arXiv subjects

Johannes Lengler

Publications and source records attributed to Johannes Lengler.

At least 55 records · Page 3Linked to original sources

arXiv2020

Large Population Sizes and Crossover Help in Dynamic Environments

Dynamic linear functions on the hypercube are functions which assign to each bit a positive weight, but the weights change over time. Throughout optimization, these functions maintain the same global optimum, and never have defecting local optima. Nevertheless, it was recently shown [Lengler, Schaller, FOCI 2019] that the $(1+1)$-Evolutionary Algorithm needs exponential time to find or approximate the optimum for some algorithm configurations. In this paper, we study the effect of larger population sizes for Dynamic BinVal, the extremal form of dynamic linear functions. We find that moderately increased population sizes extend the range of efficient algorithm configurations, and that crossover boosts this positive effect substantially. Remarkably, similar to the static setting of monotone functions in [Lengler, Zou, FOGA 2019], the hardest region of optimization for $(μ+1)$-EA is not close the optimum, but far away from it. In contrast, for the $(μ+1)$-GA, the region around the optimum is the hardest region in all studied cases.

cs.NE↗

arXiv2020

Stopping explosion by penalising transmission to hubs in scale-free spatial random graphs

We study the spread of information in finite and infinite inhomogeneous spatial random graphs. We assume that each edge has a transmission cost that is a product of an i.i.d. random variable L and a penalty factor: edges between vertices of expected degrees w_1 and w_2 are penalised by a factor of (w_1w_2)^μfor all μ>0. We study this process for scale-free percolation, for (finite and infinite) Geometric Inhomogeneous Random Graphs, and for Hyperbolic Random Graphs, all with power law degree distributions with exponent τ> 1. For τ< 3, we find a threshold behaviour, depending on how fast the cumulative distribution function of L decays at zero. If it decays at most polynomially with exponent smaller than (3-τ)/(2μ) then explosion happens, i.e., with positive probability we can reach infinitely many vertices with finite cost (for the infinite models), or reach a linear fraction of all vertices with bounded costs (for the finite models). On the other hand, if the cdf of L decays at zero at least polynomially with exponent larger than (3-τ)/(2μ), then no explosion happens. This behaviour is arguably a better representation of information spreading processes in social networks than the case without penalising factor, in which explosion always happens unless the cdf of L is doubly exponentially flat around zero. Finally, we extend the results to other penalty functions, including arbitrary polynomials in w_1 and w_2. In some cases the interesting phenomenon occurs that the model changes behaviour (from explosive to conservative and vice versa) when we reverse the role of w_1 and w_2. Intuitively, this could corresponds to reversing the flow of information: gathering information might take much longer than sending it out.

math.PR↗

arXiv2019

Random Sampling with Removal

We study randomized algorithms for constrained optimization, in abstract frameworks that include, in strictly increasing generality: convex programming; LP-type problems; violator spaces; and a setting we introduce, consistent spaces. Such algorithms typically involve a step of finding the optimal solution for a random sample of the constraints. They exploit the condition that, in finite dimension $δ$, this sample optimum violates a provably small expected fraction of the non-sampled constraints, with the fraction decreasing in the sample size~$r$. We extend such algorithms by considering the technique of removal, where a fixed number $k$ of constraints are removed from the sample according to a fixed rule, with the goal of improving the solution quality. This may have the effect of increasing the number of violated non-sampled constraints. We study this increase, and bound it in a variety of general settings. This work is motivated by, and extends, results on removal as proposed for chance-constrained optimization. For many relevant values of $r$, $δ$, and $k$, we prove matching upper and lower bounds for the expected number of constraints violated by a random sample, after the removal of $k$ constraints. For a large range of values of $k$, the new upper bounds improve the previously best bounds for LP-type problems, which moreover had only been known in special cases, and not in the generality we consider. Moreover, we show that our results extend from finite to infinite spaces, for chance-constrained optimization.

cs.CG↗

arXiv2018

Average Distance in a General Class of Scale-Free Networks with Underlying Geometry

In Chung-Lu random graphs, a classic model for real-world networks, each vertex is equipped with a weight drawn from a power-law distribution, and two vertices form an edge independently with probability proportional to the product of their weights. Chung-Lu graphs have average distance O(log log n) and thus reproduce the small-world phenomenon, a key property of real-world networks. Modern, more realistic variants of this model also equip each vertex with a random position in a specific underlying geometry. The edge probability of two vertices then depends, say, inversely polynomial on their distance. In this paper we study a generic augmented version of Chung-Lu random graphs. We analyze a model where the edge probability of two vertices can depend arbitrarily on their positions, as long as the marginal probability of forming an edge (for two vertices with fixed weights, one fixed position, and one random position) is as in Chung-Lu random graphs. The resulting class contains Chung-Lu random graphs, hyperbolic random graphs, and geometric inhomogeneous random graphs as special cases. Our main result is that every random graph model in this general class has the same average distance as Chung-Lu random graphs, up to a factor 1+o(1). This shows in particular that specific choices, such as the underlying geometry being Euclidean or the dependence on the distance being inversely polynomial, do not significantly influence the average distance. The proof also yields that our model has a giant component and polylogarithmic diameter with high probability.

