9 research outputs found
On the Minimization of Convex Functionals of Probability Distributions Under Band Constraints
The problem of minimizing convex functionals of probability distributions is
solved under the assumption that the density of every distribution is bounded
from above and below. A system of sufficient and necessary first-order
optimality conditions as well as a bound on the optimality gap of feasible
candidate solutions are derived. Based on these results, two numerical
algorithms are proposed that iteratively solve the system of optimality
conditions on a grid of discrete points. Both algorithms use a block coordinate
descent strategy and terminate once the optimality gap falls below the desired
tolerance. While the first algorithm is conceptually simpler and more
efficient, it is not guaranteed to converge for objective functions that are
not strictly convex. This shortcoming is overcome in the second algorithm,
which uses an additional outer proximal iteration, and, which is proven to
converge under mild assumptions. Two examples are given to demonstrate the
theoretical usefulness of the optimality conditions as well as the high
efficiency and accuracy of the proposed numerical algorithms.Comment: 13 pages, 5 figures, 2 tables, published in the IEEE Transactions on
Signal Processing. In previous versions, the example in Section VI.B
contained some mistakes and inaccuracies, which have been fixed in this
versio
A Linear Programming Approach to Sequential Hypothesis Testing
Under some mild Markov assumptions it is shown that the problem of designing
optimal sequential tests for two simple hypotheses can be formulated as a
linear program. The result is derived by investigating the Lagrangian dual of
the sequential testing problem, which is an unconstrained optimal stopping
problem, depending on two unknown Lagrangian multipliers. It is shown that the
derivative of the optimal cost function with respect to these multipliers
coincides with the error probabilities of the corresponding sequential test.
This property is used to formulate an optimization problem that is jointly
linear in the cost function and the Lagrangian multipliers and an be solved for
both with off-the-shelf algorithms. To illustrate the procedure, optimal
sequential tests for Gaussian random sequences with different dependency
structures are derived, including the Gaussian AR(1) process.Comment: 25 pages, 4 figures, accepted for publication in Sequential Analysi
On the Equivalence of f-Divergence Balls and Density Bands in Robust Detection
The paper deals with minimax optimal statistical tests for two composite
hypotheses, where each hypothesis is defined by a non-parametric uncertainty
set of feasible distributions. It is shown that for every pair of uncertainty
sets of the f-divergence ball type, a pair of uncertainty sets of the density
band type can be constructed, which is equivalent in the sense that it admits
the same pair of least favorable distributions. This result implies that robust
tests under -divergence ball uncertainty, which are typically only minimax
optimal for the single sample case, are also fixed sample size minimax optimal
with respect to the equivalent density band uncertainty sets.Comment: 5 pages, 1 figure, accepted for publication in the Proceedings of the
IEEE International Conference on Acoustics, Speech, and Signal Processing
(ICASSP) 201
On Optimizing the Conditional Value-at-Risk of a Maximum Cost for Risk-Averse Safety Analysis
The popularity of Conditional Value-at-Risk (CVaR), a risk functional from
finance, has been growing in the control systems community due to its intuitive
interpretation and axiomatic foundation. We consider a non-standard optimal
control problem in which the goal is to minimize the CVaR of a maximum random
cost subject to a Borel-space Markov decision process. The objective takes the
form , where is a
risk-aversion parameter representing a fraction of worst cases, is a
stage or terminal cost, and is the length of a finite
discrete-time horizon. The objective represents the maximum departure from a
desired operating region averaged over a given fraction of worst
cases. This problem provides a safety criterion for a stochastic system that is
informed by both the probability and severity of the potential consequences of
the system's trajectory. In contrast, existing safety analysis frameworks apply
stage-wise risk constraints (i.e., must be small for all , where
is a risk functional) or assess the probability of constraint violation
without quantifying its possible severity. To the best of our knowledge, the
problem of interest has not been solved. To solve the problem, we propose and
study a family of stochastic dynamic programs on an augmented state space. We
prove that the optimal CVaR of a maximum cost enjoys an equivalent
representation in terms of the solutions to this family of dynamic programs
under appropriate assumptions. We show the existence of an optimal policy that
depends on the dynamics of an augmented state under a measurable selection
condition. Moreover, we demonstrate how our safety analysis framework is useful
for assessing the severity of combined sewer overflows under precipitation
uncertainty.Comment: A shorter version is under review for IEEE Transactions on Automatic
Control, submitted December 202
Recommended from our members
Finite-Sample Bounds on the Accuracy of Plug-In Estimators of Fisher Information
Finite-sample bounds on the accuracy of Bhattacharya’s plug-in estimator for Fisher
information are derived. These bounds are further improved by introducing a clipping step that
allows for better control over the score function. This leads to superior upper bounds on the rates
of convergence, albeit under slightly different regularity conditions. The performance bounds on
both estimators are evaluated for the practically relevant case of a random variable contaminated by
Gaussian noise. Moreover, using Brown’s identity, two corresponding estimators of the minimum
mean-square error are proposed