2,566 research outputs found
Level truncation and the quartic tachyon coupling
We discuss the convergence of level truncation in bosonic open string field
theory. As a test case we consider the calculation of the quartic tachyon
coupling . We determine the exact contribution from states up to
level L=28 and discuss the extrapolation by means of the BST
algorithm. We determine in a self-consistent way both the coupling and the
exponent of the leading correction to at finite that we
assume to be . The results are and
.}Comment: 17 pages, 2 eps figure
Catalyst Acceleration for Gradient-Based Non-Convex Optimization
We introduce a generic scheme to solve nonconvex optimization problems using
gradient-based algorithms originally designed for minimizing convex functions.
Even though these methods may originally require convexity to operate, the
proposed approach allows one to use them on weakly convex objectives, which
covers a large class of non-convex functions typically appearing in machine
learning and signal processing. In general, the scheme is guaranteed to produce
a stationary point with a worst-case efficiency typical of first-order methods,
and when the objective turns out to be convex, it automatically accelerates in
the sense of Nesterov and achieves near-optimal convergence rate in function
values. These properties are achieved without assuming any knowledge about the
convexity of the objective, by automatically adapting to the unknown weak
convexity constant. We conclude the paper by showing promising experimental
results obtained by applying our approach to incremental algorithms such as
SVRG and SAGA for sparse matrix factorization and for learning neural networks
Prediction Properties of Aitken's Iterated Delta^2 Process, of Wynn's Epsilon Algorithm, and of Brezinski's Iterated Theta Algorithm
The prediction properties of Aitken's iterated Delta^2 process, Wynn's
epsilon algorithm, and Brezinski's iterated theta algorithm for (formal) power
series are analyzed. As a first step, the defining recursive schemes of these
transformations are suitably rearranged in order to permit the derivation of
accuracy-through-order relationships. On the basis of these relationships, the
rational approximants can be rewritten as a partial sum plus an appropriate
transformation term. A Taylor expansion of such a transformation term, which is
a rational function and which can be computed recursively, produces the
predictions for those coefficients of the (formal) power series which were not
used for the computation of the corresponding rational approximant.Comment: 34 pages, LaTe
Regularized Nonlinear Acceleration
We describe a convergence acceleration technique for unconstrained
optimization problems. Our scheme computes estimates of the optimum from a
nonlinear average of the iterates produced by any optimization method. The
weights in this average are computed via a simple linear system, whose solution
can be updated online. This acceleration scheme runs in parallel to the base
algorithm, providing improved estimates of the solution on the fly, while the
original optimization method is running. Numerical experiments are detailed on
classical classification problems
Accelerating Cosmic Microwave Background map-making procedure through preconditioning
Estimation of the sky signal from sequences of time ordered data is one of
the key steps in Cosmic Microwave Background (CMB) data analysis, commonly
referred to as the map-making problem. Some of the most popular and general
methods proposed for this problem involve solving generalised least squares
(GLS) equations with non-diagonal noise weights given by a block-diagonal
matrix with Toeplitz blocks. In this work we study new map-making solvers
potentially suitable for applications to the largest anticipated data sets.
They are based on iterative conjugate gradient (CG) approaches enhanced with
novel, parallel, two-level preconditioners. We apply the proposed solvers to
examples of simulated non-polarised and polarised CMB observations, and a set
of idealised scanning strategies with sky coverage ranging from nearly a full
sky down to small sky patches. We discuss in detail their implementation for
massively parallel computational platforms and their performance for a broad
range of parameters characterising the simulated data sets. We find that our
best new solver can outperform carefully-optimised standard solvers used today
by a factor of as much as 5 in terms of the convergence rate and a factor of up
to in terms of the time to solution, and to do so without significantly
increasing the memory consumption and the volume of inter-processor
communication. The performance of the new algorithms is also found to be more
stable and robust, and less dependent on specific characteristics of the
analysed data set. We therefore conclude that the proposed approaches are well
suited to address successfully challenges posed by new and forthcoming CMB data
sets.Comment: 19 pages // Final version submitted to A&
Accelerating Cosmic Microwave Background map-making procedure through preconditioning
Estimation of the sky signal from sequences of time ordered data is one of
the key steps in Cosmic Microwave Background (CMB) data analysis, commonly
referred to as the map-making problem. Some of the most popular and general
methods proposed for this problem involve solving generalised least squares
(GLS) equations with non-diagonal noise weights given by a block-diagonal
matrix with Toeplitz blocks. In this work we study new map-making solvers
potentially suitable for applications to the largest anticipated data sets.
They are based on iterative conjugate gradient (CG) approaches enhanced with
novel, parallel, two-level preconditioners. We apply the proposed solvers to
examples of simulated non-polarised and polarised CMB observations, and a set
of idealised scanning strategies with sky coverage ranging from nearly a full
sky down to small sky patches. We discuss in detail their implementation for
massively parallel computational platforms and their performance for a broad
range of parameters characterising the simulated data sets. We find that our
best new solver can outperform carefully-optimised standard solvers used today
by a factor of as much as 5 in terms of the convergence rate and a factor of up
to in terms of the time to solution, and to do so without significantly
increasing the memory consumption and the volume of inter-processor
communication. The performance of the new algorithms is also found to be more
stable and robust, and less dependent on specific characteristics of the
analysed data set. We therefore conclude that the proposed approaches are well
suited to address successfully challenges posed by new and forthcoming CMB data
sets.Comment: 19 pages // Final version submitted to A&
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