Near-Tight Runtime Guarantees for Many-Objective Evolutionary Algorithms

Despite significant progress in the field of mathematical runtime analysis of multi-objective evolutionary algorithms (MOEAs), the performance of MOEAs on discrete many-objective problems is little understood. In particular, the few existing bounds for the SEMO, global SEMO, and SMS-EMOA algorithms on classic benchmarks are all roughly quadratic in the size of the Pareto front. In this work, we prove near-tight runtime guarantees for these three algorithms on the four most common benchmark problems OneMinMax, CountingOnesCountingZeros, LeadingOnesTrailingZeros, and OneJumpZeroJump, and this for arbitrary numbers of objectives. Our bounds depend only linearly on the Pareto front size, showing that these MOEAs on these benchmarks cope much better with many objectives than what previous works suggested. Our bounds are tight apart from small polynomial factors in the number of objectives and length of bitstrings. This is the first time that such tight bounds are proven for many-objective uses of these MOEAs. While it is known that such results cannot hold for the NSGA-II, we do show that our bounds, via a recent structural result, transfer to the NSGA-III algorithm.

A Block-Coordinate Descent EMO Algorithm: Theoretical and Empirical Analysis

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.

Already Moderate Population Sizes Provably Yield Strong Robustness to Noise

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.

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.

The Runtime of Random Local Search on the Generalized Needle Problem

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.

Runtime Analysis of the SMS-EMOA for Many-Objective Optimization

The widely used multiobjective optimizer NSGA-II was recently proven to have considerable difficulties in many-objective optimization. In contrast, experimental results in the literature show a good performance of the SMS-EMOA, which can be seen as a steady-state NSGA-II that uses the hypervolume contribution instead of the crowding distance as the second selection criterion. This paper conducts the first rigorous runtime analysis of the SMS-EMOA for many-objective optimization. To this aim, we first propose a many-objective counterpart, the m-objective mOJZJ problem, of the bi-objective OJZJ benchmark, which is the first many-objective multimodal benchmark used in a mathematical runtime analysis. We prove that SMS-EMOA computes the full Pareto front of this benchmark in an expected number of $O(M^2 n^k)$ iterations, where $n$ denotes the problem size (length of the bit-string representation), $k$ the gap size (a difficulty parameter of the problem), and $M=(2n/m-2k+3)^{m/2}$ the size of the Pareto front. This result together with the existing negative result on the original NSGA-II shows that in principle, the general approach of the NSGA-II is suitable for many-objective optimization, but the crowding distance as tie-breaker has deficiencies. We obtain three additional insights on the SMS-EMOA. Different from a recent result for the bi-objective OJZJ benchmark, the stochastic population update often does not help for mOJZJ. It results in a $1/\Theta(\min\{Mk^{1/2}/2^{k/2},1\})$ speed-up, which is $\Theta(1)$ for large $m$ such as $m>k$. On the positive side, we prove that heavy-tailed mutation still results in a speed-up of order $k^{0.5+k-\beta}$. Finally, we conduct the first runtime analyses of the SMS-EMOA on the bi-objective OneMinMax and LOTZ benchmarks and show that it has a performance comparable to the GSEMO and the NSGA-II.

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.

The First Proven Performance Guarantees for the Non-Dominated Sorting Genetic Algorithm II (NSGA-II) on a Combinatorial Optimization Problem

The Non-dominated Sorting Genetic Algorithm-II (NSGA-II) is one of the most prominent algorithms to solve multi-objective optimization problems. Recently, the first mathematical runtime guarantees have been obtained for this algorithm, however only for synthetic benchmark problems. In this work, we give the first proven performance guarantees for a classic optimization problem, the NP-complete bi-objective minimum spanning tree problem. More specifically, we show that the NSGA-II with population size $N \ge 4((n-1) w_{\max} + 1)$ computes all extremal points of the Pareto front in an expected number of $O(m^2 n w_{\max} \log(n w_{\max}))$ iterations, where $n$ is the number of vertices, $m$ the number of edges, and $w_{\max}$ is the maximum edge weight in the problem instance. This result confirms, via mathematical means, the good performance of the NSGA-II observed empirically. It also shows that mathematical analyses of this algorithm are not only possible for synthetic benchmark problems, but also for more complex combinatorial optimization problems. As a side result, we also obtain a new analysis of the performance of the global SEMO algorithm on the bi-objective minimum spanning tree problem, which improves the previous best result by a factor of $|F|$, the number of extremal points of the Pareto front, a set that can be as large as $n w_{\max}$. The main reason for this improvement is our observation that both multi-objective evolutionary algorithms find the different extremal points in parallel rather than sequentially, as assumed in the previous proofs.

Runtime Analyses of Multi-Objective Evolutionary Algorithms in the Presence of Noise

In single-objective optimization, it is well known that evolutionary algorithms also without further adjustments can tolerate a certain amount of noise in the evaluation of the objective function. In contrast, this question is not at all understood for multi-objective optimization. In this work, we conduct the first mathematical runtime analysis of a simple multi-objective evolutionary algorithm (MOEA) on a classic benchmark in the presence of noise in the objective functions. We prove that when bit-wise prior noise with rate $p \le \alpha/n$, $\alpha$ a suitable constant, is present, the \emph{simple evolutionary multi-objective optimizer} (SEMO) without any adjustments to cope with noise finds the Pareto front of the OneMinMax benchmark in time $O(n^2\log n)$, just as in the case without noise. Given that the problem here is to arrive at a population consisting of $n+1$ individuals witnessing the Pareto front, this is a surprisingly strong robustness to noise (comparably simple evolutionary algorithms cannot optimize the single-objective OneMax problem in polynomial time when $p = \omega(\log(n)/n)$). Our proofs suggest that the strong robustness of the MOEA stems from its implicit diversity mechanism designed to enable it to compute a population covering the whole Pareto front. Interestingly this result only holds when the objective value of a solution is determined only once and the algorithm from that point on works with this, possibly noisy, objective value. We prove that when all solutions are reevaluated in each iteration, then any noise rate $p = \omega(\log(n)/n^2)$ leads to a super-polynomial runtime. This is very different from single-objective optimization, where it is generally preferred to reevaluate solutions whenever their fitness is important and where examples are known such that not reevaluating solutions can lead to catastrophic performance losses.