9,202 research outputs found
New advances in H∞ control and filtering for nonlinear systems
The main objective of this special issue is to
summarise recent advances in H∞ control and filtering
for nonlinear systems, including time-delay, hybrid and
stochastic systems. The published papers provide new
ideas and approaches, clearly indicating the advances
made in problem statements, methodologies or applications
with respect to the existing results. The special
issue also includes papers focusing on advanced and
non-traditional methods and presenting considerable
novelties in theoretical background or experimental
setup. Some papers present applications to newly
emerging fields, such as network-based control and
estimation
Geodesics on Calabi-Yau manifolds and winding states in nonlinear sigma models
We conjecture that a non-flat -real-dimensional compact Calabi-Yau
manifold, such as a quintic hypersurface with D=6, or a K3 manifold with D=4,
has locally length minimizing closed geodesics, and that the number of these
with length less than L grows asymptotically as L^{D}. We also outline the
physical arguments behind this conjecture, which involve the claim that all
states in a nonlinear sigma model can be identified as "momentum" and "winding"
states in the large volume limit.Comment: minor corrections, 43 pages, to appear in frontiers in mathematical
physics. Frontiers in Physics, Dec 16, 201
Newton-Type Methods for Non-Convex Optimization Under Inexact Hessian Information
We consider variants of trust-region and cubic regularization methods for
non-convex optimization, in which the Hessian matrix is approximated. Under
mild conditions on the inexact Hessian, and using approximate solution of the
corresponding sub-problems, we provide iteration complexity to achieve -approximate second-order optimality which have shown to be tight.
Our Hessian approximation conditions constitute a major relaxation over the
existing ones in the literature. Consequently, we are able to show that such
mild conditions allow for the construction of the approximate Hessian through
various random sampling methods. In this light, we consider the canonical
problem of finite-sum minimization, provide appropriate uniform and non-uniform
sub-sampling strategies to construct such Hessian approximations, and obtain
optimal iteration complexity for the corresponding sub-sampled trust-region and
cubic regularization methods.Comment: 32 page
Second-Order Optimization for Non-Convex Machine Learning: An Empirical Study
While first-order optimization methods such as stochastic gradient descent
(SGD) are popular in machine learning (ML), they come with well-known
deficiencies, including relatively-slow convergence, sensitivity to the
settings of hyper-parameters such as learning rate, stagnation at high training
errors, and difficulty in escaping flat regions and saddle points. These issues
are particularly acute in highly non-convex settings such as those arising in
neural networks. Motivated by this, there has been recent interest in
second-order methods that aim to alleviate these shortcomings by capturing
curvature information. In this paper, we report detailed empirical evaluations
of a class of Newton-type methods, namely sub-sampled variants of trust region
(TR) and adaptive regularization with cubics (ARC) algorithms, for non-convex
ML problems. In doing so, we demonstrate that these methods not only can be
computationally competitive with hand-tuned SGD with momentum, obtaining
comparable or better generalization performance, but also they are highly
robust to hyper-parameter settings. Further, in contrast to SGD with momentum,
we show that the manner in which these Newton-type methods employ curvature
information allows them to seamlessly escape flat regions and saddle points.Comment: 21 pages, 11 figures. Restructure the paper and add experiment
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