Runtime Analysis for Multi-Objective Evolutionary Algorithms in Unbounded Integer Spaces

Randomized search heuristics have been applied successfully to a plethora of problems. This success is complemented by a large body of theoretical results. Unfortunately, the vast majority of these results regard problems with binary or continuous decision variables -- the theoretical analysis of randomized search heuristics for unbounded integer domains is almost nonexistent. To resolve this shortcoming, we start the runtime analysis of multi-objective evolutionary algorithms, which are among the most successful randomized search heuristics, for unbounded integer search spaces. We analyze single- and full-dimensional mutation operators with three different mutation strengths, namely changes by plus/minus one (unit strength), random changes following a law with exponential tails, and random changes following a power-law. The performance guarantees we prove on a recently proposed natural benchmark problem suggest that unit mutation strengths can be slow when the initial solutions are far from the Pareto front. When setting the expected change right (depending on the benchmark parameter and the distance of the initial solutions), the mutation strength with exponential tails yields the best runtime guarantees in our results -- however, with a wrong choice of this expectation, the performance guarantees quickly become highly uninteresting. With power-law mutation, which is an essentially parameter-less mutation operator, we obtain good results uniformly over all problem parameters and starting points. We complement our mathematical findings with experimental results that suggest that our bounds are not always tight. Most prominently, our experiments indicate that power-law mutation outperforms the one with exponential tails even when the latter uses a near-optimal parametrization. Hence, we suggest to favor power-law mutation for unknown problems in integer spaces.

Speeding Up the NSGA-II With a Simple Tie-Breaking Rule

The non-dominated sorting genetic algorithm~II (NSGA-II) is the most popular multi-objective optimization heuristic. Recent mathematical runtime analyses have detected two shortcomings in discrete search spaces, namely, that the NSGA-II has difficulties with more than two objectives and that it is very sensitive to the choice of the population size. To overcome these difficulties, we analyze a simple tie-breaking rule in the selection of the next population. Similar rules have been proposed before, but have found only little acceptance. We prove the effectiveness of our tie-breaking rule via mathematical runtime analyses on the classic OneMinMax, LeadingOnesTrailingZeros, and OneJumpZeroJump benchmarks. We prove that this modified NSGA-II can optimize the three benchmarks efficiently also for many objectives, in contrast to the exponential lower runtime bound previously shown for OneMinMax with three or more objectives. For the bi-objective problems, we show runtime guarantees that do not increase when moderately increasing the population size over the minimum admissible size. For example, for the OneJumpZeroJump problem with representation length $n$ and gap parameter $k$, we show a runtime guarantee of $O(\max\{n^{k+1},Nn\})$ function evaluations when the population size is at least four times the size of the Pareto front. For population sizes larger than the minimal choice $N = \Theta(n)$, this result improves considerably over the $\Theta(Nn^k)$ runtime of the classic NSGA-II.

Difficulties of the NSGA-II with the Many-Objective LeadingOnes Problem

The NSGA-II is the most prominent multi-objective evolutionary algorithm (cited more than 50,000 times). Very recently, a mathematical runtime analysis has proven that this algorithm can have enormous difficulties when the number of objectives is larger than two (Zheng, Doerr. IEEE Transactions on Evolutionary Computation (2024)). However, this result was shown only for the OneMinMax benchmark problem, which has the particularity that all solutions are on the Pareto front, a fact heavily exploited in the proof of this result. In this work, we show a comparable result for the LeadingOnesTrailingZeroes benchmark. This popular benchmark problem appears more natural in that most of its solutions are not on the Pareto front. With a careful analysis of the population dynamics of the NGSA-II optimizing this benchmark, we manage to show that when the population grows on the Pareto front, then it does so much faster by creating known Pareto optima than by spreading out on the Pareto front. Consequently, already when still a constant fraction of the Pareto front is unexplored, the crowding distance becomes the crucial selection mechanism, and thus the same problems arise as in the optimization of OneMinMax. With these and some further arguments, we show that the NSGA-II, with a population size by at most a constant factor larger than the Pareto front, cannot compute the Pareto front in less than exponential time.

