5,228 research outputs found
Optimal Learning via the Fourier Transform for Sums of Independent Integer Random Variables
We study the structure and learnability of sums of independent integer random
variables (SIIRVs). For , a -SIIRV of order is the probability distribution of the sum of independent
random variables each supported on . We denote by the set of all -SIIRVs of order .
In this paper, we tightly characterize the sample and computational
complexity of learning -SIIRVs. More precisely, we design a computationally
efficient algorithm that uses samples, and learns
an arbitrary -SIIRV within error in total variation distance.
Moreover, we show that the {\em optimal} sample complexity of this learning
problem is Our algorithm
proceeds by learning the Fourier transform of the target -SIIRV in its
effective support. Its correctness relies on the {\em approximate sparsity} of
the Fourier transform of -SIIRVs -- a structural property that we establish,
roughly stating that the Fourier transform of -SIIRVs has small magnitude
outside a small set.
Along the way we prove several new structural results about -SIIRVs. As
one of our main structural contributions, we give an efficient algorithm to
construct a sparse {\em proper} -cover for in total
variation distance. We also obtain a novel geometric characterization of the
space of -SIIRVs. Our characterization allows us to prove a tight lower
bound on the size of -covers for , and is the key
ingredient in our tight sample complexity lower bound.
Our approach of exploiting the sparsity of the Fourier transform in
distribution learning is general, and has recently found additional
applications.Comment: Main differences from v1: Changed title and restructured
introduction. Added new sample optimal algorithm. Generalized sample lower
bound for any value of
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
Quantum Algorithms for Learning and Testing Juntas
In this article we develop quantum algorithms for learning and testing
juntas, i.e. Boolean functions which depend only on an unknown set of k out of
n input variables. Our aim is to develop efficient algorithms:
- whose sample complexity has no dependence on n, the dimension of the domain
the Boolean functions are defined over;
- with no access to any classical or quantum membership ("black-box")
queries. Instead, our algorithms use only classical examples generated
uniformly at random and fixed quantum superpositions of such classical
examples;
- which require only a few quantum examples but possibly many classical
random examples (which are considered quite "cheap" relative to quantum
examples).
Our quantum algorithms are based on a subroutine FS which enables sampling
according to the Fourier spectrum of f; the FS subroutine was used in earlier
work of Bshouty and Jackson on quantum learning. Our results are as follows:
- We give an algorithm for testing k-juntas to accuracy that uses
quantum examples. This improves on the number of examples used
by the best known classical algorithm.
- We establish the following lower bound: any FS-based k-junta testing
algorithm requires queries.
- We give an algorithm for learning -juntas to accuracy that
uses quantum examples and
random examples. We show that this learning algorithms is close to optimal by
giving a related lower bound.Comment: 15 pages, 1 figure. Uses synttree package. To appear in Quantum
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