It is well known that evolutionary algorithms can benefit from dynamic choices of the key parameters that control their behavior, to adjust their search strategy to the different stages of the optimization process. A prominent example where dynamic parameter choices have shown a provable super-constant speed-up is the $(1+(λ,λ))$ Genetic Algorithm optimizing the OneMax function. While optimal parameter control policies result in linear expected running times, this is not possible with static parameter choices. This result has spurred a lot of interest in parameter control policies. However, many works, in particular theoretical running time analyses, focus on controlling one single parameter. Deriving policies for controlling multiple parameters remains very challenging. In this work we reconsider the problem of the $(1+(λ,λ))$ Genetic Algorithm optimizing OneMax. We decouple its four main parameters and investigate how well state-of-the-art deep reinforcement learning techniques can approximate good control policies. We show that although making deep reinforcement learning learn effectively is a challenging task, once it works, it is very powerful and is able to find policies that outperform all previously known control policies on the same benchmark. Based on the results found through reinforcement learning, we derive a simple control policy that consistently outperforms the default theory-recommended setting by $27\%$ and the irace-tuned policy, the strongest existing control policy on this benchmark, by $13\%$, for all tested problem sizes up to $40{,}000$.

Different from single-objective evolutionary algorithms, where non-elitism is an established concept, multi-objective evolutionary algorithms almost always select the next population in a greedy fashion. In the only notable exception, Bian, Zhou, Li, and Qian (IJCAI 2023) proposed a stochastic selection mechanism for the SMS-EMOA and proved that it can speed up computing the Pareto front of the bi-objective jump benchmark with problem size $n$ and gap parameter $k$ by a factor of $\max\{1,2^{k/4}/n\}$. While this constitutes the first proven speed-up from non-elitist selection, suggesting a very interesting research direction, it has to be noted that a true speed-up only occurs for $k \ge 4\log_2(n)$, where the runtime is super-polynomial, and that the advantage reduces for larger numbers of objectives as shown in a later work. In this work, we propose a different non-elitist selection mechanism based on aging, which exempts individuals younger than a certain age from a possible removal. This remedies the two shortcomings of stochastic selection: We prove a speed-up by a factor of $\max\{1,Θ(k)^{k-1}\}$, regardless of the number of objectives. In particular, a positive speed-up can already be observed for constant $k$, the only setting for which polynomial runtimes can be witnessed. Overall, this result supports the use of non-elitist selection schemes, but suggests that aging-based mechanisms can be considerably more powerful than stochastic selection mechanisms.

Recent theoretical works have shown that the NSGA-II can have enormous difficulties to solve problems with more than two objectives. In contrast, algorithms like the NSGA-III or SMS-EMOA, differing from the NSGA-II only in the secondary selection criterion, provably perform well in these situations. To remedy this shortcoming of the NSGA-II, but at the same time keep the advantages of the widely accepted crowding distance, we use the insights of these previous work to define a variant of the crowding distance, called truthful crowding distance. Different from the classic crowding distance, it has for any number of objectives the desirable property that a small crowding distance value indicates that some other solution has a similar objective vector. Building on this property, we conduct mathematical runtime analyses for the NSGA-II with truthful crowding distance. We show that this algorithm can solve the many-objective versions of the OneMinMax, COCZ, LOTZ, and OJZJ$_k$ problems in the same (polynomial) asymptotic runtimes as the NSGA-III and the SMS-EMOA. This contrasts the exponential lower bounds previously shown for the classic NSGA-II. For the bi-objective versions of these problems, our NSGA-II has a similar performance as the classic NSGA-II, gaining however from smaller admissible population sizes. For the bi-objective OneMinMax problem, we also observe a (minimally) better performance in approximating the Pareto front. These results suggest that our truthful version of the NSGA-II has the same good performance as the classic NSGA-II in two objectives, but can resolve the drastic problems in more than two objectives.

In recent work, Lissovoi, Oliveto, and Warwicker (Artificial Intelligence (2023)) proved that the Move Acceptance Hyper-Heuristic (MAHH) leaves the local optimum of the multimodal CLIFF benchmark with remarkable efficiency. The $O(n^3)$ runtime of the MAHH, for almost all cliff widths $d\ge 2,$ is significantly better than the $\Theta(n^d)$ runtime of simple elitist evolutionary algorithms (EAs) on CLIFF. In this work, we first show that this advantage is specific to the CLIFF problem and does not extend to the JUMP benchmark, the most prominent multi-modal benchmark in the theory of randomized search heuristics. We prove that for any choice of the MAHH selection parameter $p$, the expected runtime of the MAHH on a JUMP function with gap size $m = O(n^{1/2})$ is at least $\Omega(n^{2m-1} / (2m-1)!)$. This is significantly slower than the $O(n^m)$ runtime of simple elitist EAs. Encouragingly, we also show that replacing the local one-bit mutation operator in the MAHH with the global bit-wise mutation operator, commonly used in EAs, yields a runtime of $\min\{1, O(\frac{e\ln(n)}{m})^m\} \, O(n^m)$ on JUMP functions. This is at least as good as the runtime of simple elitist EAs. For larger values of $m$, this result proves an asymptotic performance gain over simple EAs. As our proofs reveal, the MAHH profits from its ability to walk through the valley of lower objective values in moderate-size steps, always accepting inferior solutions. This is the first time that such an optimization behavior is proven via mathematical means. Generally, our result shows that combining two ways of coping with local optima, global mutation and accepting inferior solutions, can lead to considerable performance gains.

