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See detailApproximate Bayes Optimal Policy Search using Neural Networks
Castronovo, Michaël ULiege; François-Lavet, Vincent ULiege; Fonteneau, Raphaël ULiege et al

in Proceedings of the 9th International Conference on Agents and Artificial Intelligence (ICAART 2017) (2017, February)

Bayesian Reinforcement Learning (BRL) agents aim to maximise the expected collected rewards obtained when interacting with an unknown Markov Decision Process (MDP) while using some prior knowledge. State ... [more ▼]

Bayesian Reinforcement Learning (BRL) agents aim to maximise the expected collected rewards obtained when interacting with an unknown Markov Decision Process (MDP) while using some prior knowledge. State-of-the-art BRL agents rely on frequent updates of the belief on the MDP, as new observations of the environment are made. This offers theoretical guarantees to converge to an optimum, but is computationally intractable, even on small-scale problems. In this paper, we present a method that circumvents this issue by training a parametric policy able to recommend an action directly from raw observations. Artificial Neural Networks (ANNs) are used to represent this policy, and are trained on the trajectories sampled from the prior. The trained model is then used online, and is able to act on the real MDP at a very low computational cost. Our new algorithm shows strong empirical performance, on a wide range of test problems, and is robust to inaccuracies of the prior distribution. [less ▲]

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See detailApproximate Bayesian Computation reveals the crucial role of oceanic islands for the assembly of continental biodiversity
Patino, J; Carine, M; Mardulyn, P et al

in Systematic Biology (2015), 64

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See detailApproximate compositional values and tissue fatty acid profiles of Nile Tilapia (Oreachromis niloticus L.) fed azolla-diets in earthen ponds
Abou, Youssouf; Fiogbé, Emile Didier; Beckers, Yves ULiege et al

in Food and Nutrition Sciences (2011), 2

The approximate general composition and the fatty acid profile of Nile tilapia fed Azolla-diets in ponds were studied for 90 days. Six isonitrogenous (29.2% CP) and isoenergetic (16.9 kJ·g−1) diets were ... [more ▼]

The approximate general composition and the fatty acid profile of Nile tilapia fed Azolla-diets in ponds were studied for 90 days. Six isonitrogenous (29.2% CP) and isoenergetic (16.9 kJ·g−1) diets were formulated to contain 0% (A0), 10% (A10), 20% (A20), 30% (A30), 40% (A40) and 50% (A50) of Azolla meal (AM), as partial fish meal (FM) substitutes. Diet A0 without AM served as a control. Fish growth decreased as AM level exceeded 20% in diets (P < 0.05). Dry matter and crude protein showed no significant differences (P > 0.05). Crude lipid was significantly lower in fish fed A50 and significant differences were also found in crude ash (P < 0.05). Linolenic acid (LLA) decreased significantly when AM level in diets increased (P < 0.05). In contrast, arachidonic acid (ARA), eicosapentaenoic acid (EPA) and docosahex- aenoic acid (DHA) showed significantly higher values in fish fed high AM (P < 0.05). The n − 3/n − 6 ratio ranged from 0.35 to 0.49, with values being significantly higher in fish fed A20, A30 and A50. High level of the fern reduces growth without negatively affecting fatty acid in fish. Fish PUFA, especially the (n − 3) fatty acids, are affected posi- tively, even when fed 50% AM, which is good for the quality of the fish produced in regard to the benefits for the health of consumers. [less ▲]

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See detailApproximate Conditions Replacing Thin Layers
Poignard, Clair; Dular, Patrick ULiege; Perrussel, Roman et al

in IEEE Transactions on Magnetics (2008), 44(6), 1154-1157

We provide a rigorous asymptotic method to build approximate conditions equivalent to a thin layer. With these conditions the electromagnetic field may be computed in domains such as a biological cell ... [more ▼]

We provide a rigorous asymptotic method to build approximate conditions equivalent to a thin layer. With these conditions the electromagnetic field may be computed in domains such as a biological cell, without meshing the membrane. The influence of the membrane is replaced by an appropriate condition on the inner domain, while in the thin layer, the approximate field is explicitly known. We give error estimates, which validate our asymptotic method, and we present a few numerical results performed with the finite-element method. [less ▲]

