Standard Steady State Genetic Algorithms Can Hillclimb Faster than Mutation-only Evolutionary Algorithms

Explaining to what extent the real power of genetic algorithms lies in the ability of crossover to recombine individuals into higher quality solutions is an important problem in evolutionary computation. In this paper we show how the interplay between mutation and crossover can make genetic algorithms hillclimb faster than their mutation-only counterparts. We devise a Markov Chain framework that allows to rigorously prove an upper bound on the runtime of standard steady state genetic algorithms to hillclimb the ONEMAX function. The bound establishes that the steady-state genetic algorithms are 25% faster than all standard bit mutation-only evolutionary algorithms with static mutation rate up to lower order terms for moderate population sizes. The analysis also suggests that larger populations may be faster than populations of size 2. We present a lower bound for a greedy (2+1) GA that matches the upper bound for populations larger than 2, rigorously proving that 2 individuals cannot outperform larger population sizes under greedy selection and greedy crossover up to lower order terms. In complementary experiments the best population size is greater than 2 and the greedy genetic algorithms are faster than standard ones, further suggesting that the derived lower bound also holds for the standard steady state (2+1) GA

Artificial immune systems can find arbitrarily good approximations for the NP-hard number partitioning problem

Typical artificial immune system (AIS) operators such as hypermutations with mutation potential and ageing allow to efficiently overcome local optima from which evolutionary algorithms (EAs) struggle to escape. Such behaviour has been shown for artificial example functions constructed especially to show difficulties that EAs may encounter during the optimisation process. However, no evidence is available indicating that these two operators have similar behaviour also in more realistic problems. In this paper we perform an analysis for the standard NP-hard Partition problem from combinatorial optimisation and rigorously show that hypermutations and ageing allow AISs to efficiently escape from local optima where standard EAs require exponential time. As a result we prove that while EAs and random local search (RLS) may get trapped on 4/3 approximations, AISs find arbitrarily good approximate solutions of ratio (1+) within n(−(2/)−1)(1 − )−2e322/ + 2n322/ + 2n3 function evaluations in expectation. This expectation is polynomial in the problem size and exponential only in 1/

When hypermutations and ageing enable artificial immune systems to outperform evolutionary algorithms

We present a time complexity analysis of the Opt-IA artificial immune system (AIS). We first highlight the power and limitations of its distinguishing operators (i.e., hypermutations with mutation potential and ageing) by analysing them in isolation. Recent work has shown that ageing combined with local mutations can help escape local optima on a dynamic optimisation benchmark function. We generalise this result by rigorously proving that, compared to evolutionary algorithms (EAs), ageing leads to impressive speed-ups on the standard Image 1 benchmark function both when using local and global mutations. Unless the stop at first constructive mutation (FCM) mechanism is applied, we show that hypermutations require exponential expected runtime to optimise any function with a polynomial number of optima. If instead FCM is used, the expected runtime is at most a linear factor larger than the upper bound achieved for any random local search algorithm using the artificial fitness levels method. Nevertheless, we prove that algorithms using hypermutations can be considerably faster than EAs at escaping local optima. An analysis of the complete Opt-IA reveals that it is efficient on the previously considered functions and highlights problems where the use of the full algorithm is crucial. We complete the picture by presenting a class of functions for which Opt-IA fails with overwhelming probability while standard EAs are efficient

Computational complexity analysis of genetic programming

Genetic programming (GP) is an evolutionary computation technique to solve problems in an automated, domain-independent way. Rather than identifying the optimum of a function as in more traditional evolutionary optimization, the aim of GP is to evolve computer programs with a given functionality. While many GP applications have produced human competitive results, the theoretical understanding of what problem characteristics and algorithm properties allow GP to be effective is comparatively limited. Compared with traditional evolutionary algorithms for function optimization, GP applications are further complicated by two additional factors: the variable-length representation of candidate programs, and the difficulty of evaluating their quality efficiently. Such difficulties considerably impact the runtime analysis of GP, where space complexity also comes into play. As a result, initial complexity analyses of GP have focused on restricted settings such as the evolution of trees with given structures or the estimation of solution quality using only a small polynomial number of input/output examples. However, the first computational complexity analyses of GP for evolving proper functions with defined input/output behavior have recently appeared. In this chapter, we present an overview of the state of the art

On the time and space complexity of genetic programming for evolving Boolean conjunctions

