12,435 research outputs found
Differentially Private Release and Learning of Threshold Functions
We prove new upper and lower bounds on the sample complexity of differentially private algorithms for releasing approximate answers to
threshold functions. A threshold function over a totally ordered domain
evaluates to if , and evaluates to otherwise. We
give the first nontrivial lower bound for releasing thresholds with
differential privacy, showing that the task is impossible
over an infinite domain , and moreover requires sample complexity , which grows with the size of the domain. Inspired by the
techniques used to prove this lower bound, we give an algorithm for releasing
thresholds with samples. This improves the
previous best upper bound of (Beimel et al., RANDOM
'13).
Our sample complexity upper and lower bounds also apply to the tasks of
learning distributions with respect to Kolmogorov distance and of properly PAC
learning thresholds with differential privacy. The lower bound gives the first
separation between the sample complexity of properly learning a concept class
with differential privacy and learning without privacy. For
properly learning thresholds in dimensions, this lower bound extends to
.
To obtain our results, we give reductions in both directions from releasing
and properly learning thresholds and the simpler interior point problem. Given
a database of elements from , the interior point problem asks for an
element between the smallest and largest elements in . We introduce new
recursive constructions for bounding the sample complexity of the interior
point problem, as well as further reductions and techniques for proving
impossibility results for other basic problems in differential privacy.Comment: 43 page
Fast Private Data Release Algorithms for Sparse Queries
We revisit the problem of accurately answering large classes of statistical
queries while preserving differential privacy. Previous approaches to this
problem have either been very general but have not had run-time polynomial in
the size of the database, have applied only to very limited classes of queries,
or have relaxed the notion of worst-case error guarantees. In this paper we
consider the large class of sparse queries, which take non-zero values on only
polynomially many universe elements. We give efficient query release algorithms
for this class, in both the interactive and the non-interactive setting. Our
algorithms also achieve better accuracy bounds than previous general techniques
do when applied to sparse queries: our bounds are independent of the universe
size. In fact, even the runtime of our interactive mechanism is independent of
the universe size, and so can be implemented in the "infinite universe" model
in which no finite universe need be specified by the data curator
The Complexity of Datalog on Linear Orders
We study the program complexity of datalog on both finite and infinite linear
orders. Our main result states that on all linear orders with at least two
elements, the nonemptiness problem for datalog is EXPTIME-complete. While
containment of the nonemptiness problem in EXPTIME is known for finite linear
orders and actually for arbitrary finite structures, it is not obvious for
infinite linear orders. It sharply contrasts the situation on other infinite
structures; for example, the datalog nonemptiness problem on an infinite
successor structure is undecidable. We extend our upper bound results to
infinite linear orders with constants.
As an application, we show that the datalog nonemptiness problem on Allen's
interval algebra is EXPTIME-complete.Comment: 21 page
Characterizing the Sample Complexity of Private Learners
In 2008, Kasiviswanathan et al. defined private learning as a combination of
PAC learning and differential privacy. Informally, a private learner is applied
to a collection of labeled individual information and outputs a hypothesis
while preserving the privacy of each individual. Kasiviswanathan et al. gave a
generic construction of private learners for (finite) concept classes, with
sample complexity logarithmic in the size of the concept class. This sample
complexity is higher than what is needed for non-private learners, hence
leaving open the possibility that the sample complexity of private learning may
be sometimes significantly higher than that of non-private learning.
We give a combinatorial characterization of the sample size sufficient and
necessary to privately learn a class of concepts. This characterization is
analogous to the well known characterization of the sample complexity of
non-private learning in terms of the VC dimension of the concept class. We
introduce the notion of probabilistic representation of a concept class, and
our new complexity measure RepDim corresponds to the size of the smallest
probabilistic representation of the concept class.
We show that any private learning algorithm for a concept class C with sample
complexity m implies RepDim(C)=O(m), and that there exists a private learning
algorithm with sample complexity m=O(RepDim(C)). We further demonstrate that a
similar characterization holds for the database size needed for privately
computing a large class of optimization problems and also for the well studied
problem of private data release
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