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Stochastic Constraint Programming

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Abstract

To model combinatorial decision problems involving uncertainty and proba- bility, we introduce stochastic constraint programming. S tochastic constraint pro- grams contain both decision variables (which we can set) and stochastic variables (which follow a probability distribution). They combine together the best fea- tures of traditional constraint satisfaction, stochastic integer programming, and stochastic satisfiability. We give a semantics for stochast ic constraint programs, and present a complete forward checking algorithm. Finally, we discuss a number of extensions of stochastic constraint programming to rela x various assumptions like the independence between stochastic variables, and compare stochastic con- straint programming with other approaches for decision making under uncertainty like Markov decision problems and influence diagrams.

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... Dans (Koriche et al., 2016), nous introduisons une nouvelle approche au General Game Playing, dénommée MAC-UCB, basée sur la programmation par contraintes stochastiques (SCSP) (Walsh, 2009). Expérimentalement, ce dernier s'est montré compétitif toutefois à ce jour aucun programme-joueur n'utilise actuellement cet algorithme en pratique. ...
... Ainsi, dans le but de modéliser des problèmes de décision combinatoire permettant la prise en compte d'incertitudes et de probabilités, nous nous intéressons au cadre SCSP (pour Stochastic Constraint Satisfaction Problem) (Walsh, 2009) inspiré du problème de satisfaction stochastique (Littman et al., 2001). ...
... , T }, MAC-UCB recherche l'ensemble des politiques solutions en décomposant le problème en deux parties: un CSP classique et un µSCSP (plus petit que l'original). La première partie est résolue à l'aide de l'algorithme MAC (Sabin, Freuder, 1994) et la seconde partie grâce à l'algorithme SFC dédié au cadre SCSP (Walsh, 2009). Par la suite, une série d'échantillonnages avec borne de confiance est réalisée pour simuler l'utilité attendue de chaque politique solution de chaque µSCSP t . ...
... Dans (Koriche et al., 2016), nous introduisons une nouvelle approche au General Game Playing, dénommée MAC-UCB, basée sur la programmation par contraintes stochastiques (SCSP) (Walsh, 2009). Expérimentalement, ce dernier s'est montré compétitif toutefois à ce jour aucun programme-joueur n'utilise actuellement cet algorithme en pratique. ...
... Ainsi, dans le but de modéliser des problèmes de décision combinatoire permettant la prise en compte d'incertitudes et de probabilités, nous nous intéressons au cadre SCSP (pour Stochastic Constraint Satisfaction Problem) (Walsh, 2009) inspiré du problème de satisfaction stochastique (Littman et al., 2001). ...
... , T }, MAC-UCB recherche l'ensemble des politiques solutions en décomposant le problème en deux parties: un CSP classique et un µSCSP (plus petit que l'original). La première partie est résolue à l'aide de l'algorithme MAC (Sabin, Freuder, 1994) et la seconde partie grâce à l'algorithme SFC dédié au cadre SCSP (Walsh, 2009). Par la suite, une série d'échantillonnages avec borne de confiance est réalisée pour simuler l'utilité attendue de chaque politique solution de chaque µSCSP t . ...
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This article describes WoodStock, the first general game player modeling each game from the General Game Playing (GGP) by a stochastic constraint network (SCSP). Each action played is decided by the resolution of this last one by the algorithm MAC-UCB. After the translation of an instance described in Game Description Language (GDL) in a network representative of the state of the game at any time, WoodStock solves each state by the maintening arc-consistency algorithm (MAC) iteratively guided by the bandit-based stochastic sampling (UCB) of the next states. Thanks to this algorithm, WoodStock is since march 2016, the leader of the GGP Tiltyard continuous tournament. Moreover, in its last version exploiting the game symmetries finding by the constraint symmetry detection, the search space associated with a game is significatively reduced. With that, WoodStock is now the GGP champion after its victory at the International General Game Playing Competition 2016 (IGGPC 2016) organized by the Stanford University.
... Yet, these approaches cannot address random players and, more generally, the connection with GDLII. The aim of this article is to handle GGP with an original viewpoint, based on Stochastic Constraint Satisfaction Problems (SCSP) (Walsh 2002). This formalism is expressive enough to model and solve games with complete information, including those described in QCSP (Balafoutis and Stergiou 2006), but also to capture random players. ...
