arXiv subjects

Carola Doerr

Publications and source records attributed to Carola Doerr.

At least 109 records · Page 6Linked to original sources

arXiv2020

Optimal Mutation Rates for the $(1+λ)$ EA on OneMax

The OneMax problem, alternatively known as the Hamming distance problem, is often referred to as the "drosophila of evolutionary computation (EC)", because of its high relevance in theoretical and empirical analyses of EC approaches. It is therefore surprising that even for the simplest of all mutation-based algorithms, Randomized Local Search and the (1+1) EA, the optimal mutation rates were determined only very recently, in a GECCO 2019 poster. In this work, we extend the analysis of optimal mutation rates to two variants of the $(1+λ)$ EA and to the $(1+λ)$ RLS. To do this, we use dynamic programming and, for the $(1+λ)$ EA, numeric optimization, both requiring $Θ(n^3)$ time for problem dimension $n$. With this in hand, we compute for all population sizes $λ\in \{2^i \mid 0 \le i \le 18\}$ and for problem dimension $n \in \{1000, 2000, 5000\}$ which mutation rates minimize the expected running time and which ones maximize the expected progress. Our results do not only provide a lower bound against which we can measure common evolutionary approaches, but we also obtain insight into the structure of these optimal parameter choices. For example, we show that, for large population sizes, the best number of bits to flip is not monotone in the distance to the optimum. We also observe that the expected remaining running time are not necessarily unimodal for the $(1+λ)$ EA$_{0 \rightarrow 1}$ with shifted mutation.

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arXiv2020

Hybridizing the 1/5-th Success Rule with Q-Learning for Controlling the Mutation Rate of an Evolutionary Algorithm

It is well known that evolutionary algorithms (EAs) achieve peak performance only when their parameters are suitably tuned to the given problem. Even more, it is known that the best parameter values can change during the optimization process. Parameter control mechanisms are techniques developed to identify and to track these values. Recently, a series of rigorous theoretical works confirmed the superiority of several parameter control techniques over EAs with best possible static parameters. Among these results are examples for controlling the mutation rate of the $(1+λ)$~EA when optimizing the OneMax problem. However, it was shown in [Rodionova et al., GECCO'19] that the quality of these techniques strongly depends on the offspring population size $λ$. We introduce in this work a new hybrid parameter control technique, which combines the well-known one-fifth success rule with Q-learning. We demonstrate that our HQL mechanism achieves equal or superior performance to all techniques tested in [Rodionova et al., GECCO'19] and this -- in contrast to previous parameter control methods -- simultaneously for all offspring population sizes $λ$. We also show that the promising performance of HQL is not restricted to OneMax, but extends to several other benchmark problems.

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arXiv2020

Exploratory Landscape Analysis is Strongly Sensitive to the Sampling Strategy

Exploratory landscape analysis (ELA) supports supervised learning approaches for automated algorithm selection and configuration by providing sets of features that quantify the most relevant characteristics of the optimization problem at hand. In black-box optimization, where an explicit problem representation is not available, the feature values need to be approximated from a small number of sample points. In practice, uniformly sampled random point sets and Latin hypercube constructions are commonly used sampling strategies. In this work, we analyze how the sampling method and the sample size influence the quality of the feature value approximations and how this quality impacts the accuracy of a standard classification task. While, not unexpectedly, increasing the number of sample points gives more robust estimates for the feature values, to our surprise we find that the feature value approximations for different sampling strategies do not converge to the same value. This implies that approximated feature values cannot be interpreted independently of the underlying sampling strategy. As our classification experiments show, this also implies that the feature approximations used for training a classifier must stem from the same sampling strategy as those used for the actual classification tasks. As a side result we show that classifiers trained with feature values approximated by Sobol' sequences achieve higher accuracy than any of the standard sampling techniques. This may indicate improvement potential for ELA-trained machine learning models.