cs.DM↗

arXiv2018

The linear hidden subset problem for the (1+1) EA with scheduled and adaptive mutation rates

We study unbiased $(1+1)$ evolutionary algorithms on linear functions with an unknown number $n$ of bits with non-zero weight. Static algorithms achieve an optimal runtime of $O(n (\ln n)^{2+ε})$, however, it remained unclear whether more dynamic parameter policies could yield better runtime guarantees. We consider two setups: one where the mutation rate follows a fixed schedule, and one where it may be adapted depending on the history of the run. For the first setup, we give a schedule that achieves a runtime of $(1\pm o(1))βn \ln n$, where $β\approx 3.552$, which is an asymptotic improvement over the runtime of the static setup. Moreover, we show that no schedule admits a better runtime guarantee and that the optimal schedule is essentially unique. For the second setup, we show that the runtime can be further improved to $(1\pm o(1)) e n \ln n$, which matches the performance of algorithms that know $n$ in advance. Finally, we study the related model of initial segment uncertainty with static position-dependent mutation rates, and derive asymptotically optimal lower bounds. This answers a question by Doerr, Doerr, and Kötzing.

cs.NE↗

arXiv2018

When Does Hillclimbing Fail on Monotone Functions: An entropy compression argument

Hillclimbing is an essential part of any optimization algorithm. An important benchmark for hillclimbing algorithms on pseudo-Boolean functions $f: \{0,1\}^n \to \mathbb{R}$ are (strictly) montone functions, on which a surprising number of hillclimbers fail to be efficient. For example, the $(1+1)$-Evolutionary Algorithm is a standard hillclimber which flips each bit independently with probability $c/n$ in each round. Perhaps surprisingly, this algorithm shows a phase transition: it optimizes any monotone pseudo-boolean function in quasilinear time if $c<1$, but there are monotone functions for which the algorithm needs exponential time if $c>2.2$. But so far it was unclear whether the threshold is at $c=1$. In this paper we show how Moser's entropy compression argument can be adapted to this situation, that is, we show that a long runtime would allow us to encode the random steps of the algorithm with less bits than their entropy. Thus there exists a $c_0 > 1$ such that for all $0<c\le c_0$ the $(1+1)$-Evolutionary Algorithm with rate $c/n$ finds the optimum in $O(n \log^2 n)$ steps in expectation.

math.PR↗

arXiv2018

Asymptotically Optimal Amplifiers for the Moran Process

We study the Moran process as adapted by Lieberman, Hauert and Nowak. This is a model of an evolving population on a graph or digraph where certain individuals, called "mutants" have fitness r and other individuals, called non-mutants have fitness 1. We focus on the situation where the mutation is advantageous, in the sense that r>1. A family of digraphs is said to be strongly amplifying if the extinction probability tends to 0 when the Moran process is run on digraphs in this family. The most-amplifying known family of digraphs is the family of megastars of Galanis et al. We show that this family is optimal, up to logarithmic factors, since every strongly-connected n-vertex digraph has extinction probability Omega(n^(-1/2)). Next, we show that there is an infinite family of undirected graphs, called dense incubators, whose extinction probability is O(n^(-1/3)). We show that this is optimal, up to constant factors. Finally, we introduce sparse incubators, for varying edge density, and show that the extinction probability of these graphs is O(n/m), where m is the number of edges. Again, we show that this is optimal, up to constant factors.

math.PR↗

arXiv2018

Drift Analysis

Drift analysis is one of the major tools for analysing evolutionary algorithms and nature-inspired search heuristics. In this chapter we give an introduction to drift analysis and give some examples of how to use it for the analysis of evolutionary algorithms.

cs.NE↗

arXiv2018

Bounding Bloat in Genetic Programming

While many optimization problems work with a fixed number of decision variables and thus a fixed-length representation of possible solutions, genetic programming (GP) works on variable-length representations. A naturally occurring problem is that of bloat (unnecessary growth of solutions) slowing down optimization. Theoretical analyses could so far not bound bloat and required explicit assumptions on the magnitude of bloat. In this paper we analyze bloat in mutation-based genetic programming for the two test functions ORDER and MAJORITY. We overcome previous assumptions on the magnitude of bloat and give matching or close-to-matching upper and lower bounds for the expected optimization time. In particular, we show that the (1+1) GP takes (i) $Θ(T_{init} + n \log n)$ iterations with bloat control on ORDER as well as MAJORITY; and (ii) $O(T_{init} \log T_{init} + n (\log n)^3)$ and $Ω(T_{init} + n \log n)$ (and $Ω(T_{init} \log T_{init})$ for $n=1$) iterations without bloat control on MAJORITY.

