Choosing a suitable algorithm from the myriads of different search heuristics is difficult when faced with a novel optimization problem. In this work, we argue that the purely academic question of what could be the best possible algorithm in a certai
n broad class of black-box optimizers can give fruitful indications in which direction to search for good established optimization heuristics. We demonstrate this approach on the recently proposed DLB benchmark, for which the only known results are $O(n^3)$ runtimes for several classic evolutionary algorithms and an $O(n^2 log n)$ runtime for an estimation-of-distribution algorithm. Our finding that the unary unbiased black-box complexity is only $O(n^2)$ suggests the Metropolis algorithm as an interesting candidate and we prove that it solves the DLB problem in quadratic time. Since we also prove that better runtimes cannot be obtained in the class of unary unbiased algorithms, we shift our attention to algorithms that use the information of more parents to generate new solutions. An artificial algorithm of this type having an $O(n log n)$ runtime leads to the result that the significance-based compact genetic algorithm (sig-cGA) can solve the DLB problem also in time $O(n log n)$. Our experiments show a remarkably good performance of the Metropolis algorithm, clearly the best of all algorithms regarded for reasonable problem sizes.
Many real-world applications have the time-linkage property, and the only theoretical analysis is recently given by Zheng, et al. (TEVC 2021) on their proposed time-linkage OneMax problem, OneMax$_{(0,1^n)}$. However, only two elitist algorithms (1+1
)EA and ($mu$+1)EA are analyzed, and it is unknown whether the non-elitism mechanism could help to escape the local optima existed in OneMax$_{(0,1^n)}$. In general, there are few theoretical results on the benefits of the non-elitism in evolutionary algorithms. In this work, we analyze on the influence of the non-elitism via comparing the performance of the elitist (1+$lambda$)EA and its non-elitist counterpart (1,$lambda$)EA. We prove that with probability $1-o(1)$ (1+$lambda$)EA will get stuck in the local optima and cannot find the global optimum, but with probability $1$, (1,$lambda$)EA can reach the global optimum and its expected runtime is $O(n^{3+c}log n)$ with $lambda=c log_{frac{e}{e-1}} n$ for the constant $cge 1$. Noting that a smaller offspring size is helpful for escaping from the local optima, we further resort to the compact genetic algorithm where only two individuals are sampled to update the probabilistic model, and prove its expected runtime of $O(n^3log n)$. Our computational experiments also verify the efficiency of the two non-elitist algorithms.
Previous theory work on multi-objective evolutionary algorithms considers mostly easy problems that are composed of unimodal objectives. This paper takes a first step towards a deeper understanding of how evolutionary algorithms solve multi-modal mul
ti-objective problems. We propose the OneJumpZeroJump problem, a bi-objective problem whose single objectives are isomorphic to the classic jump functions benchmark. We prove that the simple evolutionary multi-objective optimizer (SEMO) cannot compute the full Pareto front. In contrast, for all problem sizes~$n$ and all jump sizes $k in [4..frac n2 - 1]$, the global SEMO (GSEMO) covers the Pareto front in $Theta((n-2k)n^{k})$ iterations in expectation. To improve the performance, we combine the GSEMO with two approaches, a heavy-tailed mutation operator and a stagnation detection strategy, that showed advantages in single-objective multi-modal problems. Runtime improvements of asymptotic order at least $k^{Omega(k)}$ are shown for both strategies. Our experiments verify the {substantial} runtime gains already for moderate problem sizes. Overall, these results show that the ideas recently developed for single-objective evolutionary algorithms can be effectively employed also in multi-objective optimization.
In real-world applications, many optimization problems have the time-linkage property, that is, the objective function value relies on the current solution as well as the historical solutions. Although the rigorous theoretical analysis on evolutionar
y algorithms has rapidly developed in recent two decades, it remains an open problem to theoretically understand the behaviors of evolutionary algorithms on time-linkage problems. This paper takes the first step to rigorously analyze evolutionary algorithms for time-linkage functions. Based on the basic OneMax function, we propose a time-linkage function where the first bit value of the last time step is integrated but has a different preference from the current first bit. We prove that with probability $1-o(1)$, randomized local search and $(1+1)$ EA cannot find the optimum, and with probability $1-o(1)$, $(mu+1)$ EA is able to reach the optimum.
