10,807 research outputs found
Weighted Polynomial Approximations: Limits for Learning and Pseudorandomness
Polynomial approximations to boolean functions have led to many positive
results in computer science. In particular, polynomial approximations to the
sign function underly algorithms for agnostically learning halfspaces, as well
as pseudorandom generators for halfspaces. In this work, we investigate the
limits of these techniques by proving inapproximability results for the sign
function.
Firstly, the polynomial regression algorithm of Kalai et al. (SIAM J. Comput.
2008) shows that halfspaces can be learned with respect to log-concave
distributions on in the challenging agnostic learning model. The
power of this algorithm relies on the fact that under log-concave
distributions, halfspaces can be approximated arbitrarily well by low-degree
polynomials. We ask whether this technique can be extended beyond log-concave
distributions, and establish a negative result. We show that polynomials of any
degree cannot approximate the sign function to within arbitrarily low error for
a large class of non-log-concave distributions on the real line, including
those with densities proportional to .
Secondly, we investigate the derandomization of Chernoff-type concentration
inequalities. Chernoff-type tail bounds on sums of independent random variables
have pervasive applications in theoretical computer science. Schmidt et al.
(SIAM J. Discrete Math. 1995) showed that these inequalities can be established
for sums of random variables with only -wise independence,
for a tail probability of . We show that their results are tight up to
constant factors.
These results rely on techniques from weighted approximation theory, which
studies how well functions on the real line can be approximated by polynomials
under various distributions. We believe that these techniques will have further
applications in other areas of computer science.Comment: 22 page
Learning Coverage Functions and Private Release of Marginals
We study the problem of approximating and learning coverage functions. A
function is a coverage function, if
there exists a universe with non-negative weights for each
and subsets of such that . Alternatively, coverage functions can be described
as non-negative linear combinations of monotone disjunctions. They are a
natural subclass of submodular functions and arise in a number of applications.
We give an algorithm that for any , given random and uniform
examples of an unknown coverage function , finds a function that
approximates within factor on all but -fraction of the
points in time . This is the first fully-polynomial
algorithm for learning an interesting class of functions in the demanding PMAC
model of Balcan and Harvey (2011). Our algorithms are based on several new
structural properties of coverage functions. Using the results in (Feldman and
Kothari, 2014), we also show that coverage functions are learnable agnostically
with excess -error over all product and symmetric
distributions in time . In contrast, we show that,
without assumptions on the distribution, learning coverage functions is at
least as hard as learning polynomial-size disjoint DNF formulas, a class of
functions for which the best known algorithm runs in time
(Klivans and Servedio, 2004).
As an application of our learning results, we give simple
differentially-private algorithms for releasing monotone conjunction counting
queries with low average error. In particular, for any , we obtain
private release of -way marginals with average error in time
Differential Privacy and the Fat-Shattering Dimension of Linear Queries
In this paper, we consider the task of answering linear queries under the
constraint of differential privacy. This is a general and well-studied class of
queries that captures other commonly studied classes, including predicate
queries and histogram queries. We show that the accuracy to which a set of
linear queries can be answered is closely related to its fat-shattering
dimension, a property that characterizes the learnability of real-valued
functions in the agnostic-learning setting.Comment: Appears in APPROX 201
Active classification with comparison queries
We study an extension of active learning in which the learning algorithm may
ask the annotator to compare the distances of two examples from the boundary of
their label-class. For example, in a recommendation system application (say for
restaurants), the annotator may be asked whether she liked or disliked a
specific restaurant (a label query); or which one of two restaurants did she
like more (a comparison query).
We focus on the class of half spaces, and show that under natural
assumptions, such as large margin or bounded bit-description of the input
examples, it is possible to reveal all the labels of a sample of size using
approximately queries. This implies an exponential improvement over
classical active learning, where only label queries are allowed. We complement
these results by showing that if any of these assumptions is removed then, in
the worst case, queries are required.
Our results follow from a new general framework of active learning with
additional queries. We identify a combinatorial dimension, called the
\emph{inference dimension}, that captures the query complexity when each
additional query is determined by examples (such as comparison queries,
each of which is determined by the two compared examples). Our results for half
spaces follow by bounding the inference dimension in the cases discussed above.Comment: 23 pages (not including references), 1 figure. The new version
contains a minor fix in the proof of Lemma 4.
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