99 research outputs found
The component model for elementary landscapes and partial neighborhoods
Theoretical Computer Science, 545, (2014), pp. 59-75Local search algorithms exploit moves on an adjacency graph of the search space. An âelementary landscapeâ exists if the objective function f is an eigenfunction of the Laplacian of the graph induced by the neighborhood operator; this allows various statistics about the neighborhood to be computed in closed form. A new component based model makes it relatively simple to prove that certain types of landscapes are elementary. The traveling salesperson problem, weighted graph (vertex) coloring and the minimum graph bisection problem yield elementary landscapes under commonly used local search operators. The component model is then used to efficiently compute the mean objective function value over partial neighborhoods for these same problems. For a traveling salesperson problem over n cities, the 2-opt neighborhood can be decomposed into ân/2â1â partial neighborhoods. For graph coloring and the minimum graph bisection problem, partial neighborhoods can be used to focus search on those moves that are capable of producing a solution with a strictly improving objective function value.Air Force Office of Scientific Research, Air Force Materiel Command, USAF, under grant number FA9550-08-1-0422
Unifying a Geometric Framework of Evolutionary Algorithms and Elementary Landscapes Theory
Evolutionary algorithms (EAs) are randomised general-purpose strategies, inspired by natural evolution, often used for finding (near) optimal solutions to problems in combinatorial optimisation. Over the last 50 years, many theoretical approaches in evolutionary computation have been developed to analyse the performance of EAs, design EAs or measure problem difficulty via fitness landscape analysis. An open challenge is to formally explain why a general class of EAs perform better, or worse, than others on a class of combinatorial problems across representations. However, the lack of a general unified theory of EAs and fitness landscapes, across problems and representations, makes it harder to characterise pairs of general classes of EAs and combinatorial problems where good performance can be guaranteed provably. This thesis explores a unification between a geometric framework of EAs and elementary landscapes theory, not tied to a specific representation nor problem, with complementary strengths in the analysis of population-based EAs and combinatorial landscapes. This unification organises around three essential aspects: search space structure induced by crossovers, search behaviour of population-based EAs and structure of fitness landscapes. First, this thesis builds a crossover classification to systematically compare crossovers in the geometric framework and elementary landscapes theory, revealing a shared general subclass of crossovers: geometric recombination P-structures, which covers well-known crossovers. The crossover classification is then extended to a general framework for axiomatically analysing the population behaviour induced by crossover classes on associated EAs. This shows the shared general class of all EAs using geometric recombination P-structures, but no mutation, always do the same abstract form of convex evolutionary search. Finally, this thesis characterises a class of globally convex combinatorial landscapes shared by the geometric framework and elementary landscapes theory: abstract convex elementary landscapes. It is formally explained why geometric recombination P-structure EAs expectedly can outperform random search on abstract convex elementary landscapes related to low-order graph Laplacian eigenvalues. Altogether, this thesis paves a way towards a general unified theory of EAs and combinatorial fitness landscapes
Analysis of combinatorial search spaces for a class of NP-hard problems, An
2011 Spring.Includes bibliographical references.Given a finite but very large set of states X and a real-valued objective function Æ defined on X, combinatorial optimization refers to the problem of finding elements of X that maximize (or minimize) Æ. Many combinatorial search algorithms employ some perturbation operator to hill-climb in the search space. Such perturbative local search algorithms are state of the art for many classes of NP-hard combinatorial optimization problems such as maximum k-satisfiability, scheduling, and problems of graph theory. In this thesis we analyze combinatorial search spaces by expanding the objective function into a (sparse) series of basis functions. While most analyses of the distribution of function values in the search space must rely on empirical sampling, the basis function expansion allows us to directly study the distribution of function values across regions of states for combinatorial problems without the need for sampling. We concentrate on objective functions that can be expressed as bounded pseudo-Boolean functions which are NP-hard to solve in general. We use the basis expansion to construct a polynomial-time algorithm for exactly computing constant-degree moments of the objective function Æ over arbitrarily large regions of the search space. On functions with restricted codomains, these moments are related to the true distribution by a system of linear equations. Given low moments supplied by our algorithm, we construct bounds of the true distribution of Æ over regions of the space using a linear programming approach. A straightforward relaxation allows us to efficiently approximate the distribution and hence quickly estimate the count of states in a given region that have certain values under the objective function. The analysis is also useful for characterizing properties of specific combinatorial problems. For instance, by connecting search space analysis to the theory of inapproximability, we prove that the bound specified by Grover's maximum principle for the Max-Ek-Lin-2 problem is sharp. Moreover, we use the framework to prove certain configurations are forbidden in regions of the Max-3-Sat search space, supplying the first theoretical confirmation of empirical results by others. Finally, we show that theoretical results can be used to drive the design of algorithms in a principled manner by using the search space analysis developed in this thesis in algorithmic applications. First, information obtained from our moment retrieving algorithm can be used to direct a hill-climbing search across plateaus in the Max-k-Sat search space. Second, the analysis can be used to control the mutation rate on a (1+1) evolutionary algorithm on bounded pseudo-Boolean functions so that the offspring of each search point is maximized in expectation. For these applications, knowledge of the search space structure supplied by the analysis translates to significant gains in the performance of search
Coverage, Continuity and Visual Cortical Architecture
The primary visual cortex of many mammals contains a continuous
representation of visual space, with a roughly repetitive aperiodic map of
orientation preferences superimposed. It was recently found that orientation
preference maps (OPMs) obey statistical laws which are apparently invariant
among species widely separated in eutherian evolution. Here, we examine whether
one of the most prominent models for the optimization of cortical maps, the
elastic net (EN) model, can reproduce this common design. The EN model
generates representations which optimally trade of stimulus space coverage and
map continuity. While this model has been used in numerous studies, no
analytical results about the precise layout of the predicted OPMs have been
obtained so far. We present a mathematical approach to analytically calculate
the cortical representations predicted by the EN model for the joint mapping of
stimulus position and orientation. We find that in all previously studied
regimes, predicted OPM layouts are perfectly periodic. An unbiased search
through the EN parameter space identifies a novel regime of aperiodic OPMs with
pinwheel densities lower than found in experiments. In an extreme limit,
aperiodic OPMs quantitatively resembling experimental observations emerge.