Genetic programming (GP) is a general purpose bio-inspired meta-heuristic for the evolution of computer programs. In contrast to the several successful applications, there is little understanding of the working principles behind GP. In this paper we present a performance analysis that sheds light on the behaviour of simple GP systems for evolving conjunctions of n variables (ANDn). The analysis of a random local search GP system with minimal terminal and function sets reveals the relationship between the number of iterations and the progress the GP makes toward finding the target function. Afterwards we consider a more realistic GP system equipped with a global mutation operator and prove that it can efficiently solve ANDn by producing programs of linear size that fit a training set to optimality and with high probability generalise well. Additionally, we consider more general problems which extend the terminal set with undesired variables or negated variables. In the presence of undesired variables, we prove that, if non-strict selection is used, then the algorithm fits the complete training set efficiently while the strict selection algorithm may fail with high probability unless the substitution operator is switched off. If negations are allowed, we show that while the algorithms fail to fit the complete training set, the constructed solutions generalise well. Finally, from a problem hardness perspective, we reveal the existence of small training sets that allow the evolution of the exact conjunctions even with access to negations or undesired variables

When is it Beneficial to Reject Improvements?

We investigate two popular trajectory-based algorithms from biology and physics to answer a question of general significance: when is it beneficial to reject improvements? A distinguishing factor of SSWM (Strong Selection Weak Mutation), a popular model from population genetics, compared to the Metropolis algorithm (MA), is that the former can reject improvements, while the latter always accepts them. We investigate when one strategy outperforms the other. Since we prove that both algorithms converge to the same stationary distribution, we concentrate on identifying a class of functions inducing large mixing times, where the algorithms will outperform each other over a long period of time. The outcome of the analysis is the definition of a function where SSWM is efficient, while Metropolis requires at least exponential time

Fast perturbative algorithm configurators

Recent work has shown that the ParamRLS and ParamILS algorithm configurators can tune some simple randomised search heuristics for standard benchmark functions in linear expected time in the size of the parameter space. In this paper we prove a linear lower bound on the expected time to optimise any parameter tuning problem for ParamRLS, ParamILS as well as for larger classes of algorithm configurators. We propose a harmonic mutation operator for perturbative algorithm configurators that provably tunes single-parameter algorithms in polylogarithmic time for unimodal and approximately unimodal (i.e., non-smooth, rugged with an underlying gradient towards the optimum) parameter spaces. It is suitable as a general-purpose operator since even on worst-case (e.g., deceptive) landscapes it is only by at most a logarithmic factor slower than the default ones used by ParamRLS and ParamILS. An experimental analysis confirms the superiority of the approach in practice for a number of configuration scenarios, including ones involving more than one parameter

Tight bounds on the expected runtime of a standard steady state genetic algorithm

Recent progress in the runtime analysis of evolutionary algorithms (EAs) has allowed the derivation of upper bounds on the expected runtime of standard steady-state genetic algorithms (GAs). These upper bounds have shown speed-ups of the GAs using crossover and mutation over the same algorithms that only use mutation operators (i.e., steady-state EAs) both for standard unimodal (i.e., ONEMAX) and multimodal (i.e., JUMP) benchmark functions. The bounds suggest that populations are beneficial to the GA as well as higher mutation rates than the default 1/n rate. However, making rigorous claims was not possible because matching lower bounds were not available. Proving lower bounds on crossover-based EAs is a notoriously difficult task as it is hard to capture the progress that a diverse population can make. We use a potential function approach to prove a tight lower bound on the expected runtime of the (2+1) GA for ONEMAX for all mutation rates c/n with c<1.422. This provides the last piece of the puzzle that completes the proof that larger population sizes improve the performance of the standard steady-state GA for ONEMAX for various mutation rates, and it proves that the optimal mutation rate for the (2+1) GA on ONEMAX is (97−−√−5)/(4n)≈1.2122/n

On inversely proportional hypermutations with mutation potential

Artificial Immune Systems (AIS) employing hypermutations with linear static mutation potential have recently been shown to be very effective at escaping local optima of combinatorial optimisation problems at the expense of being slower during the exploitation phase compared to standard evolutionary algorithms. In this paper we prove that considerable speed-ups in the exploitation phase may be achieved with dynamic inversely proportional mutation potentials (IPM) and argue that the potential should decrease inversely to the distance to the optimum rather than to the difference in fitness. Afterwards we define a simple (1+1)~Opt-IA, that uses IPM hypermutations and ageing, for realistic applications where optimal solutions are unknown. The aim of the AIS is to approximate the ideal behaviour of the inversely proportional hypermutations better and better as the search space is explored. We prove that such desired behaviour, and related speed-ups, occur for a well-studied bimodal benchmark function called \textsc{TwoMax}. Furthermore, we prove that the (1+1)~Opt-IA with IPM efficiently optimises a third bimodal function, \textsc{Cliff}, by escaping its local optima while Opt-IA with static potential cannot, thus requires exponential expected runtime in the distance between the cliff and the optimum