Denis Antipov

Denis Antipov, Aneta Neumann, Frank Neumann

The evolutionary diversity optimization aims at finding a diverse set of solutions which satisfy some constraint on their fitness. In the context of multi-objective optimization this constraint can require solutions to be Pareto-optimal. In this paper we study how the GSEMO algorithm with additional diversity-enhancing heuristic optimizes a diversity of its population on a bi-objective benchmark problem OneMinMax, for which all solutions are Pareto-optimal. We provide a rigorous runtime analysis of the last step of the optimization, when the algorithm starts with a population with a second-best diversity, and prove that it finds a population with optimal diversity in expected time $O(n^2)$, when the problem size $n$ is odd. For reaching our goal, we analyse the random walk of the population, which reflects the frequency of changes in the population and their outcomes.

Denis Antipov, Aneta Neumann, Frank Neumann

The main goal of diversity optimization is to find a diverse set of solutions which satisfy some lower bound on their fitness. Evolutionary algorithms (EAs) are often used for such tasks, since they are naturally designed to optimize populations of solutions. This approach to diversity optimization, called EDO, has been previously studied from theoretical perspective, but most studies considered only EAs with a trivial offspring population such as the $(μ+ 1)$ EA. In this paper we give an example instance of a $k$-vertex cover problem, which highlights a critical difference of the diversity optimization from the regular single-objective optimization, namely that there might be a locally optimal population from which we can escape only by replacing at least two individuals at once, which the $(μ+ 1)$ algorithms cannot do. We also show that the $(μ+ λ)$ EA with $λ\ge μ$ can effectively find a diverse population on $k$-vertex cover, if using a mutation operator inspired by Branson and Sutton (TCS 2023). To avoid the problem of subset selection which arises in the $(μ+ λ)$ EA when it optimizes diversity, we also propose the $(1_μ+ 1_μ)$ EA$_D$, which is an analogue of the $(1 + 1)$ EA for populations, and which is also efficient at optimizing diversity on the $k$-vertex cover problem.

Denis Antipov, Carola Doerr

Online algorithm selection (OAS) aims to adapt the optimization process to changes in the fitness landscape and is expected to outperform any single algorithm from a given portfolio. Although this expectation is supported by numerous empirical studies, there are currently no theoretical results proving that OAS can yield asymptotic speedups (apart from some artificial examples for hyper-heuristics). Moreover, theory-based guidelines for when and how to switch between algorithms are largely missing. In this paper, we present the first theoretical example in which switching between two algorithms -- the $(1+λ)$ EA and the $(1+(λ,λ))$ GA -- solves the OneMax problem asymptotically faster than either algorithm used in isolation. We show that an appropriate choice of population sizes for the two algorithms allows the optimum to be reached in $O(n\log\log n)$ expected time, faster than the $Θ(n\sqrt{\frac{\log n \log\log\log n}{\log\log n}})$ runtime of the best of these two algorithms with optimally tuned parameters. We first establish this bound under an idealized switching rule that changes from the $(1+λ)$ to the $(1+(λ,λ))$ GA at the optimal time. We then propose a realistic switching strategy that achieves the same performance. Our analysis combines fixed-start and fixed-target perspectives, illustrating how different algorithms dominate at different stages of the optimization process. This approach offers a promising path toward a deeper theoretical understanding of OAS.

Ishara Hewa Pathiranage, Frank Neumann, Denis Antipov et al.

Real-world optimization problems often involve stochastic and dynamic components. Evolutionary algorithms are particularly effective in these scenarios, as they can easily adapt to uncertain and changing environments but often uncertainty and dynamic changes are studied in isolation. In this paper, we explore the use of 3-objective evolutionary algorithms for the chance constrained knapsack problem with dynamic constraints. In our setting, the weights of the items are stochastic and the knapsack's capacity changes over time. We introduce a 3-objective formulation that is able to deal with the stochastic and dynamic components at the same time and is independent of the confidence level required for the constraint. This new approach is then compared to the 2-objective formulation which is limited to a single confidence level. We evaluate the approach using two different multi-objective evolutionary algorithms (MOEAs), namely the global simple evolutionary multi-objective optimizer (GSEMO) and the multi-objective evolutionary algorithm based on decomposition (MOEA/D), across various benchmark scenarios. Our analysis highlights the advantages of the 3-objective formulation over the 2-objective formulation in addressing the dynamic chance constrained knapsack problem.

