1,953 research outputs found
How to Find a Point in the Convex Hull Privately
We study the question of how to compute a point in the convex hull of an
input set of points in in a differentially private
manner. This question, which is trivial non-privately, turns out to be quite
deep when imposing differential privacy. In particular, it is known that the
input points must reside on a fixed finite subset ,
and furthermore, the size of must grow with the size of . Previous works
focused on understanding how needs to grow with , and showed that
suffices (so does not have to
grow significantly with ). However, the available constructions exhibit
running time at least , where typically for some (large)
discretization parameter , so the running time is in fact .
In this paper we give a differentially private algorithm that runs in
time, assuming that . To get this result we study
and exploit some structural properties of the Tukey levels (the regions consisting of points whose Tukey depth is at least , for ).
In particular, we derive lower bounds on their volumes for point sets in
general position, and develop a rather subtle mechanism for handling point sets
in degenerate position (where the deep Tukey regions have zero volume). A
naive approach to the construction of the Tukey regions requires
time. To reduce the cost to , we use an approximation scheme for
estimating the volumes of the Tukey regions (within their affine spans in case
of degeneracy), and for sampling a point from such a region, a scheme that is
based on the volume estimation framework of Lov\'asz and Vempala (FOCS 2003)
and of Cousins and Vempala (STOC 2015). Making this framework differentially
private raises a set of technical challenges that we address
Correlated equilibria, good and bad : an experimental study
We report results from an experiment that explores the empirical validity of correlated equilibrium, an important generalization of the Nash equilibrium concept. Specifically, we seek to
understand the conditions under which subjects playing the game of Chicken will condition their behavior on private, third–party recommendations drawn from known distributions. In a “good–recommendations” treatment, the distribution we use is a correlated equilibrium with payoffs better
than any symmetric payoff in the convex hull of Nash equilibrium payoff vectors. In a “bad–recommendations” treatment, the distribution is a correlated equilibrium with payoffs worse than any Nash equilibrium payoff vector. In a “Nash–recommendations” treatment, the distribution is a convex combination of Nash equilibrium outcomes (which is also a correlated equilibrium), and in a fourth “very–good–recommendations” treatment, the distribution yields high payoffs, but is not a correlated equilibrium. We compare behavior in all of these treatments to the case where subjects do not receive recommendations. We find that when recommendations are not given to
subjects, behavior is very close to mixed–strategy Nash equilibrium play. When recommendations are given, behavior does differ from mixed–strategy Nash equilibrium, with the nature of the differ-
ences varying according to the treatment. Our main finding is that subjects will follow third–party recommendations only if those recommendations derive from a correlated equilibrium, and further,if that correlated equilibrium is payoff–enhancing relative to the available Nash equilibria
Efficient Algorithms for Privately Releasing Marginals via Convex Relaxations
Consider a database of people, each represented by a bit-string of length
corresponding to the setting of binary attributes. A -way marginal
query is specified by a subset of attributes, and a -dimensional
binary vector specifying their values. The result for this query is a
count of the number of people in the database whose attribute vector restricted
to agrees with .
Privately releasing approximate answers to a set of -way marginal queries
is one of the most important and well-motivated problems in differential
privacy. Information theoretically, the error complexity of marginal queries is
well-understood: the per-query additive error is known to be at least
and at most
. However, no polynomial
time algorithm with error complexity as low as the information theoretic upper
bound is known for small . In this work we present a polynomial time
algorithm that, for any distribution on marginal queries, achieves average
error at most . This error
bound is as good as the best known information theoretic upper bounds for
. This bound is an improvement over previous work on efficiently releasing
marginals when is small and when error is desirable. Using private
boosting we are also able to give nearly matching worst-case error bounds.
Our algorithms are based on the geometric techniques of Nikolov, Talwar, and
Zhang. The main new ingredients are convex relaxations and careful use of the
Frank-Wolfe algorithm for constrained convex minimization. To design our
relaxations, we rely on the Grothendieck inequality from functional analysis
Prediction Markets: Alternative Mechanisms for Complex Environments with Few Traders
Double auction prediction markets have proven successful in large-scale applications such as elections and sporting events. Consequently, several large corporations have adopted these markets for smaller-scale internal applications where information may be complex and the number of traders is small. Using laboratory experiments, we test the performance of the double auction in complex environments with few traders and compare it to three alternative mechanisms. When information is complex we find that an iterated poll (or Delphi method) outperforms the double auction mechanism. We present five behavioral observations that may explain why the poll performs better in these settings
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Mathematical Structures in Group Decision-Making on Resource Allocation Distributions.
Optimal decisions on the distribution of finite resources are explicitly structured by mathematical models that specify relevant variables, constraints, and objectives. Here we report analysis and evidence that implicit mathematical structures are also involved in group decision-making on resource allocation distributions under conditions of uncertainty that disallow formal optimization. A group's array of initial distribution preferences automatically sets up a geometric decision space of alternative resource distributions. Weighted averaging mechanisms of interpersonal influence reduce the heterogeneity of the group's initial preferences on a suitable distribution. A model of opinion formation based on weighted averaging predicts a distribution that is a feasible point in the group's implicit initial decision space
Differentially Private Approximations of a Convex Hull in Low Dimensions
We give the first differentially private algorithms that estimate a variety of geometric features of points in the Euclidean space, such as diameter, width, volume of convex hull, min-bounding box, min-enclosing ball, etc. Our work relies heavily on the notion of Tukey-depth. Instead of (non-privately) approximating the convex-hull of the given set of points P, our algorithms approximate the geometric features of D_{P}(?) - the ?-Tukey region induced by P (all points of Tukey-depth ? or greater). Moreover, our approximations are all bi-criteria: for any geometric feature ? our (?,?)-approximation is a value "sandwiched" between (1-?)?(D_P(?)) and (1+?)?(D_P(?-?)).
Our work is aimed at producing a (?,?)-kernel of D_P(?), namely a set ? such that (after a shift) it holds that (1-?)D_P(?) ? CH(?) ? (1+?)D_P(?-?). We show that an analogous notion of a bi-critera approximation of a directional kernel, as originally proposed by [Pankaj K. Agarwal et al., 2004], fails to give a kernel, and so we result to subtler notions of approximations of projections that do yield a kernel. First, we give differentially private algorithms that find (?,?)-kernels for a "fat" Tukey-region. Then, based on a private approximation of the min-bounding box, we find a transformation that does turn D_P(?) into a "fat" region but only if its volume is proportional to the volume of D_P(?-?). Lastly, we give a novel private algorithm that finds a depth parameter ? for which the volume of D_P(?) is comparable to the volume of D_P(?-?). We hope our work leads to the further study of the intersection of differential privacy and computational geometry
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