arXiv subjects

Per Kristian Lehre

Publications and source records attributed to Per Kristian Lehre.

At least 19 recordsLinked to original sources

arXiv2025

Runtime Analysis of Competitive co-Evolutionary Algorithms for Maximin Optimisation of a Bilinear Function

Co-evolutionary algorithms have a wide range of applications, such as in hardware design, evolution of strategies for board games, and patching software bugs. However, these algorithms are poorly understood and applications are often limited by pathological behaviour, such as loss of gradient, relative over-generalisation, and mediocre objective stasis. It is an open challenge to develop a theory that can predict when co-evolutionary algorithms find solutions efficiently and reliable. This paper provides a first step in developing runtime analysis for population-based competitive co-evolutionary algorithms. We provide a mathematical framework for describing and reasoning about the performance of co-evolutionary processes. To illustrate the framework, we introduce a population-based co-evolutionary algorithm called \pdcoea, and prove that it obtains a solution to a bilinear maximin optimisation problem in expected polynomial time. Finally, we describe settings where \pdcoea needs exponential time with overwhelmingly high probability to obtain a solution.

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arXiv2025

A General Upper Bound for the Runtime of a Coevolutionary Algorithm on Impartial Combinatorial Games

Due to their complex dynamics, combinatorial games are a key test case and application for algorithms that train game playing agents. Among those algorithms that train using self-play are coevolutionary algorithms (CoEAs). However, the successful application of CoEAs for game playing is difficult due to pathological behaviours such as cycling, an issue especially critical for games with intransitive payoff landscapes. Insight into how to design CoEAs to avoid such behaviours can be provided by runtime analysis. In this paper, we push the scope of runtime analysis for CoEAs to combinatorial games, proving a general upper bound for the number of simulated games needed for UMDA to discover (with high probability) an optimal strategy. This result applies to any impartial combinatorial game, and for many games the implied bound is polynomial or quasipolynomial as a function of the number of game positions. After proving the main result, we provide several applications to simple well-known games: Nim, Chomp, Silver Dollar, and Turning Turtles. As the first runtime analysis for CoEAs on combinatorial games, this result is a critical step towards a comprehensive theoretical framework for coevolution.

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arXiv2024

Overcoming Binary Adversarial Optimisation with Competitive Coevolution

Co-evolutionary algorithms (CoEAs), which pair candidate designs with test cases, are frequently used in adversarial optimisation, particularly for binary test-based problems where designs and tests yield binary outcomes. The effectiveness of designs is determined by their performance against tests, and the value of tests is based on their ability to identify failing designs, often leading to more sophisticated tests and improved designs. However, CoEAs can exhibit complex, sometimes pathological behaviours like disengagement. Through runtime analysis, we aim to rigorously analyse whether CoEAs can efficiently solve test-based adversarial optimisation problems in an expected polynomial runtime. This paper carries out the first rigorous runtime analysis of $(1,λ)$ CoEA for binary test-based adversarial optimisation problems. In particular, we introduce a binary test-based benchmark problem called \Diagonal problem and initiate the first runtime analysis of competitive CoEA on this problem. The mathematical analysis shows that the $(1,λ)$-CoEA can efficiently find an $\varepsilon$ approximation to the optimal solution of the \Diagonal problem, i.e. in expected polynomial runtime assuming sufficiently low mutation rates and large offspring population size. On the other hand, the standard $(1,λ)$-EA fails to find an $\varepsilon$ approximation to the optimal solution of the \Diagonal problem in polynomial runtime. This suggests the promising potential of coevolution for solving binary adversarial optimisation problems.

