11,020 research outputs found
A generalization of the adaptive rejection sampling algorithm
The original publication is available at www.springerlink.comRejection sampling is a well-known method to generate random samples from arbitrary target probability distributions. It demands the design of a suitable proposal probability density function (pdf) from which candidate samples can be drawn. These samples are either accepted or rejected depending on a test involving the ratio of the target and proposal densities. The adaptive rejection sampling method is an efficient algorithm to sample from a log-concave target density, that attains high acceptance rates by improving the proposal density whenever a sample is rejected. In this paper we introduce a generalized adaptive rejection sampling procedure that can be applied with a broad class of target probability distributions, possibly non-log-concave and exhibiting multiple modes. The proposed technique yields a sequence of proposal densities that converge toward the target pdf, thus achieving very high acceptance rates. We provide a simple numerical example to illustrate the basic use of the proposed technique, together with a more elaborate positioning application using real data.This work has been partially supported by
the Ministry of Science and Innovation of Spain (project MONIN,
ref. TEC-2006-13514-C02-01/TCM, project DEIPRO, ref. TEC-2009-
14504-C02-01 and program Consolider-Ingenio 2010 CSD2008-
00010 COMONSENS) and the Autonomous Community of Madrid
(project PROMULTIDIS-CM, ref. S-0505/TIC/0233).Publicad
Metropolis Sampling
Monte Carlo (MC) sampling methods are widely applied in Bayesian inference,
system simulation and optimization problems. The Markov Chain Monte Carlo
(MCMC) algorithms are a well-known class of MC methods which generate a Markov
chain with the desired invariant distribution. In this document, we focus on
the Metropolis-Hastings (MH) sampler, which can be considered as the atom of
the MCMC techniques, introducing the basic notions and different properties. We
describe in details all the elements involved in the MH algorithm and the most
relevant variants. Several improvements and recent extensions proposed in the
literature are also briefly discussed, providing a quick but exhaustive
overview of the current Metropolis-based sampling's world.Comment: Wiley StatsRef-Statistics Reference Online, 201
Improved Adaptive Rejection Metropolis Sampling Algorithms
Markov Chain Monte Carlo (MCMC) methods, such as the Metropolis-Hastings (MH)
algorithm, are widely used for Bayesian inference. One of the most important
issues for any MCMC method is the convergence of the Markov chain, which
depends crucially on a suitable choice of the proposal density. Adaptive
Rejection Metropolis Sampling (ARMS) is a well-known MH scheme that generates
samples from one-dimensional target densities making use of adaptive piecewise
proposals constructed using support points taken from rejected samples. In this
work we pinpoint a crucial drawback in the adaptive procedure in ARMS: support
points might never be added inside regions where the proposal is below the
target. When this happens in many regions it leads to a poor performance of
ARMS, with the proposal never converging to the target. In order to overcome
this limitation we propose two improved adaptive schemes for constructing the
proposal. The first one is a direct modification of the ARMS procedure that
incorporates support points inside regions where the proposal is below the
target, while satisfying the diminishing adaptation property, one of the
required conditions to assure the convergence of the Markov chain. The second
one is an adaptive independent MH algorithm with the ability to learn from all
previous samples except for the current state of the chain, thus also
guaranteeing the convergence to the invariant density. These two new schemes
improve the adaptive strategy of ARMS, thus simplifying the complexity in the
construction of the proposals. Numerical results show that the new techniques
provide better performance w.r.t. the standard ARMS.Comment: Matlab code provided in http://a2rms.sourceforge.net
Active model learning and diverse action sampling for task and motion planning
The objective of this work is to augment the basic abilities of a robot by
learning to use new sensorimotor primitives to enable the solution of complex
long-horizon problems. Solving long-horizon problems in complex domains
requires flexible generative planning that can combine primitive abilities in
novel combinations to solve problems as they arise in the world. In order to
plan to combine primitive actions, we must have models of the preconditions and
effects of those actions: under what circumstances will executing this
primitive achieve some particular effect in the world?
We use, and develop novel improvements on, state-of-the-art methods for
active learning and sampling. We use Gaussian process methods for learning the
conditions of operator effectiveness from small numbers of expensive training
examples collected by experimentation on a robot. We develop adaptive sampling
methods for generating diverse elements of continuous sets (such as robot
configurations and object poses) during planning for solving a new task, so
that planning is as efficient as possible. We demonstrate these methods in an
integrated system, combining newly learned models with an efficient
continuous-space robot task and motion planner to learn to solve long horizon
problems more efficiently than was previously possible.Comment: Proceedings of the 2018 IEEE/RSJ International Conference on
Intelligent Robots and Systems (IROS), Madrid, Spain.
https://www.youtube.com/playlist?list=PLoWhBFPMfSzDbc8CYelsbHZa1d3uz-W_
What Can We Learn Privately?
Learning problems form an important category of computational tasks that
generalizes many of the computations researchers apply to large real-life data
sets. We ask: what concept classes can be learned privately, namely, by an
algorithm whose output does not depend too heavily on any one input or specific
training example? More precisely, we investigate learning algorithms that
satisfy differential privacy, a notion that provides strong confidentiality
guarantees in contexts where aggregate information is released about a database
containing sensitive information about individuals. We demonstrate that,
ignoring computational constraints, it is possible to privately agnostically
learn any concept class using a sample size approximately logarithmic in the
cardinality of the concept class. Therefore, almost anything learnable is
learnable privately: specifically, if a concept class is learnable by a
(non-private) algorithm with polynomial sample complexity and output size, then
it can be learned privately using a polynomial number of samples. We also
present a computationally efficient private PAC learner for the class of parity
functions. Local (or randomized response) algorithms are a practical class of
private algorithms that have received extensive investigation. We provide a
precise characterization of local private learning algorithms. We show that a
concept class is learnable by a local algorithm if and only if it is learnable
in the statistical query (SQ) model. Finally, we present a separation between
the power of interactive and noninteractive local learning algorithms.Comment: 35 pages, 2 figure
Two adaptive rejection sampling schemes for probability density functions log-convex tails
Monte Carlo methods are often necessary for the implementation of optimal
Bayesian estimators. A fundamental technique that can be used to generate
samples from virtually any target probability distribution is the so-called
rejection sampling method, which generates candidate samples from a proposal
distribution and then accepts them or not by testing the ratio of the target
and proposal densities. The class of adaptive rejection sampling (ARS)
algorithms is particularly interesting because they can achieve high acceptance
rates. However, the standard ARS method can only be used with log-concave
target densities. For this reason, many generalizations have been proposed.
In this work, we investigate two different adaptive schemes that can be used
to draw exactly from a large family of univariate probability density functions
(pdf's), not necessarily log-concave, possibly multimodal and with tails of
arbitrary concavity. These techniques are adaptive in the sense that every time
a candidate sample is rejected, the acceptance rate is improved. The two
proposed algorithms can work properly when the target pdf is multimodal, with
first and second derivatives analytically intractable, and when the tails are
log-convex in a infinite domain. Therefore, they can be applied in a number of
scenarios in which the other generalizations of the standard ARS fail. Two
illustrative numerical examples are shown
- …