1,288 research outputs found
Convexity properties of loss and overflow functions
We show that the fluid loss ratio in a fluid queue with finite buffer and constant link capacity is always a jointly convex function of and . This generalizes prior work [6] which shows convexity of the trade-off for large number of i.i.d. multiplexed sources, using the large deviations rate function as approximation for fluid loss. Our approach also leads to a simpler proof of the prior result, and provides a stronger basis for optimal measurement-based control of resource allocation in shared resource systems
Large deviation asymptotics for occupancy problems
In the standard formulation of the occupancy problem one considers the
distribution of r balls in n cells, with each ball assigned independently to a
given cell with probability 1/n. Although closed form expressions can be given
for the distribution of various interesting quantities (such as the fraction of
cells that contain a given number of balls), these expressions are often of
limited practical use. Approximations provide an attractive alternative, and in
the present paper we consider a large deviation approximation as r and n tend
to infinity. In order to analyze the problem we first consider a dynamical
model, where the balls are placed in the cells sequentially and ``time''
corresponds to the number of balls that have already been thrown. A complete
large deviation analysis of this ``process level'' problem is carried out, and
the rate function for the original problem is then obtained via the contraction
principle. The variational problem that characterizes this rate function is
analyzed, and a fairly complete and explicit solution is obtained. The
minimizing trajectories and minimal cost are identified up to two constants,
and the constants are characterized as the unique solution to an elementary
fixed point problem. These results are then used to solve a number of
interesting problems, including an overflow problem and the partial coupon
collector's problem.Comment: Published by the Institute of Mathematical Statistics
(http://www.imstat.org) in the Annals of Probability
(http://www.imstat.org/aop/) at http://dx.doi.org/10.1214/00911790400000013
Adaptive Normalized Risk-Averting Training For Deep Neural Networks
This paper proposes a set of new error criteria and learning approaches,
Adaptive Normalized Risk-Averting Training (ANRAT), to attack the non-convex
optimization problem in training deep neural networks (DNNs). Theoretically, we
demonstrate its effectiveness on global and local convexity lower-bounded by
the standard -norm error. By analyzing the gradient on the convexity index
, we explain the reason why to learn adaptively using
gradient descent works. In practice, we show how this method improves training
of deep neural networks to solve visual recognition tasks on the MNIST and
CIFAR-10 datasets. Without using pretraining or other tricks, we obtain results
comparable or superior to those reported in recent literature on the same tasks
using standard ConvNets + MSE/cross entropy. Performance on deep/shallow
multilayer perceptrons and Denoised Auto-encoders is also explored. ANRAT can
be combined with other quasi-Newton training methods, innovative network
variants, regularization techniques and other specific tricks in DNNs. Other
than unsupervised pretraining, it provides a new perspective to address the
non-convex optimization problem in DNNs.Comment: AAAI 2016, 0.39%~0.4% ER on MNIST with single 32-32-256-10 ConvNets,
code available at https://github.com/cauchyturing/ANRA
Robust measurement-based buffer overflow probability estimators for QoS provisioning and traffic anomaly prediction applicationm
Suitable estimators for a class of Large Deviation approximations of rare
event probabilities based on sample realizations of random processes have been
proposed in our earlier work. These estimators are expressed as non-linear
multi-dimensional optimization problems of a special structure. In this paper,
we develop an algorithm to solve these optimization problems very efficiently
based on their characteristic structure. After discussing the nature of the
objective function and constraint set and their peculiarities, we provide a
formal proof that the developed algorithm is guaranteed to always converge. The
existence of efficient and provably convergent algorithms for solving these
problems is a prerequisite for using the proposed estimators in real time
problems such as call admission control, adaptive modulation and coding with
QoS constraints, and traffic anomaly detection in high data rate communication
networks
Robust measurement-based buffer overflow probability estimators for QoS provisioning and traffic anomaly prediction applications
Suitable estimators for a class of Large Deviation approximations of rare event probabilities based on sample realizations of random processes have been proposed in our earlier work. These estimators are expressed as non-linear multi-dimensional optimization problems of a special structure. In this paper, we develop an algorithm to solve these optimization problems very efficiently based on their characteristic structure. After discussing the nature of the objective function and constraint set and their peculiarities, we provide a formal proof that the developed algorithm is guaranteed to always converge. The existence of efficient and provably convergent algorithms for solving these problems is a prerequisite for using the proposed estimators in real time problems such as call admission control, adaptive modulation and coding with QoS constraints, and traffic anomaly detection in high data rate communication networks
Many-Sources Large Deviations for Max-Weight Scheduling
In this paper, a many-sources large deviations principle (LDP) for the
transient workload of a multi-queue single-server system is established where
the service rates are chosen from a compact, convex and coordinate-convex rate
region and where the service discipline is the max-weight policy. Under the
assumption that the arrival processes satisfy a many-sources LDP, this is
accomplished by employing Garcia's extended contraction principle that is
applicable to quasi-continuous mappings.
For the simplex rate-region, an LDP for the stationary workload is also
established under the additional requirements that the scheduling policy be
work-conserving and that the arrival processes satisfy certain mixing
conditions.
The LDP results can be used to calculate asymptotic buffer overflow
probabilities accounting for the multiplexing gain, when the arrival process is
an average of \emph{i.i.d.} processes. The rate function for the stationary
workload is expressed in term of the rate functions of the finite-horizon
workloads when the arrival processes have \emph{i.i.d.} increments.Comment: 44 page
Temperature Overloads in Power Grids Under Uncertainty: a Large Deviations Approach
The advent of renewable energy has huge implications for the design and
control of power grids. Due to increasing supply-side uncertainty, traditional
reliability constraints such as strict bounds on current, voltage and
temperature in a transmission line have to be replaced by computationally
demanding chance constraints. In this paper we use large deviations techniques
to study the probability of current and temperature overloads in power grids
with stochastic power injections, and develop corresponding safe capacity
regions. In particular, we characterize the set of admissible power injections
such that the probability of overloading of any line over a given time interval
stays below a fixed target. We show how enforcing (stochastic) constraints on
temperature, rather than on current, results in a less conservative approach
and can thus lead to capacity gains.Comment: 12 pages (10 pages + 2 pages appendix), 2 figures. Revised version
with extended numerical sectio
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