Proven Runtime Guarantees for How the \moead Computes the Pareto Front From the Subproblem Solutions

The decomposition-based multi-objective evolutionary algorithm (MOEA/D) does not directly optimize a given multi-objective function $f$, but instead optimizes $N + 1$ single-objective subproblems of $f$ in a co-evolutionary manner. It maintains an archive of all non-dominated solutions found and outputs it as approximation to the Pareto front. Once the MOEA/D found all optima of the subproblems (the $g$-optima), it may still miss Pareto optima of $f$. The algorithm is then tasked to find the remaining Pareto optima directly by mutating the $g$-optima. In this work, we analyze for the first time how the MOEA/D with only standard mutation operators computes the whole Pareto front of the OneMinMax benchmark when the $g$-optima are a strict subset of the Pareto front. For standard bit mutation, we prove an expected runtime of $O(n N \log n + n^{n/(2N)} N \log n)$ function evaluations. Especially for the second, more interesting phase when the algorithm start with all $g$-optima, we prove an $\Omega(n^{(1/2)(n/N + 1)} \sqrt{N} 2^{-n/N})$ expected runtime. This runtime is super-polynomial if $N = o(n)$, since this leaves large gaps between the $g$-optima, which require costly mutations to cover. For power-law mutation with exponent $\beta \in (1, 2)$, we prove an expected runtime of $O\left(n N \log n + n^{\beta} \log n\right)$ function evaluations. The $O\left(n^{\beta} \log n\right)$ term stems from the second phase of starting with all $g$-optima, and it is independent of the number of subproblems $N$. This leads to a huge speedup compared to the lower bound for standard bit mutation. In general, our overall bound for power-law suggests that the MOEA/D performs best for $N = O(n^{\beta - 1})$, resulting in an $O(n^\beta \log n)$ bound. In contrast to standard bit mutation, smaller values of $N$ are better for power-law mutation, as it is capable of easily creating missing solutions.

A Flexible Evolutionary Algorithm With Dynamic Mutation Rate Archive

We propose a new, flexible approach for dynamically maintaining successful mutation rates in evolutionary algorithms using $k$-bit flip mutations. The algorithm adds successful mutation rates to an archive of promising rates that are favored in subsequent steps. Rates expire when their number of unsuccessful trials has exceeded a threshold, while rates currently not present in the archive can enter it in two ways: (i) via user-defined minimum selection probabilities for rates combined with a successful step or (ii) via a stagnation detection mechanism increasing the value for a promising rate after the current bit-flip neighborhood has been explored with high probability. For the minimum selection probabilities, we suggest different options, including heavy-tailed distributions. We conduct rigorous runtime analysis of the flexible evolutionary algorithm on the OneMax and Jump functions, on general unimodal functions, on minimum spanning trees, and on a class of hurdle-like functions with varying hurdle width that benefit particularly from the archive of promising mutation rates. In all cases, the runtime bounds are close to or even outperform the best known results for both stagnation detection and heavy-tailed mutations.

Superior Genetic Algorithms for the Target Set Selection Problem Based on Power-Law Parameter Choices and Simple Greedy Heuristics

The target set selection problem (TSS) asks for a set of vertices such that an influence spreading process started in these vertices reaches the whole graph. The current state of the art for this NP-hard problem are three recently proposed randomized search heuristics, namely a biased random-key genetic algorithm (BRKGA) obtained from extensive parameter tuning, a max-min ant system (MMAS), and a MMAS using Q-learning with a graph convolutional network. We show that the BRKGA with two simple modifications and without the costly parameter tuning obtains significantly better results. Our first modification is to simply choose all parameters of the BRKGA in each iteration randomly from a power-law distribution. The resulting parameterless BRKGA is already competitive with the tuned BRKGA, as our experiments on the previously used benchmarks show. We then add a natural greedy heuristic, namely to repeatedly discard small-degree vertices that are not necessary for reaching the whole graph. The resulting algorithm consistently outperforms all of the state-of-the-art algorithms. Besides providing a superior algorithm for the TSS problem, this work shows that randomized parameter choices and elementary greedy heuristics can give better results than complex algorithms and costly parameter tuning.

