A First Running Time Analysis of the Strength Pareto Evolutionary Algorithm 2 (SPEA2)

Evolutionary algorithms (EAs) have emerged as a predominant approach for addressing multi-objective optimization problems. However, the theoretical foundation of multi-objective EAs (MOEAs), particularly the fundamental aspects like running time analysis, remains largely underexplored. Existing theoretical studies mainly focus on basic MOEAs, with little attention given to practical MOEAs. In this paper, we present a running time analysis of strength Pareto evolutionary algorithm 2 (SPEA2) for the first time. Specifically, we prove that the expected running time of SPEA2 for solving three commonly used multi-objective problems, i.e., $m$OneMinMax, $m$LeadingOnesTrailingZeroes, and $m$-OneJumpZeroJump, is $O(\mu n\cdot \min\{m\log n, n\})$, $O(\mu n^2)$, and $O(\mu n^k \cdot \min\{mn, 3^{m/2}\})$, respectively. Here $m$ denotes the number of objectives, and the population size $\mu$ is required to be at least $(2n/m+1)^{m/2}$, $(2n/m+1)^{m-1}$ and $(2n/m-2k+3)^{m/2}$, respectively. The proofs are accomplished through general theorems which are also applicable for analyzing the expected running time of other MOEAs on these problems, and thus can be helpful for future theoretical analysis of MOEAs.

An Archive Can Bring Provable Speed-ups in Multi-Objective Evolutionary Algorithms

In the area of multi-objective evolutionary algorithms (MOEAs), there is a trend of using an archive to store non-dominated solutions generated during the search. This is because 1) MOEAs may easily end up with the final population containing inferior solutions that are dominated by other solutions discarded during the search process and 2) the population that has a commensurable size of the problem's Pareto front is often not practical. In this paper, we theoretically show, for the first time, that using an archive can guarantee speed-ups for MOEAs. Specifically, we prove that for two well-established MOEAs (NSGA-II and SMS-EMOA) on two commonly studied problems (OneMinMax and LeadingOnesTrailingZeroes), using an archive brings a polynomial acceleration on the expected running time. The reason is that with an archive, the size of the population can reduce to a small constant; there is no need for the population to keep all the Pareto optimal solutions found. This contrasts existing theoretical studies for MOEAs where a population with a commensurable size of the problem's Pareto front is needed. The findings in this paper not only provide a theoretical confirmation for an increasingly popular practice in the design of MOEAs, but can also be beneficial to the theory community towards studying more practical MOEAs.

Maintaining Diversity Provably Helps in Evolutionary Multimodal Optimization

In the real world, there exist a class of optimization problems that multiple (local) optimal solutions in the solution space correspond to a single point in the objective space. In this paper, we theoretically show that for such multimodal problems, a simple method that considers the diversity of solutions in the solution space can benefit the search in evolutionary algorithms (EAs). Specifically, we prove that the proposed method, working with crossover, can help enhance the exploration, leading to polynomial or even exponential acceleration on the expected running time. This result is derived by rigorous running time analysis in both single-objective and multi-objective scenarios, including $(\mu+1)$-GA solving the widely studied single-objective problem, Jump, and NSGA-II and SMS-EMOA (two well-established multi-objective EAs) solving the widely studied bi-objective problem, OneJumpZeroJump. Experiments are also conducted to validate the theoretical results. We hope that our results may encourage the exploration of diversity maintenance in the solution space for multi-objective optimization, where existing EAs usually only consider the diversity in the objective space and can easily be trapped in local optima.

