185,759 research outputs found
Implicitly Restarted Generalized Second-order Arnoldi Type Algorithms for the Quadratic Eigenvalue Problem
We investigate the generalized second-order Arnoldi (GSOAR) method, a
generalization of the SOAR method proposed by Bai and Su [{\em SIAM J. Matrix
Anal. Appl.}, 26 (2005): 640--659.], and the Refined GSOAR (RGSOAR) method for
the quadratic eigenvalue problem (QEP). The two methods use the GSOAR procedure
to generate an orthonormal basis of a given generalized second-order Krylov
subspace, and with such basis they project the QEP onto the subspace and
compute the Ritz pairs and the refined Ritz pairs, respectively. We develop
implicitly restarted GSOAR and RGSOAR algorithms, in which we propose certain
exact and refined shifts for respective use within the two algorithms.
Numerical experiments on real-world problems illustrate the efficiency of the
restarted algorithms and the superiority of the restarted RGSOAR to the
restarted GSOAR. The experiments also demonstrate that both IGSOAR and IRGSOAR
generally perform much better than the implicitly restarted Arnoldi method
applied to the corresponding linearization problems, in terms of the accuracy
and the computational efficiency.Comment: 30 pages, 6 figure
A Modular Order-sorted Equational Generalization Algorithm
Generalization, also called anti-unification, is the dual of unification. Given terms t and t
,
a generalizer is a term t of which t and t are substitution instances. The dual of
a most general unifier (mgu) is that of least general generalizer (lgg). In this work, we
extend the known untyped generalization algorithm to, first, an order-sorted typed setting
with sorts, subsorts, and subtype polymorphism; second, we extend it to work modulo
equational theories, where function symbols can obey any combination of associativity,
commutativity, and identity axioms (including the empty set of such axioms); and third, to
the combination of both, which results in a modular, order-sorted equational generalization
algorithm. Unlike the untyped case, there is in general no single lgg in our framework, due
to order-sortedness or to the equational axioms. Instead, there is a finite, minimal and
complete set of lggs, so that any other generalizer has at least one of them as an instance.
Our generalization algorithms are expressed by means of inference systems for which we
give proofs of correctness. This opens up new applications to partial evaluation, program
synthesis, and theorem proving for typed equational reasoning systems and typed rulebased
languages such as ASF+SDF, Elan, OBJ, Cafe-OBJ, and Maude.
© 2014 Elsevier Inc. All rights reserved.
1.M. Alpuente, S. Escobar, and J. Espert have been partially supported by the EU (FEDER) and the Spanish MEC/MICINN under grant TIN 2010-21062-C02-02, and by Generalitat Valenciana PROMETEO2011/052. J. Meseguer has been supported by NSF Grants CNS 09-04749, and CCF 09-05584.Alpuente Frasnedo, M.; Escobar Román, S.; Espert Real, J.; Meseguer, J. (2014). A Modular Order-sorted Equational Generalization Algorithm. Information and Computation. 235:98-136. https://doi.org/10.1016/j.ic.2014.01.006S9813623
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
PAC-Bayes and Domain Adaptation
We provide two main contributions in PAC-Bayesian theory for domain
adaptation where the objective is to learn, from a source distribution, a
well-performing majority vote on a different, but related, target distribution.
Firstly, we propose an improvement of the previous approach we proposed in
Germain et al. (2013), which relies on a novel distribution pseudodistance
based on a disagreement averaging, allowing us to derive a new tighter domain
adaptation bound for the target risk. While this bound stands in the spirit of
common domain adaptation works, we derive a second bound (introduced in Germain
et al., 2016) that brings a new perspective on domain adaptation by deriving an
upper bound on the target risk where the distributions' divergence-expressed as
a ratio-controls the trade-off between a source error measure and the target
voters' disagreement. We discuss and compare both results, from which we obtain
PAC-Bayesian generalization bounds. Furthermore, from the PAC-Bayesian
specialization to linear classifiers, we infer two learning algorithms, and we
evaluate them on real data.Comment: Neurocomputing, Elsevier, 2019. arXiv admin note: substantial text
overlap with arXiv:1503.0694
Non-local updates for quantum Monte Carlo simulations
We review the development of update schemes for quantum lattice models
simulated using world line quantum Monte Carlo algorithms. Starting from the
Suzuki-Trotter mapping we discuss limitations of local update algorithms and
highlight the main developments beyond Metropolis-style local updates: the
development of cluster algorithms, their generalization to continuous time, the
worm and directed-loop algorithms and finally a generalization of the flat
histogram method of Wang and Landau to quantum systems.Comment: 14 pages, article for the proceedings of the "The Monte Carlo Method
in the Physical Sciences: Celebrating the 50th Anniversary of the Metropolis
Algorithm", Los Alamos, June 9-11, 200
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