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See detailApproximate dynamic programming with a fuzzy parameterization
Busoniu, Lucian; Ernst, Damien ULiege; De Schutter, Bart et al

in Automatica (2010), 46(5), 804-814

Dynamic programming (DP) is a powerful paradigm for general, nonlinear optimal control. Computing exact DP solutions is in general only possible when the process states and the control actions take values ... [more ▼]

Dynamic programming (DP) is a powerful paradigm for general, nonlinear optimal control. Computing exact DP solutions is in general only possible when the process states and the control actions take values in a small discrete set. In practice, it is necessary to approximate the solutions. Therefore, we propose an algorithm for approximate DP that relies on a fuzzy partition of the state space, and on a discretization of the action space. This fuzzy Q-iteration algorithm works for deterministic processes, under the discounted return criterion. We prove that fuzzy Q-iteration asymptotically converges to a solution that lies within a bound of the optimal solution. A bound on the suboptimality of the solution obtained in a finite number of iterations is also derived. Under continuity assumptions on the dynamics and on the reward function, we show that fuzzy Q-iteration is consistent, i.e., that it asymptotically obtains the optimal solution as the approximation accuracy increases. These properties hold both when the parameters of the approximator are updated in a synchronous fashion, and when they are updated asynchronously. The asynchronous algorithm is proven to converge at least as fast as the synchronous one. The performance of fuzzy Q-iteration is illustrated in a two-link manipulator control problem. [less ▲]

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See detailApproximate local magnetic-to-electric surface operators for time-harmonic Maxwell's equations
Bouajaji, M. El; Antoine, X.; Geuzaine, Christophe ULiege

in Journal of Computational Physics (2014), 279(15), 241-260

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See detailApproximate Numerical Continuation for Aeroelastic Systems Undergoing Aperiodic Limit Cycle Oscillations
Dimitriadis, Grigorios ULiege

in Proceedings of the 2007 International Forum on Aeroelasticity and Structural Dynamics (2007, June)

This paper presents a modified numerical continuation approach for predicting the bifurcation behaviour of aeroelastic systems undergoing aperiodic limit cycles oscillations. Such oscillations can occur ... [more ▼]

This paper presents a modified numerical continuation approach for predicting the bifurcation behaviour of aeroelastic systems undergoing aperiodic limit cycles oscillations. Such oscillations can occur due to a number of nonlinear functions. Here, backlash nonlinearity in the aileron stiffness for a Generic Transport Aircraft is considered. It is shown that classical numerical continuation will fail due to the aperiodic nature of the limit cycles and the inability to perform period scaling and phase fixing. An alternative, approximate numerical continuation method is proposed, based on longer numerical integration sequences and a heuristic method for determining the period of the limit cycle oscillations. The approach is applied successfully to a simulated aeroelastic model of the Generic Transport Aircraft with backlash. [less ▲]

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See detailApproximate Policy Iteration for Closed-Loop Learning of Visual Tasks
Jodogne, Sébastien ULiege; Briquet, Cyril ULiege; Piater, Justus ULiege

in Lecture Notes in Computer Science (2006, September), 4212

Approximate Policy Iteration (API) is a reinforcement learning paradigm that is able to solve high- dimensional, continuous control problems. We propose to exploit API for the closed-loop learning of ... [more ▼]

Approximate Policy Iteration (API) is a reinforcement learning paradigm that is able to solve high- dimensional, continuous control problems. We propose to exploit API for the closed-loop learning of mappings from images to actions. This approach requires a family of function approximators that maps visual percepts to a real-valued function. For this purpose, we use Regression Extra-Trees, a fast, yet accurate and versatile machine learning algorithm. The inputs of the Extra-Trees consist of a set of visual features that digest the informative patterns in the visual signal. We also show how to parallelize the Extra-Tree learning process to further reduce the computational expense, which is often essential in visual tasks. Experimental results on real-world images are given that indicate that the combination of API with Extra-Trees is a promising framework for the interactive learning of visual tasks. [less ▲]