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arXiv2020

Landscape-Aware Fixed-Budget Performance Regression and Algorithm Selection for Modular CMA-ES Variants

Automated algorithm selection promises to support the user in the decisive task of selecting a most suitable algorithm for a given problem. A common component of these machine-trained techniques are regression models which predict the performance of a given algorithm on a previously unseen problem instance. In the context of numerical black-box optimization, such regression models typically build on exploratory landscape analysis (ELA), which quantifies several characteristics of the problem. These measures can be used to train a supervised performance regression model. First steps towards ELA-based performance regression have been made in the context of a fixed-target setting. In many applications, however, the user needs to select an algorithm that performs best within a given budget of function evaluations. Adopting this fixed-budget setting, we demonstrate that it is possible to achieve high-quality performance predictions with off-the-shelf supervised learning approaches, by suitably combining two differently trained regression models. We test this approach on a very challenging problem: algorithm selection on a portfolio of very similar algorithms, which we choose from the family of modular CMA-ES algorithms.

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arXiv2020

Towards Dynamic Algorithm Selection for Numerical Black-Box Optimization: Investigating BBOB as a Use Case

One of the most challenging problems in evolutionary computation is to select from its family of diverse solvers one that performs well on a given problem. This algorithm selection problem is complicated by the fact that different phases of the optimization process require different search behavior. While this can partly be controlled by the algorithm itself, there exist large differences between algorithm performance. It can therefore be beneficial to swap the configuration or even the entire algorithm during the run. Long deemed impractical, recent advances in Machine Learning and in exploratory landscape analysis give hope that this dynamic algorithm configuration~(dynAC) can eventually be solved by automatically trained configuration schedules. With this work we aim at promoting research on dynAC, by introducing a simpler variant that focuses only on switching between different algorithms, not configurations. Using the rich data from the Black Box Optimization Benchmark~(BBOB) platform, we show that even single-switch dynamic Algorithm selection (dynAS) can potentially result in significant performance gains. We also discuss key challenges in dynAS, and argue that the BBOB-framework can become a useful tool in overcoming these.

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arXiv2020

Benchmarking a $(μ+λ)$ Genetic Algorithm with Configurable Crossover Probability

We investigate a family of $(μ+λ)$ Genetic Algorithms (GAs) which creates offspring either from mutation or by recombining two randomly chosen parents. By scaling the crossover probability, we can thus interpolate from a fully mutation-only algorithm towards a fully crossover-based GA. We analyze, by empirical means, how the performance depends on the interplay of population size and the crossover probability. Our comparison on 25 pseudo-Boolean optimization problems reveals an advantage of crossover-based configurations on several easy optimization tasks, whereas the picture for more complex optimization problems is rather mixed. Moreover, we observe that the ``fast'' mutation scheme with its are power-law distributed mutation strengths outperforms standard bit mutation on complex optimization tasks when it is combined with crossover, but performs worse in the absence of crossover. We then take a closer look at the surprisingly good performance of the crossover-based $(μ+λ)$ GAs on the well-known LeadingOnes benchmark problem. We observe that the optimal crossover probability increases with increasing population size $μ$. At the same time, it decreases with increasing problem dimension, indicating that the advantages of the crossover are not visible in the asymptotic view classically applied in runtime analysis. We therefore argue that a mathematical investigation for fixed dimensions might help us observe effects which are not visible when focusing exclusively on asymptotic performance bounds.

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arXiv2020

Variance Reduction for Better Sampling in Continuous Domains

Design of experiments, random search, initialization of population-based methods, or sampling inside an epoch of an evolutionary algorithm use a sample drawn according to some probability distribution for approximating the location of an optimum. Recent papers have shown that the optimal search distribution, used for the sampling, might be more peaked around the center of the distribution than the prior distribution modelling our uncertainty about the location of the optimum. We confirm this statement, provide explicit values for this reshaping of the search distribution depending on the population size $λ$ and the dimension $d$, and validate our results experimentally.