Denis Antipov, Benjamin Doerr, Alexandra Ivanova

Experience shows that typical evolutionary algorithms can cope well with stochastic disturbances such as noisy function evaluations. In this first mathematical runtime analysis of the $(1+λ)$ and $(1,λ)$ evolutionary algorithms in the presence of prior bit-wise noise, we show that both algorithms can tolerate constant noise probabilities without increasing the asymptotic runtime on the OneMax benchmark. For this, a population size $λ$ suffices that is at least logarithmic in the problem size $n$. The only previous result in this direction regarded the less realistic one-bit noise model, required a population size super-linear in the problem size, and proved a runtime guarantee roughly cubic in the noiseless runtime for the OneMax benchmark. Our significantly stronger results are based on the novel proof argument that the noiseless offspring can be seen as a biased uniform crossover between the parent and the noisy offspring. We are optimistic that the technical lemmas resulting from this insight will find applications also in future mathematical runtime analyses of evolutionary algorithms.

Andre Opris, Denis Antipov

Parent selection methods are widely used in evolutionary computation to accelerate the optimization process, yet their theoretical benefits are still poorly understood. In this paper, we address this gap by incorporating different parent selection strategies into the $(μ+1)$ genetic algorithm (GA). We show that, with an appropriately chosen population size and a parent selection strategy that selects a pair of maximally distant parents with probability $Ω(1)$ for crossover, the resulting algorithm solves the Jump$_k$ problem in $O(k4^kn\log(n))$ expected time. This bound is significantly smaller than the best known bound of $O(nμ\log(μ)+n\log(n)+n^{k-1})$ for any $(μ+1)$~GA using no explicit diversity-preserving mechanism and a constant crossover probability. To establish this result, we introduce a novel diversity metric that captures both the maximum distance between pairs of individuals in the population and the number of pairs achieving this distance. The crucial point of our analysis is that it relies on crossover as a mechanism for creating and maintaining diversity throughout the run, rather than using crossover only in the final step to combine already diversified individuals, as it has been done in many previous works. The insights provided by our analysis contribute to a deeper theoretical understanding of the role of crossover in the population dynamics of genetic algorithms.

Denis Antipov, Aneta Neumann, Frank Neumann et al.

The diversity optimization is the class of optimization problems, in which we aim at finding a diverse set of good solutions. One of the frequently used approaches to solve such problems is to use evolutionary algorithms which evolve a desired diverse population. This approach is called evolutionary diversity optimization (EDO). In this paper, we analyse EDO on a 3-objective function LOTZ$_k$, which is a modification of the 2-objective benchmark function (LeadingOnes, TrailingZeros). We prove that the GSEMO computes a set of all Pareto-optimal solutions in $O(kn^3)$ expected iterations. We also analyze the runtime of the GSEMO$_D$ (a modification of the GSEMO for diversity optimization) until it finds a population with the best possible diversity for two different diversity measures, the total imbalance and the sorted imbalances vector. For the first measure we show that the GSEMO$_D$ optimizes it asymptotically faster than it finds a Pareto-optimal population, in $O(kn^2\log(n))$ expected iterations, and for the second measure we show an upper bound of $O(k^2n^3\log(n))$ expected iterations. We complement our theoretical analysis with an empirical study, which shows a very similar behavior for both diversity measures that is close to the theory predictions.

Alexandra Ivanova, Denis Antipov, Benjamin Doerr

Evolutionary algorithms are known to be robust to noise in the evaluation of the fitness. In particular, larger offspring population sizes often lead to strong robustness. We analyze to what extent the $(1+(λ,λ))$ genetic algorithm is robust to noise. This algorithm also works with larger offspring population sizes, but an intermediate selection step and a non-standard use of crossover as repair mechanism could render this algorithm less robust than, e.g., the simple $(1+λ)$ evolutionary algorithm. Our experimental analysis on several classic benchmark problems shows that this difficulty does not arise. Surprisingly, in many situations this algorithm is even more robust to noise than the $(1+λ)$~EA.

