9 research outputs found
Sorting and Selection with Imprecise Comparisons
In experimental psychology, the method of paired comparisons was proposed as a means for ranking preferences amongst n elements of a human subject. The method requires performing all (n2) comparisons then sorting elements according to the number of wins. The large number of comparisons is performed to counter the potentially faulty decision-making of the human subject, who acts as an imprecise comparator.
We consider a simple model of the imprecise comparisons: there exists some δ> 0 such that when a subject is given two elements to compare, if the values of those elements (as perceived by the subject) differ by at least δ, then the comparison will be made correctly; when the two elements have values that are within δ, the outcome of the comparison is unpredictable. This δ corresponds to the just noticeable difference unit (JND) or difference threshold in the psychophysics literature, but does not require the statistical assumptions used to define this value.
In this model, the standard method of paired comparisons minimizes the errors introduced by the imprecise comparisons at the cost of (n2) comparisons. We show that the same optimal guarantees can be achieved using 4 n 3/2 comparisons, and we prove the optimality of our method. We then explore the general tradeoff between the guarantees on the error that can be made and number of comparisons for the problems of sorting, max-finding, and selection. Our results provide close-to-optimal solutions for each of these problems.Engineering and Applied Science
Searching, Sorting, and Cake Cutting in Rounds
We study sorting and searching in rounds motivated by a cake cutting problem.
The search problem we consider is: we are given an array and an element promised to be in the array. We have access to an
oracle that answers comparison queries: "How is compared to ?",
where the answer can be "". The goal is to find the
location of with success probability at least in at most
rounds of interaction with the oracle. The problem is called ordered or
unordered search, depending on whether the array is sorted or unsorted,
respectively.
For ordered search, we show the expected query complexity of randomized
algorithms is in the worst case. In
contrast, the expected query complexity of deterministic algorithms searching
for a uniformly random element is . The uniform distribution is the worst case for deterministic
algorithms.
For unordered search, the expected query complexity of randomized algorithms
is in the worst case, while the expected
query complexity of deterministic algorithms searching for a uniformly random
element is .
We also discuss the connections of these search problems to the rank query
model, where the array can be accessed via queries of the form "Is
rank?". Unordered search is equivalent to Select with rank
queries (given , find with rank ) and ordered search to Locate with
rank queries (given , find its rank). We show an equivalence between
sorting with rank queries and proportional cake cutting with contiguous pieces
for any number of rounds, as well as an improved lower bound for deterministic
sorting in rounds with rank queries.Comment: 33 pages, 4 figure
On Connections Between Machine Learning And Information Elicitation, Choice Modeling, And Theoretical Computer Science
Machine learning, which has its origins at the intersection of computer science and statistics, is now a rapidly growing area of research that is being integrated into almost every discipline in science and business such as economics, marketing and information retrieval. As a consequence of this integration, it is necessary to understand how machine learning interacts with these disciplines and to understand fundamental questions that arise at the resulting interfaces. The goal of my thesis research is to study these interdisciplinary questions at the interface of machine learning and other disciplines including mechanism design/information elicitation, preference/choice modeling, and theoretical computer science