101 research outputs found
On The Hereditary Discrepancy of Homogeneous Arithmetic Progressions
We show that the hereditary discrepancy of homogeneous arithmetic
progressions is lower bounded by . This bound is tight up
to the constant in the exponent. Our lower bound goes via proving an
exponential lower bound on the discrepancy of set systems of subcubes of the
boolean cube .Comment: To appear in the Proceedings of the American Mathematical Societ
Approximating Hereditary Discrepancy via Small Width Ellipsoids
The Discrepancy of a hypergraph is the minimum attainable value, over
two-colorings of its vertices, of the maximum absolute imbalance of any
hyperedge. The Hereditary Discrepancy of a hypergraph, defined as the maximum
discrepancy of a restriction of the hypergraph to a subset of its vertices, is
a measure of its complexity. Lovasz, Spencer and Vesztergombi (1986) related
the natural extension of this quantity to matrices to rounding algorithms for
linear programs, and gave a determinant based lower bound on the hereditary
discrepancy. Matousek (2011) showed that this bound is tight up to a
polylogarithmic factor, leaving open the question of actually computing this
bound. Recent work by Nikolov, Talwar and Zhang (2013) showed a polynomial time
-approximation to hereditary discrepancy, as a by-product
of their work in differential privacy. In this paper, we give a direct simple
-approximation algorithm for this problem. We show that up to
this approximation factor, the hereditary discrepancy of a matrix is
characterized by the optimal value of simple geometric convex program that
seeks to minimize the largest norm of any point in a ellipsoid
containing the columns of . This characterization promises to be a useful
tool in discrepancy theory
Efficient Algorithms for Privately Releasing Marginals via Convex Relaxations
Consider a database of people, each represented by a bit-string of length
corresponding to the setting of binary attributes. A -way marginal
query is specified by a subset of attributes, and a -dimensional
binary vector specifying their values. The result for this query is a
count of the number of people in the database whose attribute vector restricted
to agrees with .
Privately releasing approximate answers to a set of -way marginal queries
is one of the most important and well-motivated problems in differential
privacy. Information theoretically, the error complexity of marginal queries is
well-understood: the per-query additive error is known to be at least
and at most
. However, no polynomial
time algorithm with error complexity as low as the information theoretic upper
bound is known for small . In this work we present a polynomial time
algorithm that, for any distribution on marginal queries, achieves average
error at most . This error
bound is as good as the best known information theoretic upper bounds for
. This bound is an improvement over previous work on efficiently releasing
marginals when is small and when error is desirable. Using private
boosting we are also able to give nearly matching worst-case error bounds.
Our algorithms are based on the geometric techniques of Nikolov, Talwar, and
Zhang. The main new ingredients are convex relaxations and careful use of the
Frank-Wolfe algorithm for constrained convex minimization. To design our
relaxations, we rely on the Grothendieck inequality from functional analysis
The Geometry of Differential Privacy: the Sparse and Approximate Cases
In this work, we study trade-offs between accuracy and privacy in the context
of linear queries over histograms. This is a rich class of queries that
includes contingency tables and range queries, and has been a focus of a long
line of work. For a set of linear queries over a database , we
seek to find the differentially private mechanism that has the minimum mean
squared error. For pure differential privacy, an approximation to
the optimal mechanism is known. Our first contribution is to give an approximation guarantee for the case of (\eps,\delta)-differential
privacy. Our mechanism is simple, efficient and adds correlated Gaussian noise
to the answers. We prove its approximation guarantee relative to the hereditary
discrepancy lower bound of Muthukrishnan and Nikolov, using tools from convex
geometry.
We next consider this question in the case when the number of queries exceeds
the number of individuals in the database, i.e. when . It is known that better mechanisms exist in this setting. Our second
main contribution is to give an (\eps,\delta)-differentially private
mechanism which is optimal up to a \polylog(d,N) factor for any given query
set and any given upper bound on . This approximation is
achieved by coupling the Gaussian noise addition approach with a linear
regression step. We give an analogous result for the \eps-differential
privacy setting. We also improve on the mean squared error upper bound for
answering counting queries on a database of size by Blum, Ligett, and Roth,
and match the lower bound implied by the work of Dinur and Nissim up to
logarithmic factors.
The connection between hereditary discrepancy and the privacy mechanism
enables us to derive the first polylogarithmic approximation to the hereditary
discrepancy of a matrix
Towards a Constructive Version of Banaszczyk's Vector Balancing Theorem
An important theorem of Banaszczyk (Random Structures & Algorithms `98)
states that for any sequence of vectors of norm at most and any
convex body of Gaussian measure in , there exists a
signed combination of these vectors which lands inside . A major open
problem is to devise a constructive version of Banaszczyk's vector balancing
theorem, i.e. to find an efficient algorithm which constructs the signed
combination.
We make progress towards this goal along several fronts. As our first
contribution, we show an equivalence between Banaszczyk's theorem and the
existence of -subgaussian distributions over signed combinations. For the
case of symmetric convex bodies, our equivalence implies the existence of a
universal signing algorithm (i.e. independent of the body), which simply
samples from the subgaussian sign distribution and checks to see if the
associated combination lands inside the body. For asymmetric convex bodies, we
provide a novel recentering procedure, which allows us to reduce to the case
where the body is symmetric.
As our second main contribution, we show that the above framework can be
efficiently implemented when the vectors have length ,
recovering Banaszczyk's results under this stronger assumption. More precisely,
we use random walk techniques to produce the required -subgaussian
signing distributions when the vectors have length , and
use a stochastic gradient ascent method to implement the recentering procedure
for asymmetric bodies
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