cs.NE↗

arXiv2018

Destructiveness of Lexicographic Parsimony Pressure and Alleviation by a Concatenation Crossover in Genetic Programming

For theoretical analyses there are two specifics distinguishing GP from many other areas of evolutionary computation. First, the variable size representations, in particular yielding a possible bloat (i.e. the growth of individuals with redundant parts). Second, the role and realization of crossover, which is particularly central in GP due to the tree-based representation. Whereas some theoretical work on GP has studied the effects of bloat, crossover had a surprisingly little share in this work. We analyze a simple crossover operator in combination with local search, where a preference for small solutions minimizes bloat (lexicographic parsimony pressure); the resulting algorithm is denoted Concatenation Crossover GP. For this purpose three variants of the well-studied MAJORITY test function with large plateaus are considered. We show that the Concatenation Crossover GP can efficiently optimize these test functions, while local search cannot be efficient for all three variants independent of employing bloat control.

cs.NE↗

arXiv2018

A General Dichotomy of Evolutionary Algorithms on Monotone Functions

It is known that the evolutionary algorithm $(1+1)$-EA with mutation rate $c/n$ optimises every monotone function efficiently if $c<1$, and needs exponential time on some monotone functions (HotTopic functions) if $c\geq 2.2$. We study the same question for a large variety of algorithms, particularly for $(1+λ)$-EA, $(μ+1)$-EA, $(μ+1)$-GA, their fast counterparts like fast $(1+1)$-EA, and for $(1+(λ,λ))$-GA. We find that all considered mutation-based algorithms show a similar dichotomy for HotTopic functions, or even for all monotone functions. For the $(1+(λ,λ))$-GA, this dichotomy is in the parameter $cγ$, which is the expected number of bit flips in an individual after mutation and crossover, neglecting selection. For the fast algorithms, the dichotomy is in $m_2/m_1$, where $m_1$ and $m_2$ are the first and second falling moment of the number of bit flips. Surprisingly, the range of efficient parameters is not affected by either population size $μ$ nor by the offspring population size $λ$. The picture changes completely if crossover is allowed. The genetic algorithms $(μ+1)$-GA and fast $(μ+1)$-GA are efficient for arbitrary mutations strengths if $μ$ is large enough.

cs.NE↗

arXiv2018

Sorting by Swaps with Noisy Comparisons

We study sorting of permutations by random swaps if each comparison gives the wrong result with some fixed probability $p<1/2$. We use this process as prototype for the behaviour of randomized, comparison-based optimization heuristics in the presence of noisy comparisons. As quality measure, we compute the expected fitness of the stationary distribution. To measure the runtime, we compute the minimal number of steps after which the average fitness approximates the expected fitness of the stationary distribution. We study the process where in each round a random pair of elements at distance at most $r$ are compared. We give theoretical results for the extreme cases $r=1$ and $r=n$, and experimental results for the intermediate cases. We find a trade-off between faster convergence (for large $r$) and better quality of the solution after convergence (for small $r$).

cs.NE↗

arXiv2017

Drift Analysis and Evolutionary Algorithms Revisited

One of the easiest randomized greedy optimization algorithms is the following evolutionary algorithm which aims at maximizing a boolean function $f:\{0,1\}^n \to {\mathbb R}$. The algorithm starts with a random search point $ξ\in \{0,1\}^n$, and in each round it flips each bit of $ξ$ with probability $c/n$ independently at random, where $c>0$ is a fixed constant. The thus created offspring $ξ'$ replaces $ξ$ if and only if $f(ξ') \ge f(ξ)$. The analysis of the runtime of this simple algorithm on monotone and on linear functions turned out to be highly non-trivial. In this paper we review known results and provide new and self-contained proofs of partly stronger results.

math.CO↗

arXiv2017

Existence of Small Separators Depends on Geometry for Geometric Inhomogeneous Random Graphs

We show that Geometric Inhomogeneous Random Graphs (GIRGs) with power law weights may either have or not have separators of linear size, depending on the underlying geometry. While it was known that for Euclidean geometry it is possible to split the giant component into two linear size components by removing at most n^{1-\eps} edges, we show that this is impossible if the geometry is given by the minimum component distance.