Bivariate Estimation-of-Distribution Algorithms Can Find an Exponential Number of Optima

Finding a large set of optima in a multimodal optimization landscape is a challenging task. Classical population-based evolutionary algorithms typically converge only to a single solution. While this can be counteracted by applying niching strategies, the number of optima is nonetheless trivially bounded by the population size. Estimation-of-distribution algorithms (EDAs) are an alternative, maintaining a probabilistic model of the solution space instead of a population. Such a model is able to implicitly represent a solution set far larger than any realistic population size. To support the study of how optimization algorithms handle large sets of optima, we propose the test function EqualBlocksOneMax (EBOM). It has an easy fitness landscape with exponentially many optima. We show that the bivariate EDA mutual-information-maximizing input clustering, without any problem-specific modification, quickly generates a model that behaves very similarly to a theoretically ideal model for EBOM, which samples each of the exponentially many optima with the same maximal probability. We also prove via mathematical means that no univariate model can come close to having this property: If the probability to sample an optimum is at least inverse-polynomial, there is a Hamming ball of logarithmic radius such that, with high probability, each sample is in this ball.

Estimation-of-Distribution Algorithms for Multi-Valued Decision Variables

With apparently all research on estimation-of-distribution algorithms (EDAs) concentrated on pseudo-Boolean optimization and permutation problems, we undertake the first steps towards using EDAs for problems in which the decision variables can take more than two values, but which are not permutation problems. To this aim, we propose a natural way to extend the known univariate EDAs to such variables. Different from a naive reduction to the binary case, it avoids additional constraints. Since understanding genetic drift is crucial for an optimal parameter choice, we extend the known quantitative analysis of genetic drift to EDAs for multi-valued variables. Roughly speaking, when the variables take $r$ different values, the time for genetic drift to become significant is $r$ times shorter than in the binary case. Consequently, the update strength of the probabilistic model has to be chosen $r$ times lower now. To investigate how desired model updates take place in this framework, we undertake a mathematical runtime analysis on the $r$-valued LeadingOnes problem. We prove that with the right parameters, the multi-valued UMDA solves this problem efficiently in $O(r\log(r)^2 n^2 \log(n))$ function evaluations. Overall, our work shows that EDAs can be adjusted to multi-valued problems, and it gives advice on how to set the main parameters.

Lasting Diversity and Superior Runtime Guarantees for the $(μ+1)$ Genetic Algorithm

Most evolutionary algorithms (EAs) used in practice employ crossover. In contrast, only for few and mostly artificial examples a runtime advantage from crossover could be proven with mathematical means. The most convincing such result shows that the $(\mu+1)$ genetic algorithm (GA) with population size $\mu=O(n)$ optimizes jump functions with gap size $k \ge 3$ in time $O(n^k / \mu + n^{k-1}\log n)$, beating the $\Theta(n^k)$ runtime of many mutation-based EAs. This result builds on a proof that the GA occasionally and then for an expected number of $\Omega(\mu^2)$ iterations has a population that is not dominated by a single genotype. In this work, we show that this diversity persist with high probability for a time exponential in $\mu$ (instead of quadratic). From this better understanding of the population diversity, we obtain stronger runtime guarantees, among them the statement that for all $c\ln(n)\le\mu \le n/\log n$, with $c$ a suitable constant, the runtime of the $(\mu+1)$ GA on $\mathrm{Jump}_k$, with $k \ge 3$, is $O(n^{k-1})$. Consequently, already with logarithmic population sizes, the GA gains a speed-up of order $\Omega(n)$ from crossover.

Run Time Analysis for Random Local Search on Generalized Majority Functions

Run time analysis of evolutionary algorithms recently makes significant progress in linking algorithm performance to algorithm parameters. However, settings that study the impact of problem parameters are rare. The recently proposed W-model provides a good framework for such analyses, generating pseudo-Boolean optimization problems with tunable properties. We initiate theoretical research of the W-model by studying how one of its properties -- neutrality -- influences the run time of random local search. Neutrality creates plateaus in the search space by first performing a majority vote for subsets of the solution candidate and then evaluating the smaller-dimensional string via a low-level fitness function. We prove upper bounds for the expected run time of random local search on this MAJORITY problem for its entire parameter spectrum. To this end, we provide a theorem, applicable to many optimization algorithms, that links the run time of MAJORITY with its symmetric version HASMAJORITY, where a sufficient majority is needed to optimize the subset. We also introduce a generalized version of classic drift theorems as well as a generalized version of Wald's equation, both of which we believe to be of independent interest.