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arXiv2020

Initial Design Strategies and their Effects on Sequential Model-Based Optimization

Sequential model-based optimization (SMBO) approaches are algorithms for solving problems that require computationally or otherwise expensive function evaluations. The key design principle of SMBO is a substitution of the true objective function by a surrogate, which is used to propose the point(s) to be evaluated next. SMBO algorithms are intrinsically modular, leaving the user with many important design choices. Significant research efforts go into understanding which settings perform best for which type of problems. Most works, however, focus on the choice of the model, the acquisition function, and the strategy used to optimize the latter. The choice of the initial sampling strategy, however, receives much less attention. Not surprisingly, quite diverging recommendations can be found in the literature. We analyze in this work how the size and the distribution of the initial sample influences the overall quality of the efficient global optimization~(EGO) algorithm, a well-known SMBO approach. While, overall, small initial budgets using Halton sampling seem preferable, we also observe that the performance landscape is rather unstructured. We furthermore identify several situations in which EGO performs unfavorably against random sampling. Both observations indicate that an adaptive SMBO design could be beneficial, making SMBO an interesting test-bed for automated algorithm design.

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arXiv2020

Sequential vs. Integrated Algorithm Selection and Configuration: A Case Study for the Modular CMA-ES

When faced with a specific optimization problem, choosing which algorithm to use is always a tough task. Not only is there a vast variety of algorithms to select from, but these algorithms often are controlled by many hyperparameters, which need to be tuned in order to achieve the best performance possible. Usually, this problem is separated into two parts: algorithm selection and algorithm configuration. With the significant advances made in Machine Learning, however, these problems can be integrated into a combined algorithm selection and hyperparameter optimization task, commonly known as the CASH problem. In this work we compare sequential and integrated algorithm selection and configuration approaches for the case of selecting and tuning the best out of 4608 variants of the Covariance Matrix Adaptation Evolution Strategy (CMA-ES) tested on the Black Box Optimization Benchmark (BBOB) suite. We first show that the ranking of the modular CMA-ES variants depends to a large extent on the quality of the hyperparameters. This implies that even a sequential approach based on complete enumeration of the algorithm space will likely result in sub-optimal solutions. In fact, we show that the integrated approach manages to provide competitive results at a much smaller computational cost. We also compare two different mixed-integer algorithm configuration techniques, called irace and Mixed-Integer Parallel Efficient Global Optimization (MIP-EGO). While we show that the two methods differ significantly in their treatment of the exploration-exploitation balance, their overall performances are very similar.

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arXiv2019

One-Shot Decision-Making with and without Surrogates

One-shot decision making is required in situations in which we can evaluate a fixed number of solution candidates but do not have any possibility for further, adaptive sampling. Such settings are frequently encountered in neural network design, hyper-parameter optimization, and many simulation-based real-world optimization tasks, in which evaluations are costly and time sparse. It seems intuitive that well-distributed samples should be more meaningful in one-shot decision making settings than uniform or grid-based samples, since they show a better coverage of the decision space. In practice, quasi-random designs such as Latin Hypercube Samples and low-discrepancy point sets form indeed the state of the art, as confirmed by a number of recent studies and competitions. In this work we take a closer look into the correlation between the distribution of the quasi-random designs and their performance in one-shot decision making tasks, with the goal to investigate whether the assumed correlation between uniform distribution and performance can be confirmed. We study three different decision tasks: classic one-shot optimization (only the best sample matters), one-shot optimization with surrogates (allowing to use surrogate models for selecting a design that need not necessarily be one of the evaluated samples), and one-shot regression (i.e., function approximation, with minimization of mean squared error as objective). Our results confirm an advantage of low-discrepancy designs for all three settings. The overall correlation, however, is rather weak. We complement our study by evolving problem-specific samples that show significantly better performance for the regression task than the standard approaches based on low-discrepancy sequences, giving strong indication that significant performance gains over state-of-the-art one-shot sampling techniques are possible.

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arXiv2019

Optimization of Chance-Constrained Submodular Functions

Submodular optimization plays a key role in many real-world problems. In many real-world scenarios, it is also necessary to handle uncertainty, and potentially disruptive events that violate constraints in stochastic settings need to be avoided. In this paper, we investigate submodular optimization problems with chance constraints. We provide a first analysis on the approximation behavior of popular greedy algorithms for submodular problems with chance constraints. Our results show that these algorithms are highly effective when using surrogate functions that estimate constraint violations based on Chernoff bounds. Furthermore, we investigate the behavior of the algorithms on popular social network problems and show that high quality solutions can still be obtained even if there are strong restrictions imposed by the chance constraint.

