26,214 research outputs found
Forming Probably Stable Communities with Limited Interactions
A community needs to be partitioned into disjoint groups; each community
member has an underlying preference over the groups that they would want to be
a member of. We are interested in finding a stable community structure: one
where no subset of members wants to deviate from the current structure. We
model this setting as a hedonic game, where players are connected by an
underlying interaction network, and can only consider joining groups that are
connected subgraphs of the underlying graph. We analyze the relation between
network structure, and one's capability to infer statistically stable (also
known as PAC stable) player partitions from data. We show that when the
interaction network is a forest, one can efficiently infer PAC stable coalition
structures. Furthermore, when the underlying interaction graph is not a forest,
efficient PAC stabilizability is no longer achievable. Thus, our results
completely characterize when one can leverage the underlying graph structure in
order to compute PAC stable outcomes for hedonic games. Finally, given an
unknown underlying interaction network, we show that it is NP-hard to decide
whether there exists a forest consistent with data samples from the network.Comment: 11 pages, full version of accepted AAAI-19 pape
A Size-Free CLT for Poisson Multinomials and its Applications
An -Poisson Multinomial Distribution (PMD) is the distribution of the
sum of independent random vectors supported on the set of standard basis vectors in . We show
that any -PMD is -close in total
variation distance to the (appropriately discretized) multi-dimensional
Gaussian with the same first two moments, removing the dependence on from
the Central Limit Theorem of Valiant and Valiant. Interestingly, our CLT is
obtained by bootstrapping the Valiant-Valiant CLT itself through the structural
characterization of PMDs shown in recent work by Daskalakis, Kamath, and
Tzamos. In turn, our stronger CLT can be leveraged to obtain an efficient PTAS
for approximate Nash equilibria in anonymous games, significantly improving the
state of the art, and matching qualitatively the running time dependence on
and of the best known algorithm for two-strategy anonymous
games. Our new CLT also enables the construction of covers for the set of
-PMDs, which are proper and whose size is shown to be essentially
optimal. Our cover construction combines our CLT with the Shapley-Folkman
theorem and recent sparsification results for Laplacian matrices by Batson,
Spielman, and Srivastava. Our cover size lower bound is based on an algebraic
geometric construction. Finally, leveraging the structural properties of the
Fourier spectrum of PMDs we show that these distributions can be learned from
samples in -time, removing
the quasi-polynomial dependence of the running time on from the
algorithm of Daskalakis, Kamath, and Tzamos.Comment: To appear in STOC 201
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