math.CO↗

arXiv2017

Tight Analysis for the 3-Majority Consensus Dynamics

We present a tight analysis for the well-studied randomized 3-majority dynamics of stabilizing consensus, hence answering the main open question of Becchetti et al. [SODA'16]. Consider a distributed system of n nodes, each initially holding an opinion in {1, 2, ..., k}. The system should converge to a setting where all (non-corrupted) nodes hold the same opinion. This consensus opinion should be \emph{valid}, meaning that it should be among the initially supported opinions, and the (fast) convergence should happen even in the presence of a malicious adversary who can corrupt a bounded number of nodes per round and in particular modify their opinions. A well-studied distributed algorithm for this problem is the 3-majority dynamics, which works as follows: per round, each node gathers three opinions --- say by taking its own and two of other nodes sampled at random --- and then it sets its opinion equal to the majority of this set; ties are broken arbitrarily, e.g., towards the node's own opinion. Becchetti et al. [SODA'16] showed that the 3-majority dynamics converges to consensus in O((k^2\sqrt{\log n} + k\log n)(k+\log n)) rounds, even in the presence of a limited adversary. We prove that, even with a stronger adversary, the convergence happens within O(k\log n) rounds. This bound is known to be optimal.

cs.DC↗

arXiv2017

Sampling Geometric Inhomogeneous Random Graphs in Linear Time

Real-world networks, like social networks or the internet infrastructure, have structural properties such as large clustering coefficients that can best be described in terms of an underlying geometry. This is why the focus of the literature on theoretical models for real-world networks shifted from classic models without geometry, such as Chung-Lu random graphs, to modern geometry-based models, such as hyperbolic random graphs. With this paper we contribute to the theoretical analysis of these modern, more realistic random graph models. Instead of studying directly hyperbolic random graphs, we use a generalization that we call geometric inhomogeneous random graphs (GIRGs). Since we ignore constant factors in the edge probabilities, GIRGs are technically simpler (specifically, we avoid hyperbolic cosines), while preserving the qualitative behaviour of hyperbolic random graphs, and we suggest to replace hyperbolic random graphs by this new model in future theoretical studies. We prove the following fundamental structural and algorithmic results on GIRGs. (1) As our main contribution we provide a sampling algorithm that generates a random graph from our model in expected linear time, improving the best-known sampling algorithm for hyperbolic random graphs by a substantial factor O(n^0.5). (2) We establish that GIRGs have clustering coefficients in Ω(1), (3) we prove that GIRGs have small separators, i.e., it suffices to delete a sublinear number of edges to break the giant component into two large pieces, and (4) we show how to compress GIRGs using an expected linear number of bits.

cs.SI↗

arXiv2017

Greedy Routing and the Algorithmic Small-World Phenomenom

The algorithmic small-world phenomenon, empirically established by Milgram's letter forwarding experiments from the 60s, was theoretically explained by Kleinberg in 2000. However, from today's perspective his model has several severe shortcomings that limit the applicability to real-world networks. In order to give a more convincing explanation of the algorithmic small-world phenomenon, we study decentralized greedy routing in a more flexible random graph model (geometric inhomogeneous random graphs) which overcomes all previous shortcomings. Apart from exhibiting good properties in theory, it has also been extensively experimentally validated that this model reasonably captures real-world networks. In this model, the greedy routing protocol is purely distributed as each vertex only needs to know information about its direct neighbors. We prove that it succeeds with constant probability, and in case of success almost surely finds an almost shortest path of length θ(loglog n), where our bound is tight including the leading constant. Moreover, we study natural local patching methods which augment greedy routing by backtracking and which do not require any global knowledge. We show that such methods can ensure success probability 1 in an asymptotically tight number of steps. These results also address the question of Krioukov et al. whether there are efficient local routing protocols for the internet graph. There were promising experimental studies, but the question remained unsolved theoretically. Our results give for the first time a rigorous and analytical affirmative answer.

cs.SI↗

arXiv2016

The (1+1) Elitist Black-Box Complexity of LeadingOnes

One important goal of black-box complexity theory is the development of complexity models allowing to derive meaningful lower bounds for whole classes of randomized search heuristics. Complementing classical runtime analysis, black-box models help us understand how algorithmic choices such as the population size, the variation operators, or the selection rules influence the optimization time. One example for such a result is the $Ω(n \log n)$ lower bound for unary unbiased algorithms on functions with a unique global optimum [Lehre/Witt, GECCO 2010], which tells us that higher arity operators or biased sampling strategies are needed when trying to beat this bound. In lack of analyzing techniques, almost no non-trivial bounds are known for other restricted models. Proving such bounds therefore remains to be one of the main challenges in black-box complexity theory. With this paper we contribute to our technical toolbox for lower bound computations by proposing a new type of information-theoretic argument. We regard the permutation- and bit-invariant version of \textsc{LeadingOnes} and prove that its (1+1) elitist black-box complexity is $Ω(n^2)$, a bound that is matched by (1+1)-type evolutionary algorithms. The (1+1) elitist complexity of \textsc{LeadingOnes} is thus considerably larger than its unrestricted one, which is known to be of order $n\log\log n$ [Afshani et al., 2013].

cs.NE↗