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 $(1+\lambda)$ and $(1,\lambda)$ 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 $\lambda$ 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.

We consider whether conditions exist under which block-coordinate descent is asymptotically efficient in evolutionary multi-objective optimization, addressing an open problem. Block-coordinate descent, where an optimization problem is decomposed into $k$ blocks of decision variables and each of the blocks is optimized (with the others fixed) in a sequence, is a technique used in some large-scale optimization problems such as airline scheduling, however its use in multi-objective optimization is less studied. We propose a block-coordinate version of GSEMO and compare its running time to the standard GSEMO algorithm. Theoretical and empirical results on a bi-objective test function, a variant of LOTZ, serve to demonstrate the existence of cases where block-coordinate descent is faster. The result may yield wider insights into this class of algorithms.

In their recent work, C. Doerr and Krejca (Transactions on Evolutionary Computation, 2023) proved upper bounds on the expected runtime of the randomized local search heuristic on generalized Needle functions. Based on these upper bounds, they deduce in a not fully rigorous manner a drastic influence of the needle radius $k$ on the runtime. In this short article, we add the missing lower bound necessary to determine the influence of parameter $k$ on the runtime. To this aim, we derive an exact description of the expected runtime, which also significantly improves the upper bound given by C. Doerr and Krejca. We also describe asymptotic estimates of the expected runtime.

Very recently, the first mathematical runtime analyses of the multi-objective evolutionary optimizer NSGA-II have been conducted. We continue this line of research with a first runtime analysis of this algorithm on a benchmark problem consisting of two multimodal objectives. We prove that if the population size $N$ is at least four times the size of the Pareto front, then the NSGA-II with four different ways to select parents and bit-wise mutation optimizes the OneJumpZeroJump benchmark with jump size~$2 \le k \le n/4$ in time $O(N n^k)$. When using fast mutation, a recently proposed heavy-tailed mutation operator, this guarantee improves by a factor of $k^{\Omega(k)}$. Overall, this work shows that the NSGA-II copes with the local optima of the OneJumpZeroJump problem at least as well as the global SEMO algorithm.

Estimation-of-distribution algorithms (EDAs) are optimization algorithms that learn a distribution on the search space from which good solutions can be sampled easily. A key parameter of most EDAs is the sample size (population size). If the population size is too small, the update of the probabilistic model builds on few samples, leading to the undesired effect of genetic drift. Too large population sizes avoid genetic drift, but slow down the process. Building on a recent quantitative analysis of how the population size leads to genetic drift, we design a smart-restart mechanism for EDAs. By stopping runs when the risk for genetic drift is high, it automatically runs the EDA in good parameter regimes. Via a mathematical runtime analysis, we prove a general performance guarantee for this smart-restart scheme. This in particular shows that in many situations where the optimal (problem-specific) parameter values are known, the restart scheme automatically finds these, leading to the asymptotically optimal performance. We also conduct an extensive experimental analysis. On four classic benchmark problems, we clearly observe the critical influence of the population size on the performance, and we find that the smart-restart scheme leads to a performance close to the one obtainable with optimal parameter values. Our results also show that previous theory-based suggestions for the optimal population size can be far from the optimal ones, leading to a performance clearly inferior to the one obtained via the smart-restart scheme. We also conduct experiments with PBIL (cross-entropy algorithm) on two combinatorial optimization problems from the literature, the max-cut problem and the bipartition problem. Again, we observe that the smart-restart mechanism finds much better values for the population size than those suggested in the literature, leading to a much better performance.

Recent theoretical works have shown that the NSGA-II efficiently computes the full Pareto front when the population size is large enough. In this work, we study how well it approximates the Pareto front when the population size is smaller. For the OneMinMax benchmark, we point out situations in which the parents and offspring cover well the Pareto front, but the next population has large gaps on the Pareto front. Our mathematical proofs suggest as reason for this undesirable behavior that the NSGA-II in the selection stage computes the crowding distance once and then removes individuals with smallest crowding distance without considering that a removal increases the crowding distance of some individuals. We then analyze two variants not prone to this problem. For the NSGA-II that updates the crowding distance after each removal (Kukkonen and Deb (2006)) and the steady-state NSGA-II (Nebro and Durillo (2009)), we prove that the gaps in the Pareto front are never more than a small constant factor larger than the theoretical minimum. This is the first mathematical work on the approximation ability of the NSGA-II and the first runtime analysis for the steady-state NSGA-II. Experiments also show the superior approximation ability of the two NSGA-II variants.