Adapting Multi-objectivized Software Configuration Tuning

When tuning software configuration for better performance (e.g., latency or throughput), an important issue that many optimizers face is the presence of local optimum traps, compounded by a highly rugged configuration landscape and expensive measurements. To mitigate these issues, a recent effort has shifted to focus on the level of optimization model (called meta multi-objectivization or MMO) instead of designing better optimizers as in traditional methods. This is done by using an auxiliary performance objective, together with the target performance objective, to help the search jump out of local optima. While effective, MMO needs a fixed weight to balance the two objectives-a parameter that has been found to be crucial as there is a large deviation of the performance between the best and the other settings. However, given the variety of configurable software systems, the "sweet spot" of the weight can vary dramatically in different cases and it is not possible to find the right setting without time-consuming trial and error. In this paper, we seek to overcome this significant shortcoming of MMO by proposing a weight adaptation method, dubbed AdMMO. Our key idea is to adaptively adjust the weight at the right time during tuning, such that a good proportion of the nondominated configurations can be maintained. Moreover, we design a partial duplicate retention mechanism to handle the issue of too many duplicate configurations without losing the rich information provided by the "good" duplicates. Experiments on several real-world systems, objectives, and budgets show that, for 71% of the cases, AdMMO is considerably superior to MMO and a wide range of state-of-the-art optimizers while achieving generally better efficiency with the best speedup between 2.2x and 20x.

Stochastic Population Update Can Provably Be Helpful in Multi-Objective Evolutionary Algorithms

Evolutionary algorithms (EAs) have been widely and successfully applied to solve multi-objective optimization problems, due to their nature of population-based search. Population update is a key component in multi-objective EAs (MOEAs), and it is performed in a greedy, deterministic manner. That is, the next-generation population is formed by selecting the first population-size ranked solutions (based on some selection criteria, e.g., non-dominated sorting, crowdedness and indicators) from the collections of the current population and newly-generated solutions. In this paper, we question this practice. We analytically present that introducing randomness into the population update procedure in MOEAs can be beneficial for the search. More specifically, we prove that the expected running time of a well-established MOEA (SMS-EMOA) for solving a commonly studied bi-objective problem, OneJumpZeroJump, can be exponentially decreased if replacing its deterministic population update mechanism by a stochastic one. Empirical studies also verify the effectiveness of the proposed stochastic population update method. This work is an attempt to challenge a common practice for the population update in MOEAs. Its positive results, which might hold more generally, should encourage the exploration of developing new MOEAs in the area.

Non-Elitist Evolutionary Multi-Objective Optimisation: Proof-of-Principle Results

Elitism, which constructs the new population by preserving best solutions out of the old population and newly-generated solutions, has been a default way for population update since its introduction into multi-objective evolutionary algorithms (MOEAs) in the late 1990s. In this paper, we take an opposite perspective to conduct the population update in MOEAs by simply discarding elitism. That is, we treat the newly-generated solutions as the new population directly (so that all selection pressure comes from mating selection). We propose a simple non-elitist MOEA (called NE-MOEA) that only uses Pareto dominance sorting to compare solutions, without involving any diversity-related selection criterion. Preliminary experimental results show that NE-MOEA can compete with well-known elitist MOEAs (NSGA-II, SMS-EMOA and NSGA-III) on several combinatorial problems. Lastly, we discuss limitations of the proposed non-elitist algorithm and suggest possible future research directions.

Multi-Objective Archiving

Most multi-objective optimisation algorithms maintain an archive explicitly or implicitly during their search. Such an archive can be solely used to store high-quality solutions presented to the decision maker, but in many cases may participate in the search process (e.g., as the population in evolutionary computation). Over the last two decades, archiving, the process of comparing new solutions with previous ones and deciding how to update the archive/population, stands as an important issue in evolutionary multi-objective optimisation (EMO). This is evidenced by constant efforts from the community on developing various effective archiving methods, ranging from conventional Pareto-based methods to more recent indicator-based and decomposition-based ones. However, the focus of these efforts is on empirical performance comparison in terms of specific quality indicators; there is lack of systematic study of archiving methods from a general theoretical perspective. In this paper, we attempt to conduct a systematic overview of multi-objective archiving, in the hope of paving the way to understand archiving algorithms from a holistic perspective of theory and practice, and more importantly providing a guidance on how to design theoretically desirable and practically useful archiving algorithms. In doing so, we also present that archiving algorithms based on weakly Pareto compliant indicators (e.g., epsilon-indicator), as long as designed properly, can achieve the same theoretical desirables as archivers based on Pareto compliant indicators (e.g., hypervolume indicator). Such desirables include the property limit-optimal, the limit form of the possible optimal property that a bounded archiving algorithm can have with respect to the most general form of superiority between solution sets.