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arXiv2019

Offspring Population Size Matters when Comparing Evolutionary Algorithms with Self-Adjusting Mutation Rates

We analyze the performance of the 2-rate $(1+λ)$ Evolutionary Algorithm (EA) with self-adjusting mutation rate control, its 3-rate counterpart, and a $(1+λ)$~EA variant using multiplicative update rules on the OneMax problem. We compare their efficiency for offspring population sizes ranging up to $λ=3,200$ and problem sizes up to $n=100,000$. Our empirical results show that the ranking of the algorithms is very consistent across all tested dimensions, but strongly depends on the population size. While for small values of $λ$ the 2-rate EA performs best, the multiplicative updates become superior for starting for some threshold value of $λ$ between 50 and 100. Interestingly, for population sizes around 50, the $(1+λ)$~EA with static mutation rates performs on par with the best of the self-adjusting algorithms. We also consider how the lower bound $p_{\min}$ for the mutation rate influences the efficiency of the algorithms. We observe that for the 2-rate EA and the EA with multiplicative update rules the more generous bound $p_{\min}=1/n^2$ gives better results than $p_{\min}=1/n$ when $λ$ is small. For both algorithms the situation reverses for large~$λ$.

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arXiv2019

Fast Re-Optimization via Structural Diversity

When a problem instance is perturbed by a small modification, one would hope to find a good solution for the new instance by building on a known good solution for the previous one. Via a rigorous mathematical analysis, we show that evolutionary algorithms, despite usually being robust problem solvers, can have unexpected difficulties to solve such re-optimization problems. When started with a random Hamming neighbor of the optimum, the (1+1) evolutionary algorithm takes $Ω(n^2)$ time to optimize the LeadingOnes benchmark function, which is the same asymptotic optimization time when started in a randomly chosen solution. There is hence no significant advantage from re-optimizing a structurally good solution. We then propose a way to overcome such difficulties. As our mathematical analysis reveals, the reason for this undesired behavior is that during the optimization structurally good solutions can easily be replaced by structurally worse solutions of equal or better fitness. We propose a simple diversity mechanism that prevents this behavior, thereby reducing the re-optimization time for LeadingOnes to $O(γδn)$, where $γ$ is the population size used by the diversity mechanism and $δ\le γ$ the Hamming distance of the new optimum from the previous solution. We show similarly fast re-optimization times for the optimization of linear functions with changing constraints and for the minimum spanning tree problem.

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arXiv2019

Online Selection of CMA-ES Variants

In the field of evolutionary computation, one of the most challenging topics is algorithm selection. Knowing which heuristics to use for which optimization problem is key to obtaining high-quality solutions. We aim to extend this research topic by taking a first step towards a selection method for adaptive CMA-ES algorithms. We build upon the theoretical work done by van Rijn \textit{et al.} [PPSN'18], in which the potential of switching between different CMA-ES variants was quantified in the context of a modular CMA-ES framework. We demonstrate in this work that their proposed approach is not very reliable, in that implementing the suggested adaptive configurations does not yield the predicted performance gains. We propose a revised approach, which results in a more robust fit between predicted and actual performance. The adaptive CMA-ES approach obtains performance gains on 18 out of 24 tested functions of the BBOB benchmark, with stable advantages of up to 23\%. An analysis of module activation indicates which modules are most crucial for the different phases of optimizing each of the 24 benchmark problems. The module activation also suggests that additional gains are possible when including the (B)IPOP modules, which we have excluded for this present work.

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arXiv2019

Hyper-Parameter Tuning for the (1+(λ,λ)) GA

It is known that the $(1+(λ,λ))$~Genetic Algorithm (GA) with self-adjusting parameter choices achieves a linear expected optimization time on OneMax if its hyper-parameters are suitably chosen. However, it is not very well understood how the hyper-parameter settings influences the overall performance of the $(1+(λ,λ))$~GA. Analyzing such multi-dimensional dependencies precisely is at the edge of what running time analysis can offer. To make a step forward on this question, we present an in-depth empirical study of the self-adjusting $(1+(λ,λ))$~GA and its hyper-parameters. We show, among many other results, that a 15\% reduction of the average running time is possible by a slightly different setup, which allows non-identical offspring population sizes of mutation and crossover phase, and more flexibility in the choice of mutation rate and crossover bias --a generalization which may be of independent interest. We also show indication that the parametrization of mutation rate and crossover bias derived by theoretical means for the static variant of the $(1+(λ,λ))$~GA extends to the non-static case.

