Optimal uncertainty quantification for legacy data observations of Lipschitz functions

We consider the problem of providing optimal uncertainty quantification (UQ) --- and hence rigorous certification --- for partially-observed functions. We present a UQ framework within which the observations may be small or large in number, and need not carry information about the probability distribution of the system in operation. The UQ objectives are posed as optimization problems, the solutions of which are optimal bounds on the quantities of interest; we consider two typical settings, namely parameter sensitivities (McDiarmid diameters) and output deviation (or failure) probabilities. The solutions of these optimization problems depend non-trivially (even non-monotonically and discontinuously) upon the specified legacy data. Furthermore, the extreme values are often determined by only a few members of the data set; in our principal physically-motivated example, the bounds are determined by just 2 out of 32 data points, and the remainder carry no information and could be neglected without changing the final answer. We propose an analogue of the simplex algorithm from linear programming that uses these observations to offer efficient and rigorous UQ for high-dimensional systems with high-cardinality legacy data. These findings suggest natural methods for selecting optimal (maximally informative) next experiments.Comment: 38 page

Implicit Hitting Set Algorithms for Constraint Optimization

Computationally hard optimization problems are commonplace not only in theory but also in practice in many real-world domains. Even determining whether a solution exists can be NP-complete or harder. Good, ideally globally optimal, solutions to instances of such problems can save money, time, or other resources. We focus on a particular generic framework for solving constraint optimization problems, the so-called implicit hitting set (IHS) approach. The approach is based on a theory of duality between solutions and sets of mutually conflicting constraints underlying a problem. Recent years have seen a number of new instantiations of the IHS approach for various problems and constraint languages. As the main contributions, we present novel instantiations of this generic algorithmic approach to four different NP-hard problem domains: maximum satisfiability (MaxSAT), learning optimal causal graphs, propositional abduction, and answer set programming (ASP). For MaxSAT, we build on an existing IHS algorithm with a fresh implementation and new methods for integrating preprocessing. We study a specific application of this IHS approach to MaxSAT for learning optimal causal graphs. In particular we develop a number of domain-specific search techniques to specialize the IHS algorithm for the problem. Furthermore, we consider two optimization settings where the corresponding decision problem is beyond NP, in these cases Σᴾ₂-hard. In the first, we compute optimal explanations for propositional abduction problems. In the second, we solve optimization problems expressed as answer set programs with disjunctive rules. For each problem domain, we empirically evaluate the resulting algorithm and contribute an open-source implementation. These implementations improve or complement the state of the art in their respective domains.Käytännön sovellutuksista kumpuavat optimointiongelmat ovat usein laskennallisesti haastavia. Deklaratiiviset menetelmät tarjoavat keskeisen tavan lähestyä erinäisiä laskennallisesti haastavia optimointiongelmia. Deklaratiivisissa lähestymistavoissa ratkaistavana oleva ongelma mallinnetaan yleisesti matemaattisina rajoitteina siten, että alkuperäisen ongelman instanssien rajoitekuvauksen rajoitteet voidaan toteuttaa jos ja vain jos ongelmainstanssille on olemassa ratkaisu. Ratkaisujen löytäminen rajoitekuvaukselle edellyttää yleisten algoritmisten ratkaisumenetelmien kehittämistä rajoitekuvauskielille. Tässä väitöskirjassa kehitetään uudentyyppisiä käytännöllisiä eksakteja deklaratiivisia ratkaisumenetelmiä jotka pohjautuvat ns. implicit hitting set (IHS) -optimointialgoritmiparadigmaan. Erityisesti työssä kehitetään ja toteutetaan IHS-pohjaisia menetelmiä neljälle laskennallisesti haastavalle, tekoälytutkimuksen näkökulmasta motivoidulle NP-kovalle optimointiongelmalle: lauselogiikan optimointilaajennukselle (MaxSAT), keskeiselle epämonotonisen päättelyn lähestymistavalle (answer set optimization, ASP), lauseloogiselle abduktiolle, sekä optimaalisten kausaaliverkkojen löytämisongelmalle. Työssä kehitetään sekä yleisiä että ongelmakohtaisia hakutekniikoita IHS-kontekstissa, kehitetään avoimen lähdekoodin implementaatioita, ja osoitetaan empiriisten evaluaatioiden kautta näiden olevan käytännössä varteenotettavia vaihtoehtoja kunkin ongelman tehokkaaseen ratkaisemiseen

PREFERENCES: OPTIMIZATION, IMPORTANCE LEARNING AND STRATEGIC BEHAVIORS

Preferences are fundamental to decision making and play an important role in artificial intelligence. Our research focuses on three group of problems based on the preference formalism Answer Set Optimization (ASO): preference aggregation problems such as computing optimal (near optimal) solutions, strategic behaviors in preference representation, and learning ranks (weights) for preferences. In the first group of problems, of interest are optimal outcomes, that is, outcomes that are optimal with respect to the preorder defined by the preference rules. In this work, we consider computational problems concerning optimal outcomes. We propose, implement and study methods to compute an optimal outcome; to compute another optimal outcome once the first one is found; to compute an optimal outcome that is similar to (or, dissimilar from) a given candidate outcome; and to compute a set of optimal answer sets each significantly different from the others. For the decision version of several of these problems we establish their computational complexity. For the second topic, the strategic behaviors such as manipulation and bribery have received much attention from the social choice community. We study these concepts for preference formalisms that identify a set of optimal outcomes rather than a single winning outcome, the case common to social choice. Such preference formalisms are of interest in the context of combinatorial domains, where preference representations are only approximations to true preferences, and seeking a single optimal outcome runs a risk of missing the one which is optimal with respect to the actual preferences. In this work, we assume that preferences may be ranked (differ in importance), and we use the Pareto principle adjusted to the case of ranked preferences as the preference aggregation rule. For two important classes of preferences, representing the extreme ends of the spectrum, we provide characterizations of situations when manipulation and bribery is possible, and establish the complexity of the problem to decide that. Finally, we study the problem of learning the importance of individual preferences in preference profiles aggregated by the ranked Pareto rule or positional scoring rules. We provide a polynomial-time algorithm that finds a ranking of preferences such that the ranked profile correctly decided all the examples, whenever such a ranking exists. We also show that the problem to learn a ranking maximizing the number of correctly decided examples is NP-hard. We obtain similar results for the case of weighted profiles