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See detailApproximate reinforcement learning: an overview
Busoniu, Lucian; Babuska, Robert; De Schutter, Bart et al

in Proceedings of the 2011 IEEE International Symposium on Adaptive Dynamic Programming and Reinforcement Learning (ADPRL-11) (2011, April)

Reinforcement learning (RL) allows agents to learn how to optimally interact with complex environments. Fueled by recent advances in approximation-based algorithms, RL has obtained impressive successes in ... [more ▼]

Reinforcement learning (RL) allows agents to learn how to optimally interact with complex environments. Fueled by recent advances in approximation-based algorithms, RL has obtained impressive successes in robotics, artificial intelligence, control, operations research, etc. However, the scarcity of survey papers about approximate RL makes it difficult for newcomers to grasp this intricate field. With the present overview, we take a step toward alleviating this situation. We review methods for approximate RL, starting from their dynamic programming roots and organizing them into three major classes: approximate value iteration, policy iteration, and policy search. Each class is subdivided into representative categories, highlighting among others offline and online algorithms, policy gradient methods, and simulation-based techniques. We also compare the different categories of methods, and outline possible ways to enhance the reviewed algorithms. [less ▲]

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See detailApproximate value iteration in the reinforcement learning context. Application to electrical power system control
Ernst, Damien ULiege; Glavic, Mevludin; Geurts, Pierre ULiege et al

in International Journal of Emerging Electrical Power Systems (2005), 3(1),

In this paper we explain how to design intelligent agents able to process the information acquired from interaction with a system to learn a good control policy and show how the methodology can be applied ... [more ▼]

In this paper we explain how to design intelligent agents able to process the information acquired from interaction with a system to learn a good control policy and show how the methodology can be applied to control some devices aimed to damp electrical power oscillations. The control problem is formalized as a discrete-time optimal control problem and the information acquired from interaction with the system is a set of samples, where each sample is composed of four elements: a state, the action taken while being in this state, the instantaneous reward observed and the successor state of the system. To process this information we consider reinforcement learning algorithms that determine an approximation of the so-called Q-function by mimicking the behavior of the value iteration algorithm. Simulations are first carried on a benchmark power system modeled with two state variables. Then we present a more complex case study on a four-machine power system where the reinforcement learning algorithm controls a Thyristor Controlled Series Capacitor (TCSC) aimed to damp power system oscillations. [less ▲]

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See detailApproximating Functions on a Mesh with Restricted Voronoï Diagrams
Nivoliers, Vincent ULiege; Lévy, Bruno

in Computer Graphics Forum (2013), 32(5), 83-92

We propose a method that computes a piecewise constant approximation of a function defined on a mesh. The approximation is associated with the cells of a restricted Voronoï diagram. Our method optimizes ... [more ▼]

We propose a method that computes a piecewise constant approximation of a function defined on a mesh. The approximation is associated with the cells of a restricted Voronoï diagram. Our method optimizes an objective function measuring the quality of the approximation. This objective function depends on the placement of the samples that define the restricted Voronoï diagram and their associated function values. We study the continuity of the objective function, derive the closed-form expression of its derivatives and use them to design a numerical solution mechanism. The method can be applied to a function that has discontinuities, and the result aligns the boundaries of the Voronoï cells with the discontinuities. Some examples are shown, suggesting potential applications in image vectorization and compact representation of lighting. [less ▲]

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See detailApproximation algorithms for integer covering problems via greedy column generation
Crama, Yves ULiege; Van de Klundert, Joris

in RAIRO : Operations Research = Recherche Opérationnelle (1994), 28

Many combinatorial optimization problems can be formulated as covering problems. In some cases, this covering formulation is not polynomial in the input size of the original problem. In order to solve the ... [more ▼]

Many combinatorial optimization problems can be formulated as covering problems. In some cases, this covering formulation is not polynomial in the input size of the original problem. In order to solve the problem approximately one can apply the greedy algorithm to the covering formulation. In this case, the column generation subproblem is to determine which column the greedy algorithm chooses. This problem might be NP-hard in itself. We propose a modification of the greedy algorithm in which the column generation subproblem is solved approximately, within a factor α. We derive upper bounds for the worst case ratio of this algorithm and related polynomial approximation algorithms. [less ▲]

