative Optimization Heuristics

Benchmarking heuristic algorithms is vital to understand under which conditions and on what kind of problems certain algorithms perform well. In most current research into heuristic optimization algorithms, only a very limited number of scenarios, algorithm configurations and hyper-parameter settings are explored, leading to incomplete and often biased insights and results. This paper presents a novel approach we call explainable benchmarking. Introducing the IOH-Xplainer software framework, for analyzing and understanding the performance of various optimization algorithms and the impact of their different components and hyper-parameters. We showcase the framework in the context of two modular optimization frameworks. Through this framework, we examine the impact of different algorithmic components and configurations, offering insights into their performance across diverse scenarios. We provide a systematic method for evaluating and interpreting the behaviour and efficiency of iterative optimization heuristics in a more transparent and comprehensible manner, allowing for better benchmarking and algorithm design.

MA-BBOB: A Problem Generator for Black-Box Optimization Using Affine Combinations and Shifts

Choosing a set of benchmark problems is often a key component of any empirical evaluation of iterative optimization heuristics. In continuous, single-objective optimization, several sets of problems have become widespread, including the well-established BBOB suite. While this suite is designed to enable rigorous benchmarking, it is also commonly used for testing methods such as algorithm selection, which the suite was never designed around. We present the MA-BBOB function generator, which uses the BBOB suite as component functions in an affine combination. In this work, we describe the full procedure to create these affine combinations and highlight the trade-offs of several design decisions, specifically the choice to place the optimum uniformly at random in the domain. We then illustrate how this generator can be used to gain more low-level insight into the function landscapes through the use of exploratory landscape analysis. Finally, we show a potential use-case of MA-BBOB in generating a wide set of training and testing data for algorithm selectors. Using this setup, we show that the basic scheme of using a set of landscape features to predict the best algorithm does not lead to optimal results, and that an algorithm selector trained purely on the BBOB functions generalizes poorly to the affine combinations.