... The stochastic CSP model that we present in this study extends the original framework of Walsh (2002) to deal with soft constraints. ...
... In a SCSP, the scope of any hard constraint is restricted to decision variables. We extend the SCSP framework defined in Walsh (Walsh 2002) to deal with soft constraints. Specifically, for any soft constraint c, the range of its value function val c is R. ...
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Among the languages used for representing goals, actions and their consequences on the world for decision making and planning, GDL (Game Description Language) has the ability to represent complex actions in potentially uncertain and competitive environments. The aim of this paper is to exploit stochastic constraint networks in order to provide compact representations of strategic games, and to identify optimal policies in those games with generic forward checking method. From this perspective, we develop a compiler allowing to translate games, described in GDL, into instances of the Stochastic Constraint Optimization Problem (SCSP). Our compiler is proved correct for the class GDL of games with complete information and oblivious environment. The interest of our approach is illustrated by solving several GDL games with a SCSP solver.
... Most of these formalisms are, however, restricted to deterministic, perfect information games: during each round of the game, players have full access to the current state and their actions have deterministic effects. This paper focuses on stochastic games, with chance events, using the framework of stochastic constraint networks [12,26,30]. ...
... From this perspective, we study a fragment of the Stochastic Constraint Satisfaction Problem (SCSP), that captures GDL games with uncertain (but complete) information. Interestingly, the SCSP for this class of games can be decomposed into a sequence of µSCSPs (also known as one-stage stochastic constraint satisfaction problems [30]). Based on this decomposition, we propose a sequential decision-making algorithm, MAC-UCB, that combines the MAC (Maintaining Arc Consistency) technique for solving each µSCSP in the sequence, and the multi-armed bandits Upper Confidence Bound (UCB) method [1] for approximating the expected utility of strategies. ...
... From a game-theoretic viewpoint, the stochastic constraint networks investigated in [12,26,30] capture one-player stochastic games, in which the chance player (defined over stochastic variables) is "oblivious": at each round of the game, the probability distribution over the chance player's moves is independent from the description of the current state. In order to encode GDL programs into stochastic constraint programs, we shall examine in this section a slight generalization of the original SCSP model that captures multiplayer and non-oblivious stochastic games. ...
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The challenge of General Game Playing (GGP) is to devise game playing programs that take as input the rules of any strategic game, described in the Game Description Language (GDL), and that effectively play without human intervention. The aim of this paper is to address the GGP challenge by casting GDL games (potentially with chance events) into the Stochastic Constraint Satisfaction Problem (SCSP). The stochastic constraint network of a game is decomposed into a sequence of µ SCSPs (also know as one-stage SCSP), each associated with a game round. Winning strategies are searched by coupling the MAC (Maintaining Arc Consistency) algorithm, used to solve each µ SCSP in turn, together with the UCB (Upper Confidence Bound) policy for approximating the values of those strategies obtained by the last µ SCSP in the sequence. Extensive experiments conducted on various GDL games with different deliberation times per round, demonstrate that the MAC-UCB algorithm significantly outperforms the state-of-the-art UCT (Upper Confidence bounds for Trees) algorithm.
... It allows the user to augment deterministic models to stochastic models without the need to commit to a particular solving strategy. To solve these stochastic models, we present transformations from Stochastic MINIZINC to deterministic MINIZINC for three different stochastic solving approaches: scenario-based [19] and policy-based [24] deterministic equivalents, as well as progressive hedging [15]. These are the only known stochastic solving techniques that can deal with both integer decision variables and non-linear constraints. ...
... All stochastic parameters and each second (and higher-)stage variable is copied for each scenario, as well as all constraints involving higher stage parameters or variables. Policy-based Search [24] treats stochastic parameters as decision variables and uses and-or-search to explore all possible scenarios. Finally, Progressive Hedging [15] solves each scenario to optimality in isolation, and then iteratively adapts the objective function to minimise the gap between the first stage variables. ...
... Policy-based search for stochastic constraint programming [24] turns stochastic parameters into decision variables and then uses backtracking search to explore the different scenarios. Instead of copying the second stage model for each scenario as in Sect. ...