Denis Antipov, Maxim Buzdalov, Benjamin Doerr

Most evolutionary algorithms have multiple parameters and their values drastically affect the performance. Due to the often complicated interplay of the parameters, setting these values right for a particular problem (parameter tuning) is a challenging task. This task becomes even more complicated when the optimal parameter values change significantly during the run of the algorithm since then a dynamic parameter choice (parameter control) is necessary. In this work, we propose a lazy but effective solution, namely choosing all parameter values (where this makes sense) in each iteration randomly from a suitably scaled power-law distribution. To demonstrate the effectiveness of this approach, we perform runtime analyses of the $(1+(λ,λ))$ genetic algorithm with all three parameters chosen in this manner. We show that this algorithm on the one hand can imitate simple hill-climbers like the $(1+1)$ EA, giving the same asymptotic runtime on problems like OneMax, LeadingOnes, or Minimum Spanning Tree. On the other hand, this algorithm is also very efficient on jump functions, where the best static parameters are very different from those necessary to optimize simple problems. We prove a performance guarantee that is comparable to the best performance known for static parameters. For the most interesting case that the jump size $k$ is constant, we prove that our performance is asymptotically better than what can be obtained with any static parameter choice. We complement our theoretical results with a rigorous empirical study confirming what the asymptotic runtime results suggest.

Denis Antipov, Maxim Buzdalov, Benjamin Doerr

The mathematical runtime analysis of evolutionary algorithms traditionally regards the time an algorithm needs to find a solution of a certain quality when initialized with a random population. In practical applications it may be possible to guess solutions that are better than random ones. We start a mathematical runtime analysis for such situations. We observe that different algorithms profit to a very different degree from a better initialization. We also show that the optimal parameterization of the algorithm can depend strongly on the quality of the initial solutions. To overcome this difficulty, self-adjusting and randomized heavy-tailed parameter choices can be profitable. Finally, we observe a larger gap between the performance of the best evolutionary algorithm we found and the corresponding black-box complexity. This could suggest that evolutionary algorithms better exploiting good initial solutions are still to be found. These first findings stem from analyzing the performance of the $(1+1)$ evolutionary algorithm and the static, self-adjusting, and heavy-tailed $(1 + (λ,λ))$ GA on the OneMax benchmark. We are optimistic that the question how to profit from good initial solutions is interesting beyond these first examples.

Denis Antipov, Benjamin Doerr

It was recently observed that the $(1+(λ,λ))$ genetic algorithm can comparably easily escape the local optimum of the jump functions benchmark. Consequently, this algorithm can optimize the jump function with jump size $k$ in an expected runtime of only $n^{(k + 1)/2}k^{-k/2}e^{O(k)}$ fitness evaluations (Antipov, Doerr, Karavaev (GECCO 2020)). To obtain this performance, however, a non-standard parameter setting depending on the jump size $k$ was used. To overcome this difficulty, we propose to choose two parameters of the $(1+(λ,λ))$ genetic algorithm randomly from a power-law distribution. Via a mathematical runtime analysis, we show that this algorithm with natural instance-independent choices of the distribution parameters on all jump functions with jump size at most $n/4$ has a performance close to what the best instance-specific parameters in the previous work obtained. This price for instance-independence can be made as small as an $O(n\log(n))$ factor. Given the difficulty of the jump problem and the runtime losses from using mildly suboptimal fixed parameters (also discussed in this work), this appears to be a fair price.

Denis Antipov, Benjamin Doerr, Vitalii Karavaev

The $(1 + (λ,λ))$ genetic algorithm is a younger evolutionary algorithm trying to profit also from inferior solutions. Rigorous runtime analyses on unimodal fitness functions showed that it can indeed be faster than classical evolutionary algorithms, though on these simple problems the gains were only moderate. In this work, we conduct the first runtime analysis of this algorithm on a multimodal problem class, the jump functions benchmark. We show that with the right parameters, the \ollga optimizes any jump function with jump size $2 \le k \le n/4$ in expected time $O(n^{(k+1)/2} e^{O(k)} k^{-k/2})$, which significantly and already for constant~$k$ outperforms standard mutation-based algorithms with their $Θ(n^k)$ runtime and standard crossover-based algorithms with their $\tilde{O}(n^{k-1})$ runtime guarantee. For the isolated problem of leaving the local optimum of jump functions, we determine provably optimal parameters that lead to a runtime of $(n/k)^{k/2} e^{Θ(k)}$. This suggests some general advice on how to set the parameters of the \ollga, which might ease the further use of this algorithm.

