114,114 research outputs found
Black-Box Complexity: Breaking the Barrier of LeadingOnes
We show that the unrestricted black-box complexity of the -dimensional
XOR- and permutation-invariant LeadingOnes function class is . This shows that the recent natural looking bound is
not tight.
The black-box optimization algorithm leading to this bound can be implemented
in a way that only 3-ary unbiased variation operators are used. Hence our bound
is also valid for the unbiased black-box complexity recently introduced by
Lehre and Witt (GECCO 2010). The bound also remains valid if we impose the
additional restriction that the black-box algorithm does not have access to the
objective values but only to their relative order (ranking-based black-box
complexity).Comment: 12 pages, to appear in the Proc. of Artificial Evolution 2011, LNCS
7401, Springer, 2012. For the unrestricted black-box complexity of
LeadingOnes there is now a tight bound, cf.
http://eccc.hpi-web.de/report/2012/087
OneMax in Black-Box Models with Several Restrictions
Black-box complexity studies lower bounds for the efficiency of
general-purpose black-box optimization algorithms such as evolutionary
algorithms and other search heuristics. Different models exist, each one being
designed to analyze a different aspect of typical heuristics such as the memory
size or the variation operators in use. While most of the previous works focus
on one particular such aspect, we consider in this work how the combination of
several algorithmic restrictions influence the black-box complexity. Our
testbed are so-called OneMax functions, a classical set of test functions that
is intimately related to classic coin-weighing problems and to the board game
Mastermind.
We analyze in particular the combined memory-restricted ranking-based
black-box complexity of OneMax for different memory sizes. While its isolated
memory-restricted as well as its ranking-based black-box complexity for bit
strings of length is only of order , the combined model does not
allow for algorithms being faster than linear in , as can be seen by
standard information-theoretic considerations. We show that this linear bound
is indeed asymptotically tight. Similar results are obtained for other memory-
and offspring-sizes. Our results also apply to the (Monte Carlo) complexity of
OneMax in the recently introduced elitist model, in which only the best-so-far
solution can be kept in the memory. Finally, we also provide improved lower
bounds for the complexity of OneMax in the regarded models.
Our result enlivens the quest for natural evolutionary algorithms optimizing
OneMax in iterations.Comment: This is the full version of a paper accepted to GECCO 201
Toward a complexity theory for randomized search heuristics : black-box models
Randomized search heuristics are a broadly used class of general-purpose algorithms. Analyzing them via classical methods of theoretical computer science is a growing field. While several strong runtime bounds exist, a powerful complexity theory for such algorithms is yet to be developed. We contribute to this goal in several aspects. In a first step, we analyze existing black-box complexity models. Our results indicate that these models are not restrictive enough. This remains true if we restrict the memory of the algorithms under consideration. These results motivate us to enrich the existing notions of black-box complexity by the additional restriction that not actual objective values, but only the relative quality of the previously evaluated solutions may be taken into account by the algorithms. Many heuristics belong to this class of algorithms. We show that our ranking-based model gives more realistic complexity estimates for some problems, while for others the low complexities of the previous models still hold. Surprisingly, our results have an interesting game-theoretic aspect as well.We show that analyzing the black-box complexity of the OneMaxn function class—a class often regarded to analyze how heuristics progress in easy parts of the search space—is the same as analyzing optimal winning strategies for the generalized Mastermind game with 2 colors and length-n codewords. This connection was seemingly overlooked so far in the search heuristics community.Randomisierte Suchheuristiken sind vielseitig einsetzbare Algorithmen, die aufgrund ihrer hohen Flexibilität nicht nur im industriellen Kontext weit verbreitet sind. Trotz zahlreicher erfolgreicher Anwendungsbeispiele steckt die Laufzeitanalyse solcher Heuristiken noch in ihren Kinderschuhen. Insbesondere fehlt es uns an einem guten Verständnis, in welchen Situationen problemunabhängige Heuristiken in kurzer Laufzeit gute Lösungen liefern können. Eine Komplexitätstheorie ähnlich wie es sie in der klassischen Algorithmik gibt, wäre wünschenswert. Mit dieser Arbeit tragen wir zur Entwicklung einer solchen Komplexitätstheorie für Suchheuristiken bei. Wir zeigen anhand verschiedener Beispiele, dass existierende Modelle die Schwierigkeit eines Problems nicht immer zufriedenstellend erfassen. Wir schlagen daher ein weiteres Modell vor. In unserem Ranking-Based Black-Box Model lernen die Algorithmen keine exakten Funktionswerte, sondern bloß die Rangordnung der bislang angefragten Suchpunkte. Dieses Modell gibt für manche Probleme eine bessere Einschätzung der Schwierigkeit. Wir zeigen jedoch auch, dass auch im neuen Modell Probleme existieren, deren Komplexität als zu gering einzuschätzen ist. Unsere Ergebnisse haben auch einen spieltheoretischen Aspekt. Optimale Gewinnstrategien für den Rater im Mastermindspiel (auch SuperHirn) mit n Positionen entsprechen genau optimalen Algorithmen zur Maximierung von OneMaxn-Funktionen. Dieser Zusammenhang wurde scheinbar bislang übersehen. Diese Arbeit ist in englischer Sprache verfasst
KL-based Control of the Learning Schedule for Surrogate Black-Box Optimization
This paper investigates the control of an ML component within the Covariance
Matrix Adaptation Evolution Strategy (CMA-ES) devoted to black-box
optimization. The known CMA-ES weakness is its sample complexity, the number of
evaluations of the objective function needed to approximate the global optimum.
This weakness is commonly addressed through surrogate optimization, learning an
estimate of the objective function a.k.a. surrogate model, and replacing most
evaluations of the true objective function with the (inexpensive) evaluation of
the surrogate model. This paper presents a principled control of the learning
schedule (when to relearn the surrogate model), based on the Kullback-Leibler
divergence of the current search distribution and the training distribution of
the former surrogate model. The experimental validation of the proposed
approach shows significant performance gains on a comprehensive set of
ill-conditioned benchmark problems, compared to the best state of the art
including the quasi-Newton high-precision BFGS method
Optimal model-free prediction from multivariate time series
© 2015 American Physical Society.Forecasting a time series from multivariate predictors constitutes a challenging problem, especially using model-free approaches. Most techniques, such as nearest-neighbor prediction, quickly suffer from the curse of dimensionality and overfitting for more than a few predictors which has limited their application mostly to the univariate case. Therefore, selection strategies are needed that harness the available information as efficiently as possible. Since often the right combination of predictors matters, ideally all subsets of possible predictors should be tested for their predictive power, but the exponentially growing number of combinations makes such an approach computationally prohibitive. Here a prediction scheme that overcomes this strong limitation is introduced utilizing a causal preselection step which drastically reduces the number of possible predictors to the most predictive set of causal drivers making a globally optimal search scheme tractable. The information-theoretic optimality is derived and practical selection criteria are discussed. As demonstrated for multivariate nonlinear stochastic delay processes, the optimal scheme can even be less computationally expensive than commonly used suboptimal schemes like forward selection. The method suggests a general framework to apply the optimal model-free approach to select variables and subsequently fit a model to further improve a prediction or learn statistical dependencies. The performance of this framework is illustrated on a climatological index of El Niño Southern Oscillation
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