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Combinatorial optimisation problems often contain uncertainty that has to be taken into account to produce realistic solutions. However, existing modelling systems either do not support uncertainty, or do not support combinatorial features, such as integer variables and non-linear constraints. This paper presents an extension of the MiniZinc modelling language that supports uncertainty. Stochastic MiniZinc enables modellers to express combinatorial stochastic problems at a high level of abstraction, independent of the stochastic solving approach. These models are translated automatically into different solver-level representations. Stochastic MiniZinc provides the first solving technology agnostic approach to stochastic modelling we are aware of.
... Such variables are typically uncontrollable (i.e., not decision variables). This is for example the case with the Stochastic CSP (SCSP) framework introduced by T. Walsh [135] to capture combinatorial decision problems involving uncertainty. In XCSP3, the domain of an integer stochastic variable is defined as usual by a sequence of integers and integer intervals, but each element of the sequence is given a probability preceded by the symbol " : " . ...
... @BULLET handle stages with an element <stages>, which alternately contains elements <decision> and <stochastic>; in each of these elements, we find a sequence of variable ids. The syntax is: Here is an illustration taken from [135] about modeling a simple m quarter production planning problem. In each quarter, we will sell between 100 and 105 copies of a book. ...
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... Mais aucun d'entre eux n'aborde les problèmes liésà l'aléatoire ni ne fait le lien avec GDL. L'objectif de cet article est d'aborder le problème GGP d'un point de vue original, fondé sur les réseaux de contraintes stochastiques (SCSP) [18]. En effet, les SCSP sont des outils puissants permettant non seulement de modéliser et de résoudre des jeuxà infor-mations complètes et certaines via QCSP [2], maiségalement de modéliser l'intervention du hasard. ...
... Le modèle des CSP stochastiques que nous présentons dans cetteétude,étend le cadre original de Walsh [18] aux contraintes valuées. De manière formelle, un Stochastic Constraint Satisfaction Problem (SCSP) est un 6-tuple X, Y, D, P, C, θ où X représente un ensemble de n variables, Y est un sous-ensemble de X représentant les variables stochastiques, D représente les domaines associés aux variables de X, P est l'ensemble des distributions de probabilités appliquées aux domaines des variables stochastiques, C est l'ensemble des contraintes et θ représente une valeur de seuil dans R. ...
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Cet article propose d’utiliser le paradigme de la programmation par contraintes stochastiques pour modéliser et identifier des politiques optimales dans les jeux à information incertaine. Nous présentons une traduction permettant de modéliser les jeux décrits dans le formalisme GDL (Game Description Language) en instances du problème d’optimisation de contraintes stochastiques (SCSP). Notre traduction est démontrée correcte pour la classe GDL des jeux à information complète et environnement « indifférent ». L’intérêt de notre approche est illustré par une première résolution d’un jeu GDL en utilisant un solveur SCSP générique.
... Cependant, la plupart de ces formalismes sont restreints aux jeuxà information complète et certaine :à chaque instant, les joueurs ont une information complète de l'état du jeu et leurs actions sont déterministes. Cetteétude se focalise sur le formalisme des réseaux de contraintes stochastiques (SCSP) [21], utilisés pour la prise de décision séquentielle, dont l'effet incertain des actions est modélisé par une distribution de probabilités. ...
... Il existe cependant une sous-classe intéressante de SCSP pour laquelle le problème de satisfaction est de complexité moindre (NP PP ). Il s'agit des one stage SCSP [21], ou micro-SCSP, que l'on note ici µSCSP. Formellement, un µSCSP est un SCSP (X, Y, D, P, C, θ), dans lequel la restriction suivante est imposée sur les politiques : un arbre d'instanciations complètes π est une politique de µSCSP si toutes les variables de décision X sont assignées avant les variables stochastiques Y . ...
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Cet article examine un fragment du problème de satisfaction de contraintes stochastiques, représentant les jeux à informations complètes et incertaines. Nous proposons un algorithme de résolution permettant de trouver des stratégies gagnantes pour cette classe de jeux. L’intérêt de ce travail est de permettre la résolution efficace d’un SCSP initial par la résolution successive d’une séquence de « micro-SCSP » à chaque étape du jeu. L’algorithme MAC, propose pour résoudre les premiers micro-SCSP de la séquence, est couplé à une heuristique de type UCB utilisée pour estimer les valeurs des stratégies sur toute la séquence. Notre approche, MAC-UCB, est validée par une série d’expérimentations sur un ensemble de jeux très divers, décrits en GDL (Game Description Language), et comparée à l’algorithme UCT, qui est la référence actuelle pour cette classe de jeux.
