11,318 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
Guidance, Flight Mechanics and Trajectory Optimization. Volume 5 - State Determination And/or Estimation
Guidance, flight mechanics, and trajectory optimizatio
Performance Limits of Stochastic Sub-Gradient Learning, Part II: Multi-Agent Case
The analysis in Part I revealed interesting properties for subgradient
learning algorithms in the context of stochastic optimization when gradient
noise is present. These algorithms are used when the risk functions are
non-smooth and involve non-differentiable components. They have been long
recognized as being slow converging methods. However, it was revealed in Part I
that the rate of convergence becomes linear for stochastic optimization
problems, with the error iterate converging at an exponential rate
to within an neighborhood of the optimizer, for some and small step-size . The conclusion was established under weaker
assumptions than the prior literature and, moreover, several important problems
(such as LASSO, SVM, and Total Variation) were shown to satisfy these weaker
assumptions automatically (but not the previously used conditions from the
literature). These results revealed that sub-gradient learning methods have
more favorable behavior than originally thought when used to enable continuous
adaptation and learning. The results of Part I were exclusive to single-agent
adaptation. The purpose of the current Part II is to examine the implications
of these discoveries when a collection of networked agents employs subgradient
learning as their cooperative mechanism. The analysis will show that, despite
the coupled dynamics that arises in a networked scenario, the agents are still
able to attain linear convergence in the stochastic case; they are also able to
reach agreement within of the optimizer
Performance Limits of Stochastic Sub-Gradient Learning, Part II: Multi-Agent Case
The analysis in Part I revealed interesting properties for subgradient
learning algorithms in the context of stochastic optimization when gradient
noise is present. These algorithms are used when the risk functions are
non-smooth and involve non-differentiable components. They have been long
recognized as being slow converging methods. However, it was revealed in Part I
that the rate of convergence becomes linear for stochastic optimization
problems, with the error iterate converging at an exponential rate
to within an neighborhood of the optimizer, for some and small step-size . The conclusion was established under weaker
assumptions than the prior literature and, moreover, several important problems
(such as LASSO, SVM, and Total Variation) were shown to satisfy these weaker
assumptions automatically (but not the previously used conditions from the
literature). These results revealed that sub-gradient learning methods have
more favorable behavior than originally thought when used to enable continuous
adaptation and learning. The results of Part I were exclusive to single-agent
adaptation. The purpose of the current Part II is to examine the implications
of these discoveries when a collection of networked agents employs subgradient
learning as their cooperative mechanism. The analysis will show that, despite
the coupled dynamics that arises in a networked scenario, the agents are still
able to attain linear convergence in the stochastic case; they are also able to
reach agreement within of the optimizer
Distributed Coupled Multi-Agent Stochastic Optimization
This work develops effective distributed strategies for the solution of
constrained multi-agent stochastic optimization problems with coupled
parameters across the agents. In this formulation, each agent is influenced by
only a subset of the entries of a global parameter vector or model, and is
subject to convex constraints that are only known locally. Problems of this
type arise in several applications, most notably in disease propagation models,
minimum-cost flow problems, distributed control formulations, and distributed
power system monitoring. This work focuses on stochastic settings, where a
stochastic risk function is associated with each agent and the objective is to
seek the minimizer of the aggregate sum of all risks subject to a set of
constraints. Agents are not aware of the statistical distribution of the data
and, therefore, can only rely on stochastic approximations in their learning
strategies. We derive an effective distributed learning strategy that is able
to track drifts in the underlying parameter model. A detailed performance and
stability analysis is carried out showing that the resulting coupled diffusion
strategy converges at a linear rate to an neighborhood of the true
penalized optimizer
CVXR: An R Package for Disciplined Convex Optimization
CVXR is an R package that provides an object-oriented modeling language for
convex optimization, similar to CVX, CVXPY, YALMIP, and Convex.jl. It allows
the user to formulate convex optimization problems in a natural mathematical
syntax rather than the restrictive form required by most solvers. The user
specifies an objective and set of constraints by combining constants,
variables, and parameters using a library of functions with known mathematical
properties. CVXR then applies signed disciplined convex programming (DCP) to
verify the problem's convexity. Once verified, the problem is converted into
standard conic form using graph implementations and passed to a cone solver
such as ECOS or SCS. We demonstrate CVXR's modeling framework with several
applications.Comment: 34 pages, 9 figure
Linear estimation in Krein spaces. Part II. Applications
We have shown that several interesting problems in Hâ-filtering, quadratic game theory, and risk sensitive control and estimation follow as special cases of the Krein-space linear estimation theory developed in Part I. We show that all these problems can be cast into the problem of calculating the stationary point of certain second-order forms, and that by considering the appropriate state space models and error Gramians, we can use the Krein-space estimation theory to calculate the stationary points and study their properties. The approach discussed here allows for interesting generalizations, such as finite memory adaptive filtering with varying sliding patterns
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