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See detailApproximation algorithms for multi-dimensional assignment problems with decomposable costs
Bandelt, Hans-Jürgen; Crama, Yves ULiege; Spieksma, Frits C.R.

in Discrete Applied Mathematics (1994), 49

The k-dimensional assignment problem with decomposable costs is formulated as follows. Given is a complete k-partite graph G = (X_0 U ... U X_{k-1}, E), with lX_il = p for each i, and a nonnegative length ... [more ▼]

The k-dimensional assignment problem with decomposable costs is formulated as follows. Given is a complete k-partite graph G = (X_0 U ... U X_{k-1}, E), with lX_il = p for each i, and a nonnegative length function defined on the edges of G. A clique of G is a subset of vertices meeting each X_i in exactly one vertex. The cost of a clique is a function of the lengths of the edges induced by the clique. Four specific cost functions are considered in this paper; namely, the cost of a clique is either the sum of the lengths of the edges induced by the clique (sum costs), or the minimum length of a spanning star (star costs) or of a traveling salesman tour (tour costs) or of a spanning tree (tree costs) of the induced subgraph. The problem is to find a minimumcost partition of the vertex set of G into cliques. We propose several simple heuristics for this problem, and we derive worst-case bounds on the ratio between the cost of the solutions produced by these heuristics and the cost of an optimal solution. The worst-case bounds are stated in terms of two parameters, viz. k and z, where the parameter z indicates how close the edge length function comes to satisfying the triangle inequality. [less ▲]

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See detailApproximation Algorithms for Multi-Dimensional Vector Assignment Problems
Dokka, Trivikram; Crama, Yves ULiege; Spieksma, Frits C.R.

E-print/Working paper (2013)

We consider a special class of axial multi-dimensional assignment problems called multi-dimensional vector assignment (MVA) problems. An instance of the MVA problem is defined by $m$ disjoint sets, each ... [more ▼]

We consider a special class of axial multi-dimensional assignment problems called multi-dimensional vector assignment (MVA) problems. An instance of the MVA problem is defined by $m$ disjoint sets, each of which contains the same number $n$ of $p$-dimensional vectors with nonnegative integral components, and a cost function defined on vectors. The cost of an $m$-tuple of vectors is defined as the cost of their component-wise maximum. The problem is now to partition the $m$ sets of vectors into $n$ $m$-tuples so that no two vectors from the same set are in the same $m$-tuple and so that the total cost of the $m$-tuples is minimized. The main motivation comes from a yield optimization problem in semi-conductor manufacturing. We consider two classes of polynomial-time heuristics for MVA, namely, hub heuristics and sequential heuristics, and we study their approximation ratio. In particular, we show that when the cost function is monotone and subadditive, hub heuristics, as well as sequential heuristics, have finite approximation ratio for every fixed $m$. Moreover, we establish better approximation ratios for certain variants of hub heuristics and sequential heuristics when the cost function is monotone and submodular, or when it is additive. We provide examples to illustrate the tightness of our analysis. Furthermore, we show that the MVA problem is APX-hard even for the case $m=3$ and for binary input vectors. Finally, we show that the problem can be solved in polynomial time in the special case of binary vectors with fixed dimension $p$. [less ▲]

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See detailApproximation algorithms for multi-dimensional vector assignment problems
Crama, Yves ULiege

Conference (2014, July 09)

Detailed reference viewed: 13 (1 ULiège)
See detailApproximation Algorithms for Multi-Dimensional Vector Assignment Problems
Crama, Yves ULiege

Conference (2013, July 04)

Detailed reference viewed: 14 (0 ULiège)
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See detailApproximation algorithms for the design of SDH/SONET networks
Brauner, Nadia; Crama, Yves ULiege; Finke, Gerd et al

in RAIRO : Operations Research = Recherche Opérationnelle (2003), 37(4, OCT-DEC), 235-247

In this paper, a graph partitioning problem that arises in the design of SONET/SDH networks is defined and formalized. Approximation algorithms with performance guarantees are presented. To solve this ... [more ▼]

In this paper, a graph partitioning problem that arises in the design of SONET/SDH networks is defined and formalized. Approximation algorithms with performance guarantees are presented. To solve this problem efficiently in practice, fast greedy algorithms and a tabu-search method are proposed and analyzed by means of an experimental study. [less ▲]

