The OneMax problem is a standard benchmark optimisation problem for a binary search space. Recent work on applying a Bandit-Based Random Mutation Hill-Climbing algorithm to the noisy OneMax Problem showed that it is important to choose a good value for the resampling number to make a careful trade off between taking more samples in order to reduce noise, and taking fewer samples to reduce the total computational cost. This paper extends that observation, by deriving an analytical expression for the running time of the RMHC algorithm with resampling applied to the noisy OneMax problem, and showing both theoretically and empirically that the optimal resampling number increases with the number of dimensions in the search space.
Droste @cite_6 defined a model for the OneMax problem, in which during every fitness evaluation of the bit-string @math , there was exactly one uniformly-chosen random bit mis-read with probability @math . Hence the measured noisy fitness values are equivalent to replacing the Gaussian noise term in by an appropriate discrete random variable taking values from @math . Under this scheme, Droste showed that (1+1)-EA with mutation probability @math could optimise a @math -bit OneMax problem corrupted by , with high probability, in polynomial time if @math is @math .
@cite_6
2914741526 2148192607 1569002781 2497618134
We study the well-known black-box optimisation algorithm (1+1)-EA on a novel type of noise model. In our noise model, the fitness function is linear with positive weights, but the absolute values of the weights may fluctuate in each round. Thus in every state, the fitness function indicates that one-bits are preferred over zero-bits. In particular, hillclimbing heuristics should be able to find the optimum fast. We show that the (1+1)-EA indeed finds the optimum in time @math if the mutation parameter is @math for a constant @math However, we also show that for @math the (1+1)-EA needs superpolynomial time to find the optimum. Thus the choice of mutation parameter is critical even for optimisation tasks in which there is a clear path to the the optimum. A similar threshold phenomenon has recently been shown for noise-free monotone fitness functions. In basic genetic algorithm (GA) applications, the fitness of a solution takes a value that is certain and unchanging. This formulation does not work for ongoing searches for better solutions in a nonstationary environment in which expected solution fitness changes with time in unpredictable ways, or for fitness evaluations corrupted by noise. In such cases, the estimated fitness has an associated uncertainty. The uncertainties due to environmental changes (process noise) and to noisy evaluations (observation noise) can be reduced, at least temporarily, by re-evaluating existing solutions. The Kalman formulation provides a formal mechanism for treating uncertainty in GA. It provides the mechanics for determining the estimated fitness and uncertainty when a new solution is generated and evaluated for the first time. It also provides the mechanics for updating the estimated fitness and uncertainty after an existing solution is re-evaluated and for increasing the uncertainty with the passage of time. A Kalman-extended GA (KGA) is developed to determine when to generate a new individual, and when to re-evaluate an existing one and which to re-evaluate. This KGA is applied to the problem of maintaining a network configuration with minimized message loss, with mobile nodes and stochastic transmission. As the nodes move, the optimal network changes, but information contained within the population of solutions allows efficient discovery of better-adapted solutions. The sensitivity of the KGA to several control parameters is explored. In practical applications evaluating a fitness function is frequently subject to noise, i. e., the “true fitness” is disturbed by some random variations. Evolutionary algorithms (EAs) are often successfully applied to noisy problems, where they have turned out to be particularly robust. Theoretical results on the behavior of EAs for noisy functions are comparatively very rare, especially for discrete search spaces. Here we present an analysis of the (1 + 1) EA for a noisy variant of OneMax and compute the maximal noise strength allowing the (1 + 1) EA a polynomial runtime asymptotically exactly. The methods used in the proofs are presented in a general form with clearly stated conditions in order to simplify further applications. Practical optimization problems frequently include uncertainty about the quality measure, for example due to noisy evaluations. Thus, they do not allow for a straightforward application of traditional optimization techniques. In these settings, randomized search heuristics such as evolutionary algorithms are a popular choice because they are often assumed to exhibit some kind of resistance to noise. Empirical evidence suggests that some algorithms, such as estimation of distribution algorithms (EDAs) are robust against a scaling of the noise intensity, even without resorting to explicit noise-handling techniques such as resampling. In this paper, we want to support such claims with mathematical rigor. We introduce the concept of graceful scaling in which the run time of an algorithm scales polynomially with noise intensity. We study a monotone fitness function over binary strings with additive noise taken from a Gaussian distribution. We show that myopic heuristics cannot efficiently optimize the function under arbitrarily intense noise without any explicit noise-handling. Furthermore, we prove that using a population does not help. Finally we show that a simple EDA called the Compact Genetic Algorithm can overcome the shortsightedness of mutation-only heuristics to scale gracefully with noise. We conjecture that recombinative genetic algorithms also have this property. This extended abstract summarizes our work "The Benefit of Recombination in Noisy Evolutionary Search," which appeared in Proceedings of International Symposium on Algorithms and Computation (ISAAC), 2015, pp. 140--150.