On multivariate quantiles under partial orders

This paper focuses on generalizing quantiles from the ordering point of view. We propose the concept of partial quantiles, which are based on a given partial order. We establish that partial quantiles are equivariant under order-preserving transformations of the data, robust to outliers, characterize the probability distribution if the partial order is sufficiently rich, generalize the concept of efficient frontier, and can measure dispersion from the partial order perspective. We also study several statistical aspects of partial quantiles. We provide estimators, associated rates of convergence, and asymptotic distributions that hold uniformly over a continuum of quantile indices. Furthermore, we provide procedures that can restore monotonicity properties that might have been disturbed by estimation error, establish computational complexity bounds, and point out a concentration of measure phenomenon (the latter under independence and the componentwise natural order). Finally, we illustrate the concepts by discussing several theoretical examples and simulations. Empirical applications to compare intake nutrients within diets, to evaluate the performance of investment funds, and to study the impact of policies on tobacco awareness are also presented to illustrate the concepts and their use.Comment: Published in at http://dx.doi.org/10.1214/10-AOS863 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Dispersion for Data-Driven Algorithm Design, Online Learning, and Private Optimization

Data-driven algorithm design, that is, choosing the best algorithm for a specific application, is a crucial problem in modern data science. Practitioners often optimize over a parameterized algorithm family, tuning parameters based on problems from their domain. These procedures have historically come with no guarantees, though a recent line of work studies algorithm selection from a theoretical perspective. We advance the foundations of this field in several directions: we analyze online algorithm selection, where problems arrive one-by-one and the goal is to minimize regret, and private algorithm selection, where the goal is to find good parameters over a set of problems without revealing sensitive information contained therein. We study important algorithm families, including SDP-rounding schemes for problems formulated as integer quadratic programs, and greedy techniques for canonical subset selection problems. In these cases, the algorithm's performance is a volatile and piecewise Lipschitz function of its parameters, since tweaking the parameters can completely change the algorithm's behavior. We give a sufficient and general condition, dispersion, defining a family of piecewise Lipschitz functions that can be optimized online and privately, which includes the functions measuring the performance of the algorithms we study. Intuitively, a set of piecewise Lipschitz functions is dispersed if no small region contains many of the functions' discontinuities. We present general techniques for online and private optimization of the sum of dispersed piecewise Lipschitz functions. We improve over the best-known regret bounds for a variety of problems, prove regret bounds for problems not previously studied, and give matching lower bounds. We also give matching upper and lower bounds on the utility loss due to privacy. Moreover, we uncover dispersion in auction design and pricing problems

PDDL2.1: An extension of PDDL for expressing temporal planning domains

In recent years research in the planning community has moved increasingly towards application of planners to realistic problems involving both time and many types of resources. For example, interest in planning demonstrated by the space research community has inspired work in observation scheduling, planetary rover ex ploration and spacecraft control domains. Other temporal and resource-intensive domains including logistics planning, plant control and manufacturing have also helped to focus the community on the modelling and reasoning issues that must be confronted to make planning technology meet the challenges of application. The International Planning Competitions have acted as an important motivating force behind the progress that has been made in planning since 1998. The third competition (held in 2002) set the planning community the challenge of handling time and numeric resources. This necessitated the development of a modelling language capable of expressing temporal and numeric properties of planning domains. In this paper we describe the language, PDDL2.1, that was used in the competition. We describe the syntax of the language, its formal semantics and the validation of concurrent plans. We observe that PDDL2.1 has considerable modelling power --- exceeding the capabilities of current planning technology --- and presents a number of important challenges to the research community

ISIPTA'07: Proceedings of the Fifth International Symposium on Imprecise Probability: Theories and Applications

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