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See detailApproximation algorithms for three-dimensional assignment problems with triangle inequalities
Crama, Yves ULiege; Spieksma, Frits C.R.

in European Journal of Operational Research (1992), 60

The three-dimensional assignment problem (3DA) is defined as follows. Given are three disjoint n-sets of points, and nonnegative costs associated with every triangle consisting of exactly one point from ... [more ▼]

The three-dimensional assignment problem (3DA) is defined as follows. Given are three disjoint n-sets of points, and nonnegative costs associated with every triangle consisting of exactly one point from each set. The problem is to find a minimum-weight collection of n triangles covering each point exactly once. We consider the special cases of 3DA where a distance (verifying the triangle inequalities) is defined on the set of points, and the cost of a triangle is either the sum of the lengths of its sides (problem TΔ ) or the sum of the lengths of its two shortest sides (problem SΔ ). We prove that TΔ and SΔ are NP-hard. For both TΔ and SΔ , we present 1/2- and 1/3-approximate algorithms, i.e. heuristics which always deliver a feasible solution whose cost is at most 3/2, resp. 4/3, of the optimal cost. Computational experiments indicate that the performance of these heuristics is excellent on randomly generated instances of TΔ and SΔ . [less ▲]

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See detailApproximation efficace de mélanges bootstrap d’arbres de Markov pour l’estimation de densité
Schnitzler, François ULiege; Ammar, Sourour; Leray, Philippe et al

in Bougrain, Laurent (Ed.) Actes de la 14e Conférence Francophone sur l'Apprentissage Automatique (CAp 2012) (2012, May 23)

Nous considérons des algorithmes pour apprendre des Mélanges bootstrap d'Arbres de Markov pour l'estimation de densité. Pour les problèmes comportant un grand nombre de variables et peu d'observations ... [more ▼]

Nous considérons des algorithmes pour apprendre des Mélanges bootstrap d'Arbres de Markov pour l'estimation de densité. Pour les problèmes comportant un grand nombre de variables et peu d'observations, ces mélanges estiment généralement mieux la densité qu'un seul arbre appris au maximum de vraisemblance, mais sont plus coûteux à apprendre. C'est pourquoi nous étudions ici un algorithme pour apprendre ces modèles de manière approchée, afin d'accélérer l'apprentissage sans sacrifier la précision. Plus spécifiquement, nous récupérons lors du calcul d'un premier arbre de Markov les arcs qui constituent de bons candidats pour la structure, et ne considérons que ceux-ci lors de l'apprentissage des arbres suivants. Nous comparons cet algorithme à l'algorithme original de mélange, à un arbre appris au maximum de vraisemblance, à un arbre régularisé et à une autre méthode approchée. [less ▲]

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See detailApproximation of reliability for multiple-trait animal models with missing data by canonical transformation
Gengler, Nicolas ULiege; Misztal, I.

in Journal of Dairy Science (1996), 79(2), 317-328

An algorithm for approximation of reliability for multiple traits by multiple diagonalization was modified to support missing data by weighting transformed contributions of records based on the pattern of ... [more ▼]

An algorithm for approximation of reliability for multiple traits by multiple diagonalization was modified to support missing data by weighting transformed contributions of records based on the pattern of missing data. The accuracy of approximation was assessed with simulated and field data by comparing approximate reliabilities with those from direct inversion. Simulated data had several levels of missing data and covariances between traits; correlations were close to those for linear type traits of dairy cattle. Field data were 1) dairy records for milk, fat, and protein yields with 26% of the observations for fat and protein removed and 2) beef records for birth weight, weaning weight, and mean gain after weaning with 43% of observations missing. These files also contained empty fixed effect classes. The algorithm worked best for simulated data, and, when covariances between traits decreased, proportion of missing traits decreased and the number of empty fixed classes decreased. For dairy data, improvement over single-trait reliability occurred only for traits with missing data; for beef data, little or no improvement occurred. The method is useful with multiple diagonalization if the proportion of missing records or number of empty fixed effect classes or covariances between traits is moderate. [less ▲]

Detailed reference viewed: 19 (1 ULiège)