... CSP Stochastiques. Présentés par Tobi Walsh en 2002 dans [58], les CSP stochastiques (ou SCSP) étendent les problèmes de contraintes traditionnels en introduisant des variables stochastiques ainsi qu'une borne inférieure de probabilite θ. Les variables stochastiques disposent d'une distribution de probabilité sur l'ensemble des valeurs de leur domaine. ...
... Dans le cadre voisin des CSP stochastiques, [58] définit aussi deux versions d'optimisation du problème. La première consiste à rechercher la politique ayant la plus grande probabilité de satisfaire toutes les contraintes, et non pas simplement une politique dépassant un certain seuil de probabilité. ...
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... Problems that involve these kinds of hard constraints on probabilities are the focus of the field of stochas- tic constraint programming (SCP) (Walsh 2002), which combines probabilistic inference and constraint program- ming to solve Stochastic Constraint Optimization Prob- lems (SCOPs). SCP is closely related to chance constraint programming (Charnes and Cooper 1959) and probabilistic constraint programming (Tarim et al. 2009). ...
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... The techniques here presented generalise the discussion in [12], in which statistical inference is applied in the context of stochastic constraint satisfaction to identify approximate solutions featuring given statistical properties. However, stochastic constraint programming [17] works with decision and random variables over a set of decision stages; random variable distributions are assumed to be known. Statistical constraints instead operate under the assumption that distribution of random variables is only partially specified (parametric statistical constraints) or not specified at all (non-parametric statistical constraints); furthermore, statistical constraints do not model explicitly random variables, they model instead sets of random variates as decision variables. ...
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... However, the schedule table size grows exponentially with the number of conditions. Therefore, the problem of finding an optimal schedule table becomes P-Space complete [5]. ...
... Empruntant la terminologie de [23] ...
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Le game description language avec informations incomplètes (GDL-II) est assez expressif pour représenter les jeux stochastiques multi-agents avec observation partielle. Malheureusement, une telle expressivité n’est pas possible sans un prix : le problème consistant à trouver une stratégie gagnante est NExp^(NP)-hard, une classe de complexité qui est bien au-delà de la portée des solvers actuels. Dans ce papier, nous identifions un fragment Pspace-complete de GDL-II, où les agents partagent les mêmes observations (partielles). Nous montrons que ce fragment peut être encapsulé dans un problème de satisfaction de contraintes stochastiques décomposable (SCSP) qui, par tour, peut être résolu en utilisant des techniques de programmation par contraintes usuelles. Dès lors, nous avons développé un algorithme de décisions séquentielles fondé sur les contraintes pour les jeux GDL-II exploitant la propagation par contraintes, l’évaluation Monte-Carlo et la détection de symétries. Notre algorithme, vérifié sur une large variété de jeux, surpasse aisément l’état de l’art des algorithmes du general game playing (GGP).
... A prominent example is stochastic two-stage optimisation [18], where the objective is to find optimal first-stage variable assignments (master problem) such that all second-stage variable assignments (sub-problems) are optimal for each scenario. AND/OR search for stochastic optimisation is called policy based search [27]. AND/OR search is also applied in other applications, such as graphical models [11]. ...
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Much of the power of CP comes from the ability to create complex hybrid search algorithms specific to an application. Unfortunately there is no widely accepted standard for specifying search, and each solver typically requires detailed knowledge in order to build complex searches. This makes the barrier to entry for exploring different search methods quite high. Furthermore, search is a core part of the solver and usually highly optimised. Any imposition on the solver writer to change this part of their system is significant. In this paper we investigate how powerful we can make a uniform language for meta-search without placing any burden on the solver writer. The key to this is to only interact with the solver when a solution is found. We present MINISEARCH, a meta-search language that can directly use any FLATZINC solver. Optionally, it can interact with solvers through an efficient C++ API. We illustrate the expressiveness of the language and performance using different solvers on a number of examples.