Do Performance Aspirations Matter for Guiding Software Configuration Tuning?

Configurable software systems can be tuned for better performance. Leveraging on some Pareto optimizers, recent work has shifted from tuning for a single, time-related performance objective to two intrinsically different objectives that assess distinct performance aspects of the system, each with varying aspirations. Before we design better optimizers, a crucial engineering decision to make therein is how to handle the performance requirements with clear aspirations in the tuning process. For this, the community takes two alternative optimization models: either quantifying and incorporating the aspirations into the search objectives that guide the tuning, or not considering the aspirations during the search but purely using them in the later decision-making process only. However, despite being a crucial decision that determines how an optimizer can be designed and tailored, there is a rather limited understanding of which optimization model should be chosen under what particular circumstance, and why. In this paper, we seek to close this gap. Firstly, we do that through a review of over 426 papers in the literature and 14 real-world requirements datasets. Drawing on these, we then conduct a comprehensive empirical study that covers 15 combinations of the state-of-the-art performance requirement patterns, four types of aspiration space, three Pareto optimizers, and eight real-world systems/environments, leading to 1,296 cases of investigation. We found that (1) the realism of aspirations is the key factor that determines whether they should be used to guide the tuning; (2) the given patterns and the position of the realistic aspirations in the objective landscape are less important for the choice, but they do matter to the extents of improvement; (3) the available tuning budget can also influence the choice for unrealistic aspirations but it is insignificant under realistic ones.

Multi-agent Dynamic Algorithm Configuration

Automated algorithm configuration relieves users from tedious, trial-and-error tuning tasks. A popular algorithm configuration tuning paradigm is dynamic algorithm configuration (DAC), in which an agent learns dynamic configuration policies across instances by reinforcement learning (RL). However, in many complex algorithms, there may exist different types of configuration hyperparameters, and such heterogeneity may bring difficulties for classic DAC which uses a single-agent RL policy. In this paper, we aim to address this issue and propose multi-agent DAC (MA-DAC), with one agent working for one type of configuration hyperparameter. MA-DAC formulates the dynamic configuration of a complex algorithm with multiple types of hyperparameters as a contextual multi-agent Markov decision process and solves it by a cooperative multi-agent RL (MARL) algorithm. To instantiate, we apply MA-DAC to a well-known optimization algorithm for multi-objective optimization problems. Experimental results show the effectiveness of MA-DAC in not only achieving superior performance compared with other configuration tuning approaches based on heuristic rules, multi-armed bandits, and single-agent RL, but also being capable of generalizing to different problem classes. Furthermore, we release the environments in this paper as a benchmark for testing MARL algorithms, with the hope of facilitating the application of MARL.

An Effective and Efficient Evolutionary Algorithm for Many-Objective Optimization

In evolutionary multi-objective optimization, effectiveness refers to how an evolutionary algorithm performs in terms of converging its solutions into the Pareto front and also diversifying them over the front. This is not an easy job, particularly for optimization problems with more than three objectives, dubbed many-objective optimization problems. In such problems, classic Pareto-based algorithms fail to provide sufficient selection pressure towards the Pareto front, whilst recently developed algorithms, such as decomposition-based ones, may struggle to maintain a set of well-distributed solutions on certain problems (e.g., those with irregular Pareto fronts). Another issue in some many-objective optimizers is rapidly increasing computational requirement with the number of objectives, such as hypervolume-based algorithms and shift-based density estimation (SDE) methods. In this paper, we aim to address this problem and develop an effective and efficient evolutionary algorithm (E3A) that can handle various many-objective problems. In E3A, inspired by SDE, a novel population maintenance method is proposed. We conduct extensive experiments and show that E3A performs better than 11 state-of-the-art many-objective evolutionary algorithms in quickly finding a set of well-converged and well-diversified solutions.