43 research outputs found
A Direct Integral Pseudospectral Method for Solving a Class of Infinite-Horizon Optimal Control Problems Using Gegenbauer Polynomials and Certain Parametric Maps
We present a novel direct integral pseudospectral (PS) method (a direct IPS
method) for solving a class of continuous-time infinite-horizon optimal control
problems (IHOCs). The method transforms the IHOCs into finite-horizon optimal
control problems (FHOCs) in their integral forms by means of certain parametric
mappings, which are then approximated by finite-dimensional nonlinear
programming problems (NLPs) through rational collocations based on Gegenbauer
polynomials and Gegenbauer-Gauss-Radau (GGR) points. The paper also analyzes
the interplay between the parametric maps, barycentric rational collocations
based on Gegenbauer polynomials and GGR points, and the convergence properties
of the collocated solutions for IHOCs. Some novel formulas for the construction
of the rational interpolation weights and the GGR-based integration and
differentiation matrices in barycentric-trigonometric forms are derived. A
rigorous study on the error and convergence of the proposed method is
presented. A stability analysis based on the Lebesgue constant for GGR-based
rational interpolation is investigated. Two easy-to-implement pseudocodes of
computational algorithms for computing the barycentric-trigonometric rational
weights are described. Two illustrative test examples are presented to support
the theoretical results. We show that the proposed collocation method leveraged
with a fast and accurate NLP solver converges exponentially to near-optimal
approximations for a coarse collocation mesh grid size. The paper also shows
that typical direct spectral/PS- and IPS-methods based on classical Jacobi
polynomials and certain parametric maps usually diverge as the number of
collocation points grow large, if the computations are carried out using
floating-point arithmetic and the discretizations use a single mesh grid
whether they are of Gauss/Gauss-Radau (GR) type or equally-spaced.Comment: 33 pages, 19 figure
Dynamical problems and phase transitions
Issued as Financial status report, Technical reports [nos. 1-12], and Final report, Project B-06-68
A Size-Free CLT for Poisson Multinomials and its Applications
An -Poisson Multinomial Distribution (PMD) is the distribution of the
sum of independent random vectors supported on the set of standard basis vectors in . We show
that any -PMD is -close in total
variation distance to the (appropriately discretized) multi-dimensional
Gaussian with the same first two moments, removing the dependence on from
the Central Limit Theorem of Valiant and Valiant. Interestingly, our CLT is
obtained by bootstrapping the Valiant-Valiant CLT itself through the structural
characterization of PMDs shown in recent work by Daskalakis, Kamath, and
Tzamos. In turn, our stronger CLT can be leveraged to obtain an efficient PTAS
for approximate Nash equilibria in anonymous games, significantly improving the
state of the art, and matching qualitatively the running time dependence on
and of the best known algorithm for two-strategy anonymous
games. Our new CLT also enables the construction of covers for the set of
-PMDs, which are proper and whose size is shown to be essentially
optimal. Our cover construction combines our CLT with the Shapley-Folkman
theorem and recent sparsification results for Laplacian matrices by Batson,
Spielman, and Srivastava. Our cover size lower bound is based on an algebraic
geometric construction. Finally, leveraging the structural properties of the
Fourier spectrum of PMDs we show that these distributions can be learned from
samples in -time, removing
the quasi-polynomial dependence of the running time on from the
algorithm of Daskalakis, Kamath, and Tzamos.Comment: To appear in STOC 201
Combined optimization algorithms applied to pattern classification
Accurate classification by minimizing the error on test samples is the main
goal in pattern classification. Combinatorial optimization is a well-known
method for solving minimization problems, however, only a few examples of
classifiers axe described in the literature where combinatorial optimization is
used in pattern classification. Recently, there has been a growing interest
in combining classifiers and improving the consensus of results for a greater
accuracy. In the light of the "No Ree Lunch Theorems", we analyse the combination
of simulated annealing, a powerful combinatorial optimization method
that produces high quality results, with the classical perceptron algorithm.
This combination is called LSA machine. Our analysis aims at finding paradigms
for problem-dependent parameter settings that ensure high classifica,
tion results. Our computational experiments on a large number of benchmark
problems lead to results that either outperform or axe at least competitive to
results published in the literature. Apart from paxameter settings, our analysis
focuses on a difficult problem in computation theory, namely the network
complexity problem. The depth vs size problem of neural networks is one of
the hardest problems in theoretical computing, with very little progress over
the past decades. In order to investigate this problem, we introduce a new
recursive learning method for training hidden layers in constant depth circuits.
Our findings make contributions to a) the field of Machine Learning, as the
proposed method is applicable in training feedforward neural networks, and to
b) the field of circuit complexity by proposing an upper bound for the number
of hidden units sufficient to achieve a high classification rate. One of the major
findings of our research is that the size of the network can be bounded by
the input size of the problem and an approximate upper bound of 8 + √2n/n
threshold gates as being sufficient for a small error rate, where n := log/SL
and SL is the training set