7,091 research outputs found
Convex Learning of Multiple Tasks and their Structure
Reducing the amount of human supervision is a key problem in machine learning
and a natural approach is that of exploiting the relations (structure) among
different tasks. This is the idea at the core of multi-task learning. In this
context a fundamental question is how to incorporate the tasks structure in the
learning problem.We tackle this question by studying a general computational
framework that allows to encode a-priori knowledge of the tasks structure in
the form of a convex penalty; in this setting a variety of previously proposed
methods can be recovered as special cases, including linear and non-linear
approaches. Within this framework, we show that tasks and their structure can
be efficiently learned considering a convex optimization problem that can be
approached by means of block coordinate methods such as alternating
minimization and for which we prove convergence to the global minimum.Comment: 26 pages, 1 figure, 2 table
A Simple and Efficient Algorithm for Nonlinear Model Predictive Control
We present PANOC, a new algorithm for solving optimal control problems
arising in nonlinear model predictive control (NMPC). A usual approach to this
type of problems is sequential quadratic programming (SQP), which requires the
solution of a quadratic program at every iteration and, consequently, inner
iterative procedures. As a result, when the problem is ill-conditioned or the
prediction horizon is large, each outer iteration becomes computationally very
expensive. We propose a line-search algorithm that combines forward-backward
iterations (FB) and Newton-type steps over the recently introduced
forward-backward envelope (FBE), a continuous, real-valued, exact merit
function for the original problem. The curvature information of Newton-type
methods enables asymptotic superlinear rates under mild assumptions at the
limit point, and the proposed algorithm is based on very simple operations:
access to first-order information of the cost and dynamics and low-cost direct
linear algebra. No inner iterative procedure nor Hessian evaluation is
required, making our approach computationally simpler than SQP methods. The
low-memory requirements and simple implementation make our method particularly
suited for embedded NMPC applications
A second derivative SQP method: theoretical issues
Sequential quadratic programming (SQP) methods form a class of highly efficient algorithms for solving nonlinearly constrained optimization problems. Although second derivative information may often be calculated, there is little practical theory that justifies exact-Hessian SQP methods. In particular, the resulting quadratic programming (QP) subproblems are often nonconvex, and thus finding their global solutions may be computationally nonviable. This paper presents a second-derivative SQP method based on quadratic subproblems that are either convex, and thus may be solved efficiently, or need not be solved globally. Additionally, an explicit descent-constraint is imposed on certain QP subproblems, which “guides” the iterates through areas in which nonconvexity is a concern. Global convergence of the resulting algorithm is established
Geometric approach to Fletcher's ideal penalty function
Original article can be found at: www.springerlink.com Copyright Springer. [Originally produced as UH Technical Report 280, 1993]In this note, we derive a geometric formulation of an ideal penalty function for equality constrained problems. This differentiable penalty function requires no parameter estimation or adjustment, has numerical conditioning similar to that of the target function from which it is constructed, and also has the desirable property that the strict second-order constrained minima of the target function are precisely those strict second-order unconstrained minima of the penalty function which satisfy the constraints. Such a penalty function can be used to establish termination properties for algorithms which avoid ill-conditioned steps. Numerical values for the penalty function and its derivatives can be calculated efficiently using automatic differentiation techniques.Peer reviewe
Joint Reconstruction of Multi-channel, Spectral CT Data via Constrained Total Nuclear Variation Minimization
We explore the use of the recently proposed "total nuclear variation" (TNV)
as a regularizer for reconstructing multi-channel, spectral CT images. This
convex penalty is a natural extension of the total variation (TV) to
vector-valued images and has the advantage of encouraging common edge locations
and a shared gradient direction among image channels. We show how it can be
incorporated into a general, data-constrained reconstruction framework and
derive update equations based on the first-order, primal-dual algorithm of
Chambolle and Pock. Early simulation studies based on the numerical XCAT
phantom indicate that the inter-channel coupling introduced by the TNV leads to
better preservation of image features at high levels of regularization,
compared to independent, channel-by-channel TV reconstructions.Comment: Submitted to Physics in Medicine and Biolog
Learning Multiple Visual Tasks while Discovering their Structure
Multi-task learning is a natural approach for computer vision applications
that require the simultaneous solution of several distinct but related
problems, e.g. object detection, classification, tracking of multiple agents,
or denoising, to name a few. The key idea is that exploring task relatedness
(structure) can lead to improved performances.
In this paper, we propose and study a novel sparse, non-parametric approach
exploiting the theory of Reproducing Kernel Hilbert Spaces for vector-valued
functions. We develop a suitable regularization framework which can be
formulated as a convex optimization problem, and is provably solvable using an
alternating minimization approach. Empirical tests show that the proposed
method compares favorably to state of the art techniques and further allows to
recover interpretable structures, a problem of interest in its own right.Comment: 19 pages, 3 figures, 3 table
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