54,684 research outputs found
A comparison of sample-based Stochastic Optimal Control methods
In this paper, we compare the performance of two scenario-based numerical
methods to solve stochastic optimal control problems: scenario trees and
particles. The problem consists in finding strategies to control a dynamical
system perturbed by exogenous noises so as to minimize some expected cost along
a discrete and finite time horizon. We introduce the Mean Squared Error (MSE)
which is the expected -distance between the strategy given by the
algorithm and the optimal strategy, as a performance indicator for the two
models. We study the behaviour of the MSE with respect to the number of
scenarios used for discretization. The first model, widely studied in the
Stochastic Programming community, consists in approximating the noise diffusion
using a scenario tree representation. On a numerical example, we observe that
the number of scenarios needed to obtain a given precision grows exponentially
with the time horizon. In that sense, our conclusion on scenario trees is
equivalent to the one in the work by Shapiro (2006) and has been widely noticed
by practitioners. However, in the second part, we show using the same example
that, by mixing Stochastic Programming and Dynamic Programming ideas, the
particle method described by Carpentier et al (2009) copes with this numerical
difficulty: the number of scenarios needed to obtain a given precision now does
not depend on the time horizon. Unfortunately, we also observe that serious
obstacles still arise from the system state space dimension
On The Power of Tree Projections: Structural Tractability of Enumerating CSP Solutions
The problem of deciding whether CSP instances admit solutions has been deeply
studied in the literature, and several structural tractability results have
been derived so far. However, constraint satisfaction comes in practice as a
computation problem where the focus is either on finding one solution, or on
enumerating all solutions, possibly projected to some given set of output
variables. The paper investigates the structural tractability of the problem of
enumerating (possibly projected) solutions, where tractability means here
computable with polynomial delay (WPD), since in general exponentially many
solutions may be computed. A general framework based on the notion of tree
projection of hypergraphs is considered, which generalizes all known
decomposition methods. Tractability results have been obtained both for classes
of structures where output variables are part of their specification, and for
classes of structures where computability WPD must be ensured for any possible
set of output variables. These results are shown to be tight, by exhibiting
dichotomies for classes of structures having bounded arity and where the tree
decomposition method is considered
Online Convex Optimization for Sequential Decision Processes and Extensive-Form Games
Regret minimization is a powerful tool for solving large-scale extensive-form
games. State-of-the-art methods rely on minimizing regret locally at each
decision point. In this work we derive a new framework for regret minimization
on sequential decision problems and extensive-form games with general compact
convex sets at each decision point and general convex losses, as opposed to
prior work which has been for simplex decision points and linear losses. We
call our framework laminar regret decomposition. It generalizes the CFR
algorithm to this more general setting. Furthermore, our framework enables a
new proof of CFR even in the known setting, which is derived from a perspective
of decomposing polytope regret, thereby leading to an arguably simpler
interpretation of the algorithm. Our generalization to convex compact sets and
convex losses allows us to develop new algorithms for several problems:
regularized sequential decision making, regularized Nash equilibria in
extensive-form games, and computing approximate extensive-form perfect
equilibria. Our generalization also leads to the first regret-minimization
algorithm for computing reduced-normal-form quantal response equilibria based
on minimizing local regrets. Experiments show that our framework leads to
algorithms that scale at a rate comparable to the fastest variants of
counterfactual regret minimization for computing Nash equilibrium, and
therefore our approach leads to the first algorithm for computing quantal
response equilibria in extremely large games. Finally we show that our
framework enables a new kind of scalable opponent exploitation approach
Robust Machine Learning Applied to Astronomical Datasets I: Star-Galaxy Classification of the SDSS DR3 Using Decision Trees
We provide classifications for all 143 million non-repeat photometric objects
in the Third Data Release of the Sloan Digital Sky Survey (SDSS) using decision
trees trained on 477,068 objects with SDSS spectroscopic data. We demonstrate
that these star/galaxy classifications are expected to be reliable for
approximately 22 million objects with r < ~20. The general machine learning
environment Data-to-Knowledge and supercomputing resources enabled extensive
investigation of the decision tree parameter space. This work presents the
first public release of objects classified in this way for an entire SDSS data
release. The objects are classified as either galaxy, star or nsng (neither
star nor galaxy), with an associated probability for each class. To demonstrate
how to effectively make use of these classifications, we perform several
important tests. First, we detail selection criteria within the probability
space defined by the three classes to extract samples of stars and galaxies to
a given completeness and efficiency. Second, we investigate the efficacy of the
classifications and the effect of extrapolating from the spectroscopic regime
by performing blind tests on objects in the SDSS, 2dF Galaxy Redshift and 2dF
QSO Redshift (2QZ) surveys. Given the photometric limits of our spectroscopic
training data, we effectively begin to extrapolate past our star-galaxy
training set at r ~ 18. By comparing the number counts of our training sample
with the classified sources, however, we find that our efficiencies appear to
remain robust to r ~ 20. As a result, we expect our classifications to be
accurate for 900,000 galaxies and 6.7 million stars, and remain robust via
extrapolation for a total of 8.0 million galaxies and 13.9 million stars.
[Abridged]Comment: 27 pages, 12 figures, to be published in ApJ, uses emulateapj.cl
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