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arXiv2024

Concentration Tail-Bound Analysis of Coevolutionary and Bandit Learning Algorithms

Runtime analysis, as a branch of the theory of AI, studies how the number of iterations algorithms take before finding a solution (its runtime) depends on the design of the algorithm and the problem structure. Drift analysis is a state-of-the-art tool for estimating the runtime of randomised algorithms, such as evolutionary and bandit algorithms. Drift refers roughly to the expected progress towards the optimum per iteration. This paper considers the problem of deriving concentration tail-bounds on the runtime/regret of algorithms. It provides a novel drift theorem that gives precise exponential tail-bounds given positive, weak, zero and even negative drift. Previously, such exponential tail bounds were missing in the case of weak, zero, or negative drift. Our drift theorem can be used to prove a strong concentration of the runtime/regret of algorithms in AI. For example, we prove that the regret of the \rwab bandit algorithm is highly concentrated, while previous analyses only considered the expected regret. This means that the algorithm obtains the optimum within a given time frame with high probability, i.e. a form of algorithm reliability. Moreover, our theorem implies that the time needed by the co-evolutionary algorithm RLS-PD to obtain a Nash equilibrium in a \bilinear max-min-benchmark problem is highly concentrated. However, we also prove that the algorithm forgets the Nash equilibrium, and the time until this occurs is highly concentrated. This highlights a weakness in the RLS-PD which should be addressed by future work.

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arXiv2023

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.

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arXiv2020

Self-adaptation in non-Elitist Evolutionary Algorithms on Discrete Problems with Unknown Structure

A key challenge to make effective use of evolutionary algorithms is to choose appropriate settings for their parameters. However, the appropriate parameter setting generally depends on the structure of the optimisation problem, which is often unknown to the user. Non-deterministic parameter control mechanisms adjust parameters using information obtained from the evolutionary process. Self-adaptation -- where parameter settings are encoded in the chromosomes of individuals and evolve through mutation and crossover -- is a popular parameter control mechanism in evolutionary strategies. However, there is little theoretical evidence that self-adaptation is effective, and self-adaptation has largely been ignored by the discrete evolutionary computation community. Here we show through a theoretical runtime analysis that a non-elitist, discrete evolutionary algorithm which self-adapts its mutation rate not only outperforms EAs which use static mutation rates on \leadingones, but also improves asymptotically on an EA using a state-of-the-art control mechanism. The structure of this problem depends on a parameter $k$, which is \emph{a priori} unknown to the algorithm, and which is needed to appropriately set a fixed mutation rate. The self-adaptive EA achieves the same asymptotic runtime as if this parameter was known to the algorithm beforehand, which is an asymptotic speedup for this problem compared to all other EAs previously studied. An experimental study of how the mutation-rates evolve show that they respond adequately to a diverse range of problem structures. These results suggest that self-adaptation should be adopted more broadly as a parameter control mechanism in discrete, non-elitist evolutionary algorithms.

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arXiv2019

Runtime Analysis of Fitness-Proportionate Selection on Linear Functions

This paper extends the runtime analysis of non-elitist evolutionary algorithms (EAs) with fitness-proportionate selection from the simple OneMax function to the linear functions. Not only does our analysis cover a larger class of fitness functions, it also holds for a wider range of mutation rates. We show that with overwhelmingly high probability, no linear function can be optimised in less than exponential time, assuming bitwise mutation rate $Θ(1/n)$ and population size $λ=n^k$ for any constant $k>2$. In contrast to this negative result, we also show that for any linear function with polynomially bounded weights, the EA achieves a polynomial expected runtime if the mutation rate is reduced to $Θ(1/n^2)$ and the population size is sufficiently large. Furthermore, the EA with mutation rate $χ/n=Θ(1/n)$ and modest population size $λ=Ω(\ln n)$ optimises the scaled fitness function $e^{(χ+\varepsilon)f(x)}$ for any linear function $f$ and any $\varepsilon>0$ in expected time $O(nλ\lnλ+n^2)$. These upper bounds also extend to some additively decomposed fitness functions, such as the Royal Road functions. We expect that the obtained results may be useful not only for the development of the theory of evolutionary algorithms, but also for biological applications, such as the directed evolution.

