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Register a new userConcordance and the Smallest Covering Set of Preference Orderings
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Zhiwei Lin
Publication date
2016
fields
Informatics Engineering
and research's language is
English
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Preference orderings are orderings of a set of items according to the preferences (of judges). Such orderings arise in a variety of domains, including group decision making, consumer marketing, voting and machine learning. Measuring the mutual information and extracting the common patterns in a set of preference orderings are key to these areas. In this paper we deal with the representation of sets of preference orderings, the quantification of the degree to which judges agree on their ordering of the items (i.e. the concordance), and the efficient, meaningful description of such sets. We propose to represent the orderings in a subsequence-based feature space and present a new algorithm to calculate the size of the set of all common subsequences - the basis of a quantification of concordance, not only for pairs of orderings but also for sets of orderings. The new algorithm is fast and storage efficient with a time complexity of only $O(Nn^2)$ for the orderings of $n$ items by $N$ judges and a space complexity of only $O(min{Nn,n^2})$. Also, we propose to represent the set of all $N$ orderings through a smallest set of covering preferences and present an algorithm to construct this smallest covering set. The source code for the algorithms is available at https://github.com/zhiweiuu/secs
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Learning of user preferences, as represented by, for example, Conditional Preference Networks (CP-nets), has become a core issue in AI research. Recent studies investigate learning of CP-nets from randomly chosen examples or from membership and equivalence queries. To assess the optimality of learning algorithms as well as to better understand the combinatorial structure of classes of CP-nets, it is helpful to calculate certain learning-theoretic information complexity parameters. This article focuses on the frequently studied case of learning from so-called swap examples, which express preferences among objects that differ in only one attribute. It presents bounds on or exact values of some well-studied information complexity parameters, namely the VC dimension, the teaching dimension, and the recursive teaching dimension, for classes of acyclic CP-nets. We further provide algorithms that learn tree-structured and general acyclic CP-nets from membership queries. Using our results on complexity parameters, we assess the optimality of our algorithms as well as that of another query learning algorithm for acyclic CP-nets presented in the literature. Our algorithms are near-optimal, and can, under certain assumptions, be adapted to the case when the membership oracle is faulty.
We describe the adjacency of vertices of the (unbounded version of the) set covering polyhedron, in a similar way to the description given by Chvatal for the stable set polytope. We find a sufficient condition for adjacency, and characterize it with similar conditions in the case where the underlying matrix is row circular. We apply our findings to show a new infinite family of minimally nonideal matrices.
In this note we show that for each Latin square $L$ of order $ngeq 2$, there exists a Latin square $L eq L$ of order $n$ such that $L$ and $L$ differ in at most $8sqrt{n}$ cells. Equivalently, each Latin square of order $n$ contains a Latin trade of size at most $8sqrt{n}$. We also show that the size of the smallest defining set in a Latin square is $Omega(n^{3/2})$. %That is, there are constants $c$ and $n_0$ such that for any $n>n_0$ the size of the smallest defining %set of order $n$ is at least $cn^{3/2}$.
The binary relation framework has been shown to be applicable to many real-life preference handling scenarios. Here we study preference contraction: the problem of discarding selected preferences. We argue that the property of minimality and the preservation of strict partial orders are crucial for contractions. Contractions can be further constrained by specifying which preferences should be protected. We consider two classes of preference relations: finite and finitely representable. We present algorithms for computing minimal and preference-protecting minimal contractions for finite as well as finitely representable preference relations. We study relationships between preference change in the binary relation framework and belief change in the belief revision theory. We also introduce some preference query optimization techniques which can be used in the presence of contraction. We evaluate the proposed algorithms experimentally and present the results.
The last few years have seen an explosion of academic and popular interest in algorithmic fairness. Despite this interest and the volume and velocity of work that has been produced recently, the fundamental science of fairness in machine learning is still in a nascent state. In March 2018, we convened a group of experts as part of a CCC visioning workshop to assess the state of the field, and distill the most promising research directions going forward. This report summarizes the findings of that workshop. Along the way, it surveys recent theoretical work in the field and points towards promising directions for research.
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