One of the key difficulties in using estimation-of-distribution algorithms is choosing the population size(s) appropriately: Too small values lead to genetic drift, which can cause enormous difficulties. In the regime with no genetic drift, however,
often the runtime is roughly proportional to the population size, which renders large population sizes inefficient. Based on a recent quantitative analysis which population sizes lead to genetic drift, we propose a parameter-less version of the compact genetic algorithm that automatically finds a suitable population size without spending too much time in situations unfavorable due to genetic drift. We prove a mathematical runtime guarantee for this algorithm and conduct an extensive experimental analysis on four classic benchmark problems both without and with additive centered Gaussian posterior noise. The former shows that under a natural assumption, our algorithm has a performance very similar to the one obtainable from the best problem-specific population size. The latter confirms that missing the right population size in the original cGA can be detrimental and that previous theory-based suggestions for the population size can be far away from the right values; it also shows that our algorithm as well as a previously proposed parameter-less variant of the cGA based on parallel runs avoid such pitfalls. Comparing the two parameter-less approaches, ours profits from its ability to abort runs which are likely to be stuck in a genetic drift situation.
Estimation of Distribution Algorithms (EDAs) are one branch of Evolutionary Algorithms (EAs) in the broad sense that they evolve a probabilistic model instead of a population. Many existing algorithms fall into this category. Analogous to genetic dri
ft in EAs, EDAs also encounter the phenomenon that updates of the probabilistic model not justified by the fitness move the sampling frequencies to the boundary values. This can result in a considerable performance loss. This paper proves the first sharp estimates of the boundary hitting time of the sampling frequency of a neutral bit for several univariate EDAs. For the UMDA that selects $mu$ best individuals from $lambda$ offspring each generation, we prove that the expected first iteration when the frequency of the neutral bit leaves the middle range $[tfrac 14, tfrac 34]$ and the expected first time it is absorbed in 0 or 1 are both $Theta(mu)$. The corresponding hitting times are $Theta(K^2)$ for the cGA with hypothetical population size $K$. This paper further proves that for PBIL with parameters $mu$, $lambda$, and $rho$, in an expected number of $Theta(mu/rho^2)$ iterations the sampling frequency of a neutral bit leaves the interval $[Theta(rho/mu),1-Theta(rho/mu)]$ and then always the same value is sampled for this bit, that is, the frequency approaches the corresponding boundary value with maximum speed. For the lower bounds implicit in these statements, we also show exponential tail bounds. If a bit is not neutral, but neutral or has a preference for ones, then the lower bounds on the times to reach a low frequency value still hold. An analogous statement holds for bits that are neutral or prefer the value zero.
We conduct a first fundamental analysis of the working principles of binary differential evolution (BDE), an optimization heuristic for binary decision variables that was derived by Gong and Tuson (2007) from the very successful classic differential
evolution (DE) for continuous optimization. We show that unlike most other optimization paradigms, it is stable in the sense that neutral bit values are sampled with probability close to $1/2$ for a long time. This is generally a desirable property, however, it makes it harder to find the optima for decision variables with small influence on the objective function. This can result in an optimization time exponential in the dimension when optimizing simple symmetric functions like OneMax. On the positive side, BDE quickly detects and optimizes the most important decision variables. For example, dominant bits converge to the optimal value in time logarithmic in the population size. This enables BDE to optimize the most important bits very fast. Overall, our results indicate that BDE is an interesting optimization paradigm having characteristics significantly different from classic evolutionary algorithms or estimation-of-distribution algorithms (EDAs). On the technical side, we observe that the strong stochastic dependencies in the random experiment describing a run of BDE prevent us from proving all desired results with the mathematical rigor that was successfully used in the analysis of other evolutionary algorithms. Inspired by mean-field approaches in statistical physics we propose a more independent variant of BDE, show experimentally its similarity to BDE, and prove some statements rigorously only for the independent variant. Such a semi-rigorous approach might be interesting for other problems in evolutionary computation where purely mathematical methods failed so far.
