15,422 research outputs found
Dispersion for Data-Driven Algorithm Design, Online Learning, and Private Optimization
Data-driven algorithm design, that is, choosing the best algorithm for a
specific application, is a crucial problem in modern data science.
Practitioners often optimize over a parameterized algorithm family, tuning
parameters based on problems from their domain. These procedures have
historically come with no guarantees, though a recent line of work studies
algorithm selection from a theoretical perspective. We advance the foundations
of this field in several directions: we analyze online algorithm selection,
where problems arrive one-by-one and the goal is to minimize regret, and
private algorithm selection, where the goal is to find good parameters over a
set of problems without revealing sensitive information contained therein. We
study important algorithm families, including SDP-rounding schemes for problems
formulated as integer quadratic programs, and greedy techniques for canonical
subset selection problems. In these cases, the algorithm's performance is a
volatile and piecewise Lipschitz function of its parameters, since tweaking the
parameters can completely change the algorithm's behavior. We give a sufficient
and general condition, dispersion, defining a family of piecewise Lipschitz
functions that can be optimized online and privately, which includes the
functions measuring the performance of the algorithms we study. Intuitively, a
set of piecewise Lipschitz functions is dispersed if no small region contains
many of the functions' discontinuities. We present general techniques for
online and private optimization of the sum of dispersed piecewise Lipschitz
functions. We improve over the best-known regret bounds for a variety of
problems, prove regret bounds for problems not previously studied, and give
matching lower bounds. We also give matching upper and lower bounds on the
utility loss due to privacy. Moreover, we uncover dispersion in auction design
and pricing problems
The Effect of Quadrature Errors in the Computation of L^2 Piecewise Polynomial Approximations
In this paper we investigate the L^2 piecewise polynomial approximation problem. L^2 bounds for the derivatives of the error in approximating sufficiently smooth functions by polynomial splines follow immediately from the analogous results for polynomial spline interpolation. We derive L^2 bounds for the errors introduced by the use of two types of quadrature rules for the numerical computation of L^2 piecewise polynomial approximations. These bounds enable us to present some asymptotic results and to examine the consistent convergence of appropriately chosen sequences of such approximations. Some numerical results are also included
Development and Analysis of Deterministic Privacy-Preserving Policies Using Non-Stochastic Information Theory
A deterministic privacy metric using non-stochastic information theory is
developed. Particularly, minimax information is used to construct a measure of
information leakage, which is inversely proportional to the measure of privacy.
Anyone can submit a query to a trusted agent with access to a non-stochastic
uncertain private dataset. Optimal deterministic privacy-preserving policies
for responding to the submitted query are computed by maximizing the measure of
privacy subject to a constraint on the worst-case quality of the response
(i.e., the worst-case difference between the response by the agent and the
output of the query computed on the private dataset). The optimal
privacy-preserving policy is proved to be a piecewise constant function in the
form of a quantization operator applied on the output of the submitted query.
The measure of privacy is also used to analyze the performance of -anonymity
methodology (a popular deterministic mechanism for privacy-preserving release
of datasets using suppression and generalization techniques), proving that it
is in fact not privacy-preserving.Comment: improved introduction and numerical exampl
Effect of Chaotic Noise on Multistable Systems
In a recent letter [Phys.Rev.Lett. {\bf 30}, 3269 (1995), chao-dyn/9510011],
we reported that a macroscopic chaotic determinism emerges in a multistable
system: the unidirectional motion of a dissipative particle subject to an
apparently symmetric chaotic noise occurs even if the particle is in a
spatially symmetric potential. In this paper, we study the global dynamics of a
dissipative particle by investigating the barrier crossing probability of the
particle between two basins of the multistable potential. We derive
analytically an expression of the barrier crossing probability of the particle
subject to a chaotic noise generated by a general piecewise linear map. We also
show that the obtained analytical barrier crossing probability is applicable to
a chaotic noise generated not only by a piecewise linear map with a uniform
invariant density but also by a non-piecewise linear map with non-uniform
invariant density. We claim, from the viewpoint of the noise induced motion in
a multistable system, that chaotic noise is a first realization of the effect
of {\em dynamical asymmetry} of general noise which induces the symmetry
breaking dynamics.Comment: 14 pages, 9 figures, to appear in Phys.Rev.
Stabilised finite element methods for ill-posed problems with conditional stability
In this paper we discuss the adjoint stabilised finite element method
introduced in, E. Burman, Stabilized finite element methods for nonsymmetric,
noncoercive and ill-posed problems. Part I: elliptic equations, SIAM Journal on
Scientific Computing, and how it may be used for the computation of solutions
to problems for which the standard stability theory given by the Lax-Milgram
Lemma or the Babuska-Brezzi Theorem fails. We pay particular attention to
ill-posed problems that have some conditional stability property and prove
(conditional) error estimates in an abstract framework. As a model problem we
consider the elliptic Cauchy problem and provide a complete numerical analysis
for this case. Some numerical examples are given to illustrate the theory.Comment: Accepted in the proceedings from the EPSRC Durham Symposium Building
Bridges: Connections and Challenges in Modern Approaches to Numerical Partial
Differential Equation
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