Denis Antipov, Maxim Buzdalov, Benjamin Doerr

The heavy-tailed mutation operator proposed in Doerr, Le, Makhmara, and Nguyen (GECCO 2017), called \emph{fast mutation} to agree with the previously used language, so far was proven to be advantageous only in mutation-based algorithms. There, it can relieve the algorithm designer from finding the optimal mutation rate and nevertheless obtain a performance close to the one that the optimal mutation rate gives. In this first runtime analysis of a crossover-based algorithm using a heavy-tailed choice of the mutation rate, we show an even stronger impact. For the $(1+(λ,λ))$ genetic algorithm optimizing the OneMax benchmark function, we show that with a heavy-tailed mutation rate a linear runtime can be achieved. This is asymptotically faster than what can be obtained with any static mutation rate, and is asymptotically equivalent to the runtime of the self-adjusting version of the parameters choice of the $(1+(λ,λ))$ genetic algorithm. This result is complemented by an empirical study which shows the effectiveness of the fast mutation also on random satisfiable Max-3SAT instances.

Denis Antipov, Benjamin Doerr, Quentin Yang

Understanding when evolutionary algorithms are efficient or not, and how they efficiently solve problems, is one of the central research tasks in evolutionary computation. In this work, we make progress in understanding the interplay between parent and offspring population size of the $(μ,λ)$ EA. Previous works, roughly speaking, indicate that for $λ\ge (1+\varepsilon) e μ$, this EA easily optimizes the OneMax function, whereas an offspring population size $λ\le (1 -\varepsilon) e μ$ leads to an exponential runtime. Motivated also by the observation that in the efficient regime the $(μ,λ)$ EA loses its ability to escape local optima, we take a closer look into this phase transition. Among other results, we show that when $μ\le n^{1/2 - c}$ for any constant $c > 0$, then for any $λ\le e μ$ we have a super-polynomial runtime. However, if $μ\ge n^{2/3 + c}$, then for any $λ\ge e μ$, the runtime is polynomial. For the latter result we observe that the $(μ,λ)$ EA profits from better individuals also because these, by creating slightly worse offspring, stabilize slightly sub-optimal sub-populations. While these first results close to the phase transition do not yet give a complete picture, they indicate that the boundary between efficient and super-polynomial is not just the line $λ= e μ$, and that the reasons for efficiency or not are more complex than what was known so far.

Denis Antipov, Benjamin Doerr

Despite significant progress in the theory of evolutionary algorithms, the theoretical understanding of evolutionary algorithms which use non-trivial populations remains challenging and only few rigorous results exist. Already for the most basic problem, the determination of the asymptotic runtime of the $(μ+λ)$ evolutionary algorithm on the simple OneMax benchmark function, only the special cases $μ=1$ and $λ=1$ have been solved. In this work, we analyze this long-standing problem and show the asymptotically tight result that the runtime $T$, the number of iterations until the optimum is found, satisfies \[E[T] = Θ\bigg(\frac{n\log n}λ+\frac{n}{λ/ μ} + \frac{n\log^+\log^+ λ/ μ}{\log^+ λ/ μ}\bigg),\] where $\log^+ x := \max\{1, \log x\}$ for all $x > 0$. The same methods allow to improve the previous-best $O(\frac{n \log n}λ + n \log λ)$ runtime guarantee for the $(λ+λ)$~EA with fair parent selection to a tight $Θ(\frac{n \log n}λ + n)$ runtime result.

Denis Antipov, Benjamin Doerr

To gain a better theoretical understanding of how evolutionary algorithms (EAs) cope with plateaus of constant fitness, we propose the $n$-dimensional Plateau$_k$ function as natural benchmark and analyze how different variants of the $(1 + 1)$ EA optimize it. The Plateau$_k$ function has a plateau of second-best fitness in a ball of radius $k$ around the optimum. As evolutionary algorithm, we regard the $(1 + 1)$ EA using an arbitrary unbiased mutation operator. Denoting by $α$ the random number of bits flipped in an application of this operator and assuming that $\Pr[α= 1]$ has at least some small sub-constant value, we show the surprising result that for all constant $k \ge 2$, the runtime $T$ follows a distribution close to the geometric one with success probability equal to the probability to flip between $1$ and $k$ bits divided by the size of the plateau. Consequently, the expected runtime is the inverse of this number, and thus only depends on the probability to flip between $1$ and $k$ bits, but not on other characteristics of the mutation operator. Our result also implies that the optimal mutation rate for standard bit mutation here is approximately $k/(en)$. Our main analysis tool is a combined analysis of the Markov chains on the search point space and on the Hamming level space, an approach that promises to be useful also for other plateau problems.