12,727 research outputs found
A Stochastic Interpretation of Stochastic Mirror Descent: Risk-Sensitive Optimality
Stochastic mirror descent (SMD) is a fairly new family of algorithms that has
recently found a wide range of applications in optimization, machine learning,
and control. It can be considered a generalization of the classical stochastic
gradient algorithm (SGD), where instead of updating the weight vector along the
negative direction of the stochastic gradient, the update is performed in a
"mirror domain" defined by the gradient of a (strictly convex) potential
function. This potential function, and the mirror domain it yields, provides
considerable flexibility in the algorithm compared to SGD. While many
properties of SMD have already been obtained in the literature, in this paper
we exhibit a new interpretation of SMD, namely that it is a risk-sensitive
optimal estimator when the unknown weight vector and additive noise are
non-Gaussian and belong to the exponential family of distributions. The
analysis also suggests a modified version of SMD, which we refer to as
symmetric SMD (SSMD). The proofs rely on some simple properties of Bregman
divergence, which allow us to extend results from quadratics and Gaussians to
certain convex functions and exponential families in a rather seamless way
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
Maximum Persistency in Energy Minimization
We consider discrete pairwise energy minimization problem (weighted
constraint satisfaction, max-sum labeling) and methods that identify a globally
optimal partial assignment of variables. When finding a complete optimal
assignment is intractable, determining optimal values for a part of variables
is an interesting possibility. Existing methods are based on different
sufficient conditions. We propose a new sufficient condition for partial
optimality which is: (1) verifiable in polynomial time (2) invariant to
reparametrization of the problem and permutation of labels and (3) includes
many existing sufficient conditions as special cases. We pose the problem of
finding the maximum optimal partial assignment identifiable by the new
sufficient condition. A polynomial method is proposed which is guaranteed to
assign same or larger part of variables than several existing approaches. The
core of the method is a specially constructed linear program that identifies
persistent assignments in an arbitrary multi-label setting.Comment: Extended technical report for the CVPR 2014 paper. Update: correction
to the proof of characterization theore
Conic Optimization Theory: Convexification Techniques and Numerical Algorithms
Optimization is at the core of control theory and appears in several areas of
this field, such as optimal control, distributed control, system
identification, robust control, state estimation, model predictive control and
dynamic programming. The recent advances in various topics of modern
optimization have also been revamping the area of machine learning. Motivated
by the crucial role of optimization theory in the design, analysis, control and
operation of real-world systems, this tutorial paper offers a detailed overview
of some major advances in this area, namely conic optimization and its emerging
applications. First, we discuss the importance of conic optimization in
different areas. Then, we explain seminal results on the design of hierarchies
of convex relaxations for a wide range of nonconvex problems. Finally, we study
different numerical algorithms for large-scale conic optimization problems.Comment: 18 page
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