... La classification intuitive des algorithmes de recherche semble grossièrement correspondre aux classes de complexité. Les algorithmes n'effectuant pas de recherche correspondentà la classe P. Les Le model-checking de formules CTL ou de sous-ensembles de cette logique temporelle [Clarke et al., 2000], le calcul de plans d'actions pour des problèmes exprimés en STRIPS propositionnel [Fikes et Nilsson, 1971], certains jeux et certains problèmes de décision en présence de données incertaines [Walsh, 2002] sont parmi les nombreux exemples de problèmes PSPACE-complets, et il est donc effectivement impossible de les exprimer comme des problèmes de satisfaisabilité propositionnelle 5 . Ceci ne signifie pas que les résolveurs de contraintes ne soient d'aucune utilité pour ces problèmes : l'une des approches les plus compétitives pour la résolution de ces problèmes consiste précisémentà exprimer des versions bornées des problèmes en question sous forme de problèmes SAT ou CSP (existence d'un plan d'une longueur donnée [Kautz et Selman, 1992], vérification de séquences d'exécution de longueur bornée [Clarke et al., 2001]). ...
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... Here, the outcome sought is a single robust solution that covers as many realisations as possible. As such, there are links not only to anytime methods for robust solutions to CSPs [8], but also to solving mixed CSPs with probability distributions over the parameters [4], which are an instance of the stochastic CSP framework [18]. For instance, the scenario sampling methods for stochastic CSPs give the opportunity for an anytime algorithm [10]. ...
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An algorithm with the anytime property has an approximate solution always available; and the longer the algorithm runs, the better the solution be-comes. Anytime solving is important in domains such as aerospace, where time for reasoning is limited and a viable (if suboptimal) course of action must be al-ways available. In this paper we study the anytime behaviour of solving a mixed CSP, an extension of classical CSP that accounts for uncontrollable parameters, using a benchmark problem from aerospace sub-system control. We propose two enhancements to the existing decomposition algorithm: heuristics for selecting the next uncertain environment to decompose, and solving of incrementally larger subproblems. We evaluate these enhancements empirically, showing that a heuris-tic on uncertainty analogous to 'first fail' gives the best performance. We also show that incremental subproblem solving provides effective anytime behaviour, and can be combined with the decomposition heuristics.
... The Uncertain CSP model (UCSP) is an extension of MCSP whose main innovation is that it considers continuous domains (Yorke-Smith & Gervet, 2009). The Stochastic CSP model (SCSP) (Walsh, 2002) assumes a probability distribution associated with the uncertain domain of each uncontrollable variable. The Branching CSP model (BCSP) considers the possible addition of variables to the current problem (Fowler & Brown, 2000). ...
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Because of the dynamism and uncertainty associated with many real life problems, these problems and their associated Constraint Satisfaction Problem (CSP) models may change over time; thus an earlier solution found for the latter may become invalid. Moreover, many approaches proposed in the literature cannot be applied when the required information about dynamism is unknown ([9], [4], [5], [11], [10], etc.). This fact has motivated us to consider dynamic situations where, in addition, only limited assumptions about changes can be made. Our analysis focuses on CSPs with ordered and discrete domains that model problems for which the order over the elements of the domain is significant. In these cases, a common type of change that problems may undergo is restrictive modifications over the bounds of the solution space. A discussion of these assumptions, their motivation and real life examples can be found in [3]. In this paper, we present an algorithm that meets the goal of combining solution stability (meaning that solutions can often be repaired using other similar values if they undergo a value loss) and robustness (meaning that solutions have a high likelihood of remaining solutions after changes). The desireability of this combination of features was noted in the survey [8]. Furthermore, in this work we have extended both concepts to apply to the type of CSP analyzed. The paper is organized as follows. Section 2 presents the new conceptions of robustness and stability. Section 3 describes our approach for finding solutions that simultaneously meet both these criteria. In Section 4 we present some experimental results. Section 5 gives conclusions.
... Brown et al. propose adversarial CSPs [6], which focuses on the case where two opponents take turns to assign variables, each trying to direct the solution towards their own objectives. Another related work is Stochastic CSPs [26], which can represent adversaries by known probability distributions. We seek actions to minimize/maximize the expected cost for all the possible scenarios. ...
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