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arXiv2019

Interpolating Local and Global Search by Controlling the Variance of Standard Bit Mutation

A key property underlying the success of evolutionary algorithms (EAs) is their global search behavior, which allows the algorithms to `jump' from a current state to other parts of the search space, thereby avoiding to get stuck in local optima. This property is obtained through a random choice of the radius at which offspring are sampled from previously evaluated solutions. It is well known that, thanks to this global search behavior, the probability that an EA using standard bit mutation finds a global optimum of an arbitrary function $f:\{0,1\}^n \to \mathbb{R}$ tends to one as the number of function evaluations grows. This advantage over heuristics using a fixed search radius, however, comes at the cost of using non-optimal step sizes also in those regimes in which the optimal rate is stable for a long time. This downside results in significant performance losses for many standard benchmark problems. We introduce in this work a simple way to interpolate between the random global search of EAs and their deterministic counterparts which sample from a fixed radius only. To this end, we introduce \emph{normalized standard bit mutation}, in which the binomial choice of the search radius is replaced by a normal distribution. Normalized standard bit mutation allows a straightforward way to control its variance, and hence the degree of randomness involved. We experiment with a self-adjusting choice of this variance, and demonstrate its effectiveness for the two classic benchmark problems LeadingOnes and OneMax. Our work thereby also touches a largely ignored question in discrete evolutionary computation: multi-dimensional parameter control.

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arXiv2018

The Query Complexity of a Permutation-Based Variant of Mastermind

We study the query complexity of a permutation-based variant of the guessing game Mastermind. In this variant, the secret is a pair $(z,π)$ which consists of a binary string $z \in \{0,1\}^n$ and a permutation $π$ of $[n]$. The secret must be unveiled by asking queries of the form $x \in \{0,1\}^n$. For each such query, we are returned the score \[ f_{z,π}(x):= \max \{ i \in [0..n]\mid \forall j \leq i: z_{π(j)} = x_{π(j)}\}\,;\] i.e., the score of $x$ is the length of the longest common prefix of $x$ and $z$ with respect to the order imposed by $π$. The goal is to minimize the number of queries needed to identify $(z,π)$. This problem originates from the study of black-box optimization heuristics, where it is known as the \textsc{LeadingOnes} problem. In this work, we prove matching upper and lower bounds for the deterministic and randomized query complexity of this game, which are $Θ(n \log n)$ and $Θ(n \log \log n)$, respectively.

cs.DS↗

arXiv2018

Towards a More Practice-Aware Runtime Analysis of Evolutionary Algorithms

Theory of evolutionary computation (EC) aims at providing mathematically founded statements about the performance of evolutionary algorithms (EAs). The predominant topic in this research domain is runtime analysis, which studies the time it takes a given EA to solve a given optimization problem. Runtime analysis has witnessed significant advances in the last couple of years, allowing us to compute precise runtime estimates for several EAs and several problems. Runtime analysis is, however (and unfortunately!), often judged by practitioners to be of little relevance for real applications of EAs. Several reasons for this claim exist. We address two of them in this present work: (1) EA implementations often differ from their vanilla pseudocode description, which, in turn, typically form the basis for runtime analysis. To close the resulting gap between empirically observed and theoretically derived performance estimates, we therefore suggest to take this discrepancy into account in the mathematical analysis and to adjust, for example, the cost assigned to the evaluation of search points that equal one of their direct parents (provided that this is easy to verify as is the case in almost all standard EAs). (2) Most runtime analysis results make statements about the expected time to reach an optimal solution (and possibly the distribution of this optimization time) only, thus explicitly or implicitly neglecting the importance of understanding how the function values evolve over time. We suggest to extend runtime statements to runtime profiles, covering the expected time needed to reach points of intermediate fitness values. As a direct consequence, we obtain a result showing that the greedy (2+1) GA of Sudholt [GECCO 2012] outperforms any unary unbiased black-box algorithm on OneMax.

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