24,076 research outputs found
Bayesian optimization using sequential Monte Carlo
We consider the problem of optimizing a real-valued continuous function
using a Bayesian approach, where the evaluations of are chosen sequentially
by combining prior information about , which is described by a random
process model, and past evaluation results. The main difficulty with this
approach is to be able to compute the posterior distributions of quantities of
interest which are used to choose evaluation points. In this article, we decide
to use a Sequential Monte Carlo (SMC) approach
Bayesian Subset Simulation: a kriging-based subset simulation algorithm for the estimation of small probabilities of failure
The estimation of small probabilities of failure from computer simulations is
a classical problem in engineering, and the Subset Simulation algorithm
proposed by Au & Beck (Prob. Eng. Mech., 2001) has become one of the most
popular method to solve it. Subset simulation has been shown to provide
significant savings in the number of simulations to achieve a given accuracy of
estimation, with respect to many other Monte Carlo approaches. The number of
simulations remains still quite high however, and this method can be
impractical for applications where an expensive-to-evaluate computer model is
involved. We propose a new algorithm, called Bayesian Subset Simulation, that
takes the best from the Subset Simulation algorithm and from sequential
Bayesian methods based on kriging (also known as Gaussian process modeling).
The performance of this new algorithm is illustrated using a test case from the
literature. We are able to report promising results. In addition, we provide a
numerical study of the statistical properties of the estimator.Comment: 11th International Probabilistic Assessment and Management Conference
(PSAM11) and The Annual European Safety and Reliability Conference (ESREL
2012), Helsinki : Finland (2012
Optimization Monte Carlo: Efficient and Embarrassingly Parallel Likelihood-Free Inference
We describe an embarrassingly parallel, anