702 research outputs found
Approximating the Noise Sensitivity of a Monotone Boolean Function
The noise sensitivity of a Boolean function f: {0,1}^n - > {0,1} is one of its fundamental properties. For noise parameter delta, the noise sensitivity is denoted as NS_{delta}[f]. This quantity is defined as follows: First, pick x = (x_1,...,x_n) uniformly at random from {0,1}^n, then pick z by flipping each x_i independently with probability delta. NS_{delta}[f] is defined to equal Pr [f(x) != f(z)]. Much of the existing literature on noise sensitivity explores the following two directions: (1) Showing that functions with low noise-sensitivity are structured in certain ways. (2) Mathematically showing that certain classes of functions have low noise sensitivity. Combined, these two research directions show that certain classes of functions have low noise sensitivity and therefore have useful structure.
The fundamental importance of noise sensitivity, together with this wealth of structural results, motivates the algorithmic question of approximating NS_{delta}[f] given an oracle access to the function f. We show that the standard sampling approach is essentially optimal for general Boolean functions. Therefore, we focus on estimating the noise sensitivity of monotone functions, which form an important subclass of Boolean functions, since many functions of interest are either monotone or can be simply transformed into a monotone function (for example the class of unate functions consists of all the functions that can be made monotone by reorienting some of their coordinates [O\u27Donnell, 2014]).
Specifically, we study the algorithmic problem of approximating NS_{delta}[f] for monotone f, given the promise that NS_{delta}[f] >= 1/n^{C} for constant C, and for delta in the range 1/n <= delta <= 1/2. For such f and delta, we give a randomized algorithm performing O((min(1,sqrt{n} delta log^{1.5} n))/(NS_{delta}[f]) poly (1/epsilon)) queries and approximating NS_{delta}[f] to within a multiplicative factor of (1 +/- epsilon). Given the same constraints on f and delta, we also prove a lower bound of Omega((min(1,sqrt{n} delta))/(NS_{delta}[f] * n^{xi})) on the query complexity of any algorithm that approximates NS_{delta}[f] to within any constant factor, where xi can be any positive constant. Thus, our algorithm\u27s query complexity is close to optimal in terms of its dependence on n.
We introduce a novel descending-ascending view of noise sensitivity, and use it as a central tool for the analysis of our algorithm. To prove lower bounds on query complexity, we develop a technique that reduces computational questions about query complexity to combinatorial questions about the existence of "thin" functions with certain properties. The existence of such "thin" functions is proved using the probabilistic method. These techniques also yield new lower bounds on the query complexity of approximating other fundamental properties of Boolean functions: the total influence and the bias
Learning circuits with few negations
Monotone Boolean functions, and the monotone Boolean circuits that compute
them, have been intensively studied in complexity theory. In this paper we
study the structure of Boolean functions in terms of the minimum number of
negations in any circuit computing them, a complexity measure that interpolates
between monotone functions and the class of all functions. We study this
generalization of monotonicity from the vantage point of learning theory,
giving near-matching upper and lower bounds on the uniform-distribution
learnability of circuits in terms of the number of negations they contain. Our
upper bounds are based on a new structural characterization of negation-limited
circuits that extends a classical result of A. A. Markov. Our lower bounds,
which employ Fourier-analytic tools from hardness amplification, give new
results even for circuits with no negations (i.e. monotone functions)
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
Privately Releasing Conjunctions and the Statistical Query Barrier
Suppose we would like to know all answers to a set of statistical queries C
on a data set up to small error, but we can only access the data itself using
statistical queries. A trivial solution is to exhaustively ask all queries in
C. Can we do any better?
+ We show that the number of statistical queries necessary and sufficient for
this task is---up to polynomial factors---equal to the agnostic learning
complexity of C in Kearns' statistical query (SQ) model. This gives a complete
answer to the question when running time is not a concern.
+ We then show that the problem can be solved efficiently (allowing arbitrary
error on a small fraction of queries) whenever the answers to C can be
described by a submodular function. This includes many natural concept classes,
such as graph cuts and Boolean disjunctions and conjunctions.
While interesting from a learning theoretic point of view, our main
applications are in privacy-preserving data analysis:
Here, our second result leads to the first algorithm that efficiently
releases differentially private answers to of all Boolean conjunctions with 1%
average error. This presents significant progress on a key open problem in
privacy-preserving data analysis.
Our first result on the other hand gives unconditional lower bounds on any
differentially private algorithm that admits a (potentially
non-privacy-preserving) implementation using only statistical queries. Not only
our algorithms, but also most known private algorithms can be implemented using
only statistical queries, and hence are constrained by these lower bounds. Our
result therefore isolates the complexity of agnostic learning in the SQ-model
as a new barrier in the design of differentially private algorithms
Agnostic Learning of Disjunctions on Symmetric Distributions
We consider the problem of approximating and learning disjunctions (or
equivalently, conjunctions) on symmetric distributions over .
Symmetric distributions are distributions whose PDF is invariant under any
permutation of the variables. We give a simple proof that for every symmetric
distribution , there exists a set of
functions , such that for every disjunction , there is function
, expressible as a linear combination of functions in , such
that -approximates in distance on or
. This directly
gives an agnostic learning algorithm for disjunctions on symmetric
distributions that runs in time . The best known
previous bound is and follows from approximation of the
more general class of halfspaces (Wimmer, 2010). We also show that there exists
a symmetric distribution , such that the minimum degree of a
polynomial that -approximates the disjunction of all variables is
distance on is . Therefore the
learning result above cannot be achieved via -regression with a
polynomial basis used in most other agnostic learning algorithms.
Our technique also gives a simple proof that for any product distribution
and every disjunction , there exists a polynomial of
degree such that -approximates in
distance on . This was first proved by Blais et al.
(2008) via a more involved argument
Geometric Influences II: Correlation Inequalities and Noise Sensitivity
In a recent paper, we presented a new definition of influences in product
spaces of continuous distributions, and showed that analogues of the most
fundamental results on discrete influences, such as the KKL theorem, hold for
the new definition in Gaussian space. In this paper we prove Gaussian analogues
of two of the central applications of influences: Talagrand's lower bound on
the correlation of increasing subsets of the discrete cube, and the
Benjamini-Kalai-Schramm (BKS) noise sensitivity theorem. We then use the
Gaussian results to obtain analogues of Talagrand's bound for all discrete
probability spaces and to reestablish analogues of the BKS theorem for biased
two-point product spaces.Comment: 20 page
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