7,276 research outputs found
Information-Geometric Optimization Algorithms: A Unifying Picture via Invariance Principles
We present a canonical way to turn any smooth parametric family of
probability distributions on an arbitrary search space into a
continuous-time black-box optimization method on , the
\emph{information-geometric optimization} (IGO) method. Invariance as a design
principle minimizes the number of arbitrary choices. The resulting \emph{IGO
flow} conducts the natural gradient ascent of an adaptive, time-dependent,
quantile-based transformation of the objective function. It makes no
assumptions on the objective function to be optimized.
The IGO method produces explicit IGO algorithms through time discretization.
It naturally recovers versions of known algorithms and offers a systematic way
to derive new ones. The cross-entropy method is recovered in a particular case,
and can be extended into a smoothed, parametrization-independent maximum
likelihood update (IGO-ML). For Gaussian distributions on , IGO
is related to natural evolution strategies (NES) and recovers a version of the
CMA-ES algorithm. For Bernoulli distributions on , we recover the
PBIL algorithm. From restricted Boltzmann machines, we obtain a novel algorithm
for optimization on . All these algorithms are unified under a
single information-geometric optimization framework.
Thanks to its intrinsic formulation, the IGO method achieves invariance under
reparametrization of the search space , under a change of parameters of the
probability distributions, and under increasing transformations of the
objective function.
Theory strongly suggests that IGO algorithms have minimal loss in diversity
during optimization, provided the initial diversity is high. First experiments
using restricted Boltzmann machines confirm this insight. Thus IGO seems to
provide, from information theory, an elegant way to spontaneously explore
several valleys of a fitness landscape in a single run.Comment: Final published versio
Filling in CMB map missing data using constrained Gaussian realizations
For analyzing maps of the cosmic microwave background sky, it is necessary to
mask out the region around the galactic equator where the parasitic foreground
emission is strongest as well as the brightest compact sources. Since many of
the analyses of the data, particularly those searching for non-Gaussianity of a
primordial origin, are most straightforwardly carried out on full-sky maps, it
is of great interest to develop efficient algorithms for filling in the missing
information in a plausible way. We explore practical algorithms for filling in
based on constrained Gaussian realizations. Although carrying out such
realizations is in principle straightforward, for finely pixelized maps as will
be required for the Planck analysis a direct brute force method is not
numerically tractable. We present some concrete solutions to this problem, both
on a spatially flat sky with periodic boundary conditions and on the pixelized
sphere. One approach is to solve the linear system with an appropriately
preconditioned conjugate gradient method. While this approach was successfully
implemented on a rectangular domain with periodic boundary conditions and
worked even for very wide masked regions, we found that the method failed on
the pixelized sphere for reasons that we explain here. We present an approach
that works for full-sky pixelized maps on the sphere involving a kernel-based
multi-resolution Laplace solver followed by a series of conjugate gradient
corrections near the boundary of the mask.Comment: 22 pages, 14 figures, minor changes, a few missing references adde
A scalable line-independent design algorithm for voltage and frequency control in AC islanded microgrids
We propose a decentralized control synthesis procedure for stabilizing
voltage and frequency in AC Islanded microGrids (ImGs) composed of Distributed
Generation Units (DGUs) and loads interconnected through power lines. The
presented approach enables Plug-and-Play (PnP) operations, meaning that DGUs
can be added or removed without compromising the overall ImG stability. The
main feature of our approach is that the proposed design algorithm is
line-independent. This implies that (i) the synthesis of each local controller
requires only the parameters of the corresponding DGU and not the model of
power lines connecting neighboring DGUs, and (ii) whenever a new DGU is plugged
in, DGUs physically coupled with it do not have to retune their regulators
because of the new power line connected to them. Moreover, we formally prove
that stabilizing local controllers can be always computed, independently of the
electrical parameters. Theoretical results are validated by simulating in PSCAD
the behavior of a 10-DGUs ImG
Subdiffusive heat-kernel decay in four-dimensional i.i.d. random conductance models
We study the diagonal heat-kernel decay for the four-dimensional
nearest-neighbor random walk (on ) among i.i.d. random conductances that
are positive, bounded from above but can have arbitrarily heavy tails at zero.
It has been known that the quenched return probability \cmss
P_\omega^{2n}(0,0) after steps is at most , but
the best lower bound till now has been . Here we will show
that the term marks a real phenomenon by constructing an environment,
for each sequence , such that \cmss
P_\omega^{2n}(0,0)\ge C(\omega)\log(n)n^{-2}/\lambda_n, with
a.s., along a deterministic subsequence of 's. Notably, this holds
simultaneously with a (non-degenerate) quenched invariance principle. As for
the cases studied earlier, the source of the anomalous decay is a
trapping phenomenon although the contribution is in this case collected from a
whole range of spatial scales.Comment: 28 pages, version to appear in J. Lond. Math. So
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