628 research outputs found
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
Information Processin
On active and passive testing
Given a property of Boolean functions, what is the minimum number of queries
required to determine with high probability if an input function satisfies this
property or is "far" from satisfying it? This is a fundamental question in
Property Testing, where traditionally the testing algorithm is allowed to pick
its queries among the entire set of inputs. Balcan, Blais, Blum and Yang have
recently suggested to restrict the tester to take its queries from a smaller
random subset of polynomial size of the inputs. This model is called active
testing, and in the extreme case when the size of the set we can query from is
exactly the number of queries performed it is known as passive testing.
We prove that passive or active testing of k-linear functions (that is, sums
of k variables among n over Z_2) requires Theta(k*log n) queries, assuming k is
not too large. This extends the case k=1, (that is, dictator functions),
analyzed by Balcan et. al.
We also consider other classes of functions including low degree polynomials,
juntas, and partially symmetric functions. Our methods combine algebraic,
combinatorial, and probabilistic techniques, including the Talagrand
concentration inequality and the Erdos--Rado theorem on Delta-systems.Comment: 16 page
Optimal Bounds on Approximation of Submodular and XOS Functions by Juntas
We investigate the approximability of several classes of real-valued
functions by functions of a small number of variables ({\em juntas}). Our main
results are tight bounds on the number of variables required to approximate a
function within -error over
the uniform distribution: 1. If is submodular, then it is -close
to a function of variables.
This is an exponential improvement over previously known results. We note that
variables are necessary even for linear
functions. 2. If is fractionally subadditive (XOS) it is -close
to a function of variables. This result holds for all
functions with low total -influence and is a real-valued analogue of
Friedgut's theorem for boolean functions. We show that
variables are necessary even for XOS functions.
As applications of these results, we provide learning algorithms over the
uniform distribution. For XOS functions, we give a PAC learning algorithm that
runs in time . For submodular functions we give
an algorithm in the more demanding PMAC learning model (Balcan and Harvey,
2011) which requires a multiplicative factor approximation with
probability at least over the target distribution. Our uniform
distribution algorithm runs in time .
This is the first algorithm in the PMAC model that over the uniform
distribution can achieve a constant approximation factor arbitrarily close to 1
for all submodular functions. As follows from the lower bounds in (Feldman et
al., 2013) both of these algorithms are close to optimal. We also give
applications for proper learning, testing and agnostic learning with value
queries of these classes.Comment: Extended abstract appears in proceedings of FOCS 201
Partially Symmetric Functions are Efficiently Isomorphism-Testable
Given a function f: {0,1}^n \to {0,1}, the f-isomorphism testing problem
requires a randomized algorithm to distinguish functions that are identical to
f up to relabeling of the input variables from functions that are far from
being so. An important open question in property testing is to determine for
which functions f we can test f-isomorphism with a constant number of queries.
Despite much recent attention to this question, essentially only two classes of
functions were known to be efficiently isomorphism testable: symmetric
functions and juntas.
We unify and extend these results by showing that all partially symmetric
functions---functions invariant to the reordering of all but a constant number
of their variables---are efficiently isomorphism-testable. This class of
functions, first introduced by Shannon, includes symmetric functions, juntas,
and many other functions as well. We conjecture that these functions are
essentially the only functions efficiently isomorphism-testable.
To prove our main result, we also show that partial symmetry is efficiently
testable. In turn, to prove this result we had to revisit the junta testing
problem. We provide a new proof of correctness of the nearly-optimal junta
tester. Our new proof replaces the Fourier machinery of the original proof with
a purely combinatorial argument that exploits the connection between sets of
variables with low influence and intersecting families.
Another important ingredient in our proofs is a new notion of symmetric
influence. We use this measure of influence to prove that partial symmetry is
efficiently testable and also to construct an efficient sample extractor for
partially symmetric functions. We then combine the sample extractor with the
testing-by-implicit-learning approach to complete the proof that partially
symmetric functions are efficiently isomorphism-testable.Comment: 22 page
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