34,761 research outputs found
Probabilistic analysis of a differential equation for linear programming
In this paper we address the complexity of solving linear programming
problems with a set of differential equations that converge to a fixed point
that represents the optimal solution. Assuming a probabilistic model, where the
inputs are i.i.d. Gaussian variables, we compute the distribution of the
convergence rate to the attracting fixed point. Using the framework of Random
Matrix Theory, we derive a simple expression for this distribution in the
asymptotic limit of large problem size. In this limit, we find that the
distribution of the convergence rate is a scaling function, namely it is a
function of one variable that is a combination of three parameters: the number
of variables, the number of constraints and the convergence rate, rather than a
function of these parameters separately. We also estimate numerically the
distribution of computation times, namely the time required to reach a vicinity
of the attracting fixed point, and find that it is also a scaling function.
Using the problem size dependence of the distribution functions, we derive high
probability bounds on the convergence rates and on the computation times.Comment: 1+37 pages, latex, 5 eps figures. Version accepted for publication in
the Journal of Complexity. Changes made: Presentation reorganized for
clarity, expanded discussion of measure of complexity in the non-asymptotic
regime (added a new section
Feynman-Kac representation of fully nonlinear PDEs and applications
The classical Feynman-Kac formula states the connection between linear
parabolic partial differential equations (PDEs), like the heat equation, and
expectation of stochastic processes driven by Brownian motion. It gives then a
method for solving linear PDEs by Monte Carlo simulations of random processes.
The extension to (fully)nonlinear PDEs led in the recent years to important
developments in stochastic analysis and the emergence of the theory of backward
stochastic differential equations (BSDEs), which can be viewed as nonlinear
Feynman-Kac formulas. We review in this paper the main ideas and results in
this area, and present implications of these probabilistic representations for
the numerical resolution of nonlinear PDEs, together with some applications to
stochastic control problems and model uncertainty in finance
Quantifying Differential Privacy under Temporal Correlations
Differential Privacy (DP) has received increased attention as a rigorous
privacy framework. Existing studies employ traditional DP mechanisms (e.g., the
Laplace mechanism) as primitives, which assume that the data are independent,
or that adversaries do not have knowledge of the data correlations. However,
continuously generated data in the real world tend to be temporally correlated,
and such correlations can be acquired by adversaries. In this paper, we
investigate the potential privacy loss of a traditional DP mechanism under
temporal correlations in the context of continuous data release. First, we
model the temporal correlations using Markov model and analyze the privacy
leakage of a DP mechanism when adversaries have knowledge of such temporal
correlations. Our analysis reveals that the privacy leakage of a DP mechanism
may accumulate and increase over time. We call it temporal privacy leakage.
Second, to measure such privacy leakage, we design an efficient algorithm for
calculating it in polynomial time. Although the temporal privacy leakage may
increase over time, we also show that its supremum may exist in some cases.
Third, to bound the privacy loss, we propose mechanisms that convert any
existing DP mechanism into one against temporal privacy leakage. Experiments
with synthetic data confirm that our approach is efficient and effective.Comment: appears at ICDE 201
Singularly perturbed forward-backward stochastic differential equations: application to the optimal control of bilinear systems
We study linear-quadratic stochastic optimal control problems with bilinear
state dependence for which the underlying stochastic differential equation
(SDE) consists of slow and fast degrees of freedom. We show that, in the same
way in which the underlying dynamics can be well approximated by a reduced
order effective dynamics in the time scale limit (using classical
homogenziation results), the associated optimal expected cost converges in the
time scale limit to an effective optimal cost. This entails that we can well
approximate the stochastic optimal control for the whole system by the reduced
order stochastic optimal control, which is clearly easier to solve because of
lower dimensionality. The approach uses an equivalent formulation of the
Hamilton-Jacobi-Bellman (HJB) equation, in terms of forward-backward SDEs
(FBSDEs). We exploit the efficient solvability of FBSDEs via a least squares
Monte Carlo algorithm and show its applicability by a suitable numerical
example
A forward--backward stochastic algorithm for quasi-linear PDEs
We propose a time-space discretization scheme for quasi-linear parabolic
PDEs. The algorithm relies on the theory of fully coupled forward--backward
SDEs, which provides an efficient probabilistic representation of this type of
equation. The derivated algorithm holds for strong solutions defined on any
interval of arbitrary length. As a bypass product, we obtain a discretization
procedure for the underlying FBSDE. In particular, our work provides an
alternative to the method described in [Douglas, Ma and Protter (1996) Ann.
Appl. Probab. 6 940--968] and weakens the regularity assumptions required in
this reference.Comment: Published at http://dx.doi.org/10.1214/105051605000000674 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Quantifying Differential Privacy in Continuous Data Release under Temporal Correlations
Differential Privacy (DP) has received increasing attention as a rigorous
privacy framework. Many existing studies employ traditional DP mechanisms
(e.g., the Laplace mechanism) as primitives to continuously release private
data for protecting privacy at each time point (i.e., event-level privacy),
which assume that the data at different time points are independent, or that
adversaries do not have knowledge of correlation between data. However,
continuously generated data tend to be temporally correlated, and such
correlations can be acquired by adversaries. In this paper, we investigate the
potential privacy loss of a traditional DP mechanism under temporal
correlations. First, we analyze the privacy leakage of a DP mechanism under
temporal correlation that can be modeled using Markov Chain. Our analysis
reveals that, the event-level privacy loss of a DP mechanism may
\textit{increase over time}. We call the unexpected privacy loss
\textit{temporal privacy leakage} (TPL). Although TPL may increase over time,
we find that its supremum may exist in some cases. Second, we design efficient
algorithms for calculating TPL. Third, we propose data releasing mechanisms
that convert any existing DP mechanism into one against TPL. Experiments
confirm that our approach is efficient and effective.Comment: accepted in TKDE special issue "Best of ICDE 2017". arXiv admin note:
substantial text overlap with arXiv:1610.0754
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