Stabilization of these layouts results from strong nonlocal interactions rather
than from a coverage-continuity-compromise. Our results demonstrate that
optimization models for stimulus representations dominated by nonlocal
suppressive interactions are in principle capable of correctly predicting the
common OPM design. They question that visual cortical feature representations
can be explained by a coverage-continuity-compromise.Comment: 100 pages, including an Appendix, 21 + 7 figure
Design of Heuristic Algorithms for Hard Optimization
This open access book demonstrates all the steps required to design heuristic algorithms for difficult optimization. The classic problem of the travelling salesman is used as a common thread to illustrate all the techniques discussed. This problem is ideal for introducing readers to the subject because it is very intuitive and its solutions can be graphically represented. The book features a wealth of illustrations that allow the concepts to be understood at a glance. The book approaches the main metaheuristics from a new angle, deconstructing them into a few key concepts presented in separate chapters: construction, improvement, decomposition, randomization and learning methods. Each metaheuristic can then be presented in simplified form as a combination of these concepts. This approach avoids giving the impression that metaheuristics is a non-formal discipline, a kind of cloud sculpture. Moreover, it provides concrete applications of the travelling salesman problem, which illustrate in just a few lines of code how to design a new heuristic and remove all ambiguities left by a general framework. Two chapters reviewing the basics of combinatorial optimization and complexity theory make the book self-contained. As such, even readers with a very limited background in the field will be able to follow all the content
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
Syntax-based machine translation using dependency grammars and discriminative machine learning
Machine translation underwent huge improvements since the groundbreaking
introduction of statistical methods in the early 2000s, going from very
domain-specific systems that still performed relatively poorly despite the
painstakingly crafting of thousands of ad-hoc rules, to general-purpose
systems automatically trained on large collections of bilingual texts which
manage to deliver understandable translations that convey the general
meaning of the original input.
These approaches however still perform quite below the level of human
translators, typically failing to convey detailed meaning and register, and
producing translations that, while readable, are often ungrammatical and
unidiomatic.
This quality gap, which is considerably large compared to most other
natural language processing tasks, has been the focus of the research in
recent years, with the development of increasingly sophisticated models that
attempt to exploit the syntactical structure of human languages, leveraging
the technology of statistical parsers, as well as advanced machine learning
methods such as marging-based structured prediction algorithms and neural
networks.
The translation software itself became more complex in order to accommodate
for the sophistication of these advanced models: the main translation
engine (the decoder) is now often combined with a pre-processor which
reorders the words of the source sentences to a target language word order, or
with a post-processor that ranks and selects a translation according according
to fine model from a list of candidate translations generated by a coarse
model.
In this thesis we investigate the statistical machine translation problem
from various angles, focusing on translation from non-analytic languages
whose syntax is best described by fluid non-projective dependency grammars
rather than the relatively strict phrase-structure grammars or projectivedependency
grammars which are most commonly used in the literature.
We propose a framework for modeling word reordering phenomena
between language pairs as transitions on non-projective source dependency
parse graphs. We quantitatively characterize reordering phenomena for the
German-to-English language pair as captured by this framework, specifically
investigating the incidence and effects of the non-projectivity of source
syntax and the non-locality of word movement w.r.t. the graph structure.
We evaluated several variants of hand-coded pre-ordering rules in order to
assess the impact of these phenomena on translation quality.
We propose a class of dependency-based source pre-ordering approaches
that reorder sentences based on a flexible models trained by SVMs and and
several recurrent neural network architectures.
We also propose a class of translation reranking models, both syntax-free
and source dependency-based, which make use of a type of neural networks
known as graph echo state networks which is highly flexible and requires
extremely little training resources, overcoming one of the main limitations
of neural network models for natural language processing tasks
BNAIC 2008:Proceedings of BNAIC 2008, the twentieth Belgian-Dutch Artificial Intelligence Conference
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