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arXiv2019

On the Limitations of the Univariate Marginal Distribution Algorithm to Deception and Where Bivariate EDAs might help

We introduce a new benchmark problem called Deceptive Leading Blocks (DLB) to rigorously study the runtime of the Univariate Marginal Distribution Algorithm (UMDA) in the presence of epistasis and deception. We show that simple Evolutionary Algorithms (EAs) outperform the UMDA unless the selective pressure $μ/λ$ is extremely high, where $μ$ and $λ$ are the parent and offspring population sizes, respectively. More precisely, we show that the UMDA with a parent population size of $μ=Ω(\log n)$ has an expected runtime of $e^{Ω(μ)}$ on the DLB problem assuming any selective pressure $\fracμλ \geq \frac{14}{1000}$, as opposed to the expected runtime of $\mathcal{O}(nλ\log λ+n^3)$ for the non-elitist $(μ,λ)~\text{EA}$ with $μ/λ\leq 1/e$. These results illustrate inherent limitations of univariate EDAs against deception and epistasis, which are common characteristics of real-world problems. In contrast, empirical evidence reveals the efficiency of the bi-variate MIMIC algorithm on the DLB problem. Our results suggest that one should consider EDAs with more complex probabilistic models when optimising problems with some degree of epistasis and deception.

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arXiv2019

Runtime Analysis of the Univariate Marginal Distribution Algorithm under Low Selective Pressure and Prior Noise

We perform a rigorous runtime analysis for the Univariate Marginal Distribution Algorithm on the LeadingOnes function, a well-known benchmark function in the theory community of evolutionary computation with a high correlation between decision variables. For a problem instance of size $n$, the currently best known upper bound on the expected runtime is $\mathcal{O}(nλ\logλ+n^2)$ (Dang and Lehre, GECCO 2015), while a lower bound necessary to understand how the algorithm copes with variable dependencies is still missing. Motivated by this, we show that the algorithm requires a $e^{Ω(μ)}$ runtime with high probability and in expectation if the selective pressure is low; otherwise, we obtain a lower bound of $Ω(\frac{nλ}{\log(λ-μ)})$ on the expected runtime. Furthermore, we for the first time consider the algorithm on the function under a prior noise model and obtain an $\mathcal{O}(n^2)$ expected runtime for the optimal parameter settings. In the end, our theoretical results are accompanied by empirical findings, not only matching with rigorous analyses but also providing new insights into the behaviour of the algorithm.

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arXiv2019

Parallel Black-Box Complexity with Tail Bounds

We propose a new black-box complexity model for search algorithms evaluating $λ$ search points in parallel. The parallel unary unbiased black-box complexity gives lower bounds on the number of function evaluations every parallel unary unbiased black-box algorithm needs to optimise a given problem. It captures the inertia caused by offspring populations in evolutionary algorithms and the total computational effort in parallel metaheuristics. We present complexity results for LeadingOnes and OneMax. Our main result is a general performance limit: we prove that on every function every $λ$-parallel unary unbiased algorithm needs at least $Ω(\frac{λn}{\ln λ} + n \log n)$ evaluations to find any desired target set of up to exponential size, with an overwhelming probability. This yields lower bounds for the typical optimisation time on unimodal and multimodal problems, for the time to find any local optimum, and for the time to even get close to any optimum. The power and versatility of this approach is shown for a wide range of illustrative problems from combinatorial optimisation. Our performance limits can guide parameter choice and algorithm design; we demonstrate the latter by presenting an optimal $λ$-parallel algorithm for OneMax that uses parallelism most effectively.

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arXiv2018

Level-Based Analysis of the Univariate Marginal Distribution Algorithm

Estimation of Distribution Algorithms (EDAs) are stochastic heuristics that search for optimal solutions by learning and sampling from probabilistic models. Despite their popularity in real-world applications, there is little rigorous understanding of their performance. Even for the Univariate Marginal Distribution Algorithm (UMDA) -- a simple population-based EDA assuming independence between decision variables -- the optimisation time on the linear problem OneMax was until recently undetermined. The incomplete theoretical understanding of EDAs is mainly due to lack of appropriate analytical tools. We show that the recently developed level-based theorem for non-elitist populations combined with anti-concentration results yield upper bounds on the expected optimisation time of the UMDA. This approach results in the bound $\mathcal{O}(nλ\log λ+n^2)$ on two problems, LeadingOnes and BinVal, for population sizes $λ>μ=Ω(\log n)$, where $μ$ and $λ$ are parameters of the algorithm. We also prove that the UMDA with population sizes $μ\in \mathcal{O}(\sqrt{n}) \cap Ω(\log n)$ optimises OneMax in expected time $\mathcal{O}(λn)$, and for larger population sizes $μ=Ω(\sqrt{n}\log n)$, in expected time $\mathcal{O}(λ\sqrt{n})$. The facility and generality of our arguments suggest that this is a promising approach to derive bounds on the expected optimisation time of EDAs.

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arXiv2018

Level-Based Analysis of the Population-Based Incremental Learning Algorithm

The Population-Based Incremental Learning (PBIL) algorithm uses a convex combination of the current model and the empirical model to construct the next model, which is then sampled to generate offspring. The Univariate Marginal Distribution Algorithm (UMDA) is a special case of the PBIL, where the current model is ignored. Dang and Lehre (GECCO 2015) showed that UMDA can optimise LeadingOnes efficiently. The question still remained open if the PBIL performs equally well. Here, by applying the level-based theorem in addition to Dvoretzky--Kiefer--Wolfowitz inequality, we show that the PBIL optimises function LeadingOnes in expected time $\mathcal{O}(nλ\log λ+ n^2)$ for a population size $λ= Ω(\log n)$, which matches the bound of the UMDA. Finally, we show that the result carries over to BinVal, giving the fist runtime result for the PBIL on the BinVal problem.

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arXiv2018

General Drift Analysis with Tail Bounds

Drift analysis is one of the state-of-the-art techniques for the runtime analysis of randomized search heuristics (RSHs) such as evolutionary algorithms (EAs), simulated annealing etc. The vast majority of existing drift theorems yield bounds on the expected value of the hitting time for a target state, e.g., the set of optimal solutions, without making additional statements on the distribution of this time. We address this lack by providing a general drift theorem that includes bounds on the upper and lower tail of the hitting time distribution. The new tail bounds are applied to prove very precise sharp-concentration results on the running time of a simple EA on standard benchmark problems, including the class of general linear functions. Surprisingly, the probability of deviating by an $r$-factor in lower order terms of the expected time decreases exponentially with $r$ on all these problems. The usefulness of the theorem outside the theory of RSHs is demonstrated by deriving tail bounds on the number of cycles in random permutations. All these results handle a position-dependent (variable) drift that was not covered by previous drift theorems with tail bounds. Moreover, our theorem can be specialized into virtually all existing drift theorems with drift towards the target from the literature. Finally, user-friendly specializations of the general drift theorem are given.

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arXiv2018

Improved Runtime Bounds for the Univariate Marginal Distribution Algorithm via Anti-Concentration

Unlike traditional evolutionary algorithms which produce offspring via genetic operators, Estimation of Distribution Algorithms (EDAs) sample solutions from probabilistic models which are learned from selected individuals. It is hoped that EDAs may improve optimisation performance on epistatic fitness landscapes by learning variable interactions. However, hardly any rigorous results are available to support claims about the performance of EDAs, even for fitness functions without epistasis. The expected runtime of the Univariate Marginal Distribution Algorithm (UMDA) on OneMax was recently shown to be in $\mathcal{O}\left(nλ\log λ\right)$ by Dang and Lehre (GECCO 2015). Later, Krejca and Witt (FOGA 2017) proved the lower bound $Ω\left(λ\sqrt{n}+n\log n\right)$ via an involved drift analysis. We prove a $\mathcal{O}\left(nλ\right)$ bound, given some restrictions on the population size. This implies the tight bound $Θ\left(n\log n\right)$ when $λ=\mathcal{O}\left(\log n\right)$, matching the runtime of classical EAs. Our analysis uses the level-based theorem and anti-concentration properties of the Poisson-Binomial distribution. We expect that these generic methods will facilitate further analysis of EDAs.

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arXiv2017

Theoretical Analysis of Stochastic Search Algorithms

Theoretical analyses of stochastic search algorithms, albeit few, have always existed since these algorithms became popular. Starting in the nineties a systematic approach to analyse the performance of stochastic search heuristics has been put in place. This quickly increasing basis of results allows, nowadays, the analysis of sophisticated algorithms such as population-based evolutionary algorithms, ant colony optimisation and artificial immune systems. Results are available concerning problems from various domains including classical combinatorial and continuous optimisation, single and multi-objective optimisation, and noisy and dynamic optimisation. This chapter introduces the mathematical techniques that are most commonly used in the runtime analysis of stochastic search heuristics. Careful attention is given to the very popular artificial fitness levels and drift analyses techniques for which several variants are presented. To aid the reader's comprehension of the presented mathematical methods, these are applied to the analysis of simple evolutionary algorithms for artificial example functions. The chapter is concluded by providing references to more complex applications and further extensions of the techniques for the obtainment of advanced results.

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arXiv2016

Level-based Analysis of Genetic Algorithms and other Search Processes

Understanding how the time-complexity of evolutionary algorithms (EAs) depend on their parameter settings and characteristics of fitness landscapes is a fundamental problem in evolutionary computation. Most rigorous results were derived using a handful of key analytic techniques, including drift analysis. However, since few of these techniques apply effortlessly to population-based EAs, most time-complexity results concern simplified EAs, such as the (1+1) EA. This paper describes the level-based theorem, a new technique tailored to population-based processes. It applies to any non-elitist process where offspring are sampled independently from a distribution depending only on the current population. Given conditions on this distribution, our technique provides upper bounds on the expected time until the process reaches a target state. We demonstrate the technique on several pseudo-Boolean functions, the sorting problem, and approximation of optimal solutions in combinatorial optimisation. The conditions of the theorem are often straightforward to verify, even for Genetic Algorithms and Estimation of Distribution Algorithms which were considered highly non-trivial to analyse. Finally, we prove that the theorem is nearly optimal for the processes considered. Given the information the theorem requires about the process, a much tighter bound cannot be proved.

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arXiv2016

Escaping Local Optima using Crossover with Emergent or Reinforced Diversity

Population diversity is essential for avoiding premature convergence in Genetic Algorithms (GAs) and for the effective use of crossover. Yet the dynamics of how diversity emerges in populations are not well understood. We use rigorous runtime analysis to gain insight into population dynamics and GA performance for the ($μ$+1) GA and the $\text{Jump}_k$ test function. We show that the interplay of crossover and mutation may serve as a catalyst leading to a sudden burst of diversity. This leads to improvements of the expected optimisation time of order $Ω(n/\log n)$ compared to mutation-only algorithms like (1+1) EA. Moreover, increasing the mutation rate by an arbitrarily small constant factor can facilitate the generation of diversity, leading to speedups of order $Ω(n)$. We also compare seven commonly used diversity mechanisms and evaluate their impact on runtime bounds for the ($μ$+1) GA. All previous results in this context only hold for unrealistically low crossover probability $p_c=O(k/n)$, while we give analyses for the setting of constant $p_c < 1$ in all but one case. For the typical case of constant $k > 2$ and constant $p_c$, we can compare the resulting expected runtimes for different diversity mechanisms assuming an optimal choice of $μ$: $O(n^{k-1})$ for duplicate elimination/minim., $O(n^2\log n)$ for maximising the convex hull, $O(n\log n)$ for deterministic crowding (assuming $p_c = k/n$), $O(n\log n)$ for maximising Hamming distance, $O(n\log n)$ for fitness sharing, $O(n\log n)$ for single-receiver island model. This proves a sizeable advantage of all variants of the ($μ$+1) GA compared to (1+1) EA, which requires time $Θ(n^k)$. Experiments complement our theoretical findings and further highlight the benefits of crossover and diversity on $\text{Jump}_k$.

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arXiv2016

Populations can be essential in tracking dynamic optima

Real-world optimisation problems are often dynamic. Previously good solutions must be updated or replaced due to changes in objectives and constraints. It is often claimed that evolutionary algorithms are particularly suitable for dynamic optimisation because a large population can contain different solutions that may be useful in the future. However, rigorous theoretical demonstrations for how populations in dynamic optimisation can be essential are sparse and restricted to special cases. This paper provides theoretical explanations of how populations can be essential in evolutionary dynamic optimisation in a general and natural setting. We describe a natural class of dynamic optimisation problems where a sufficiently large population is necessary to keep track of moving optima reliably. We establish a relationship between the population-size and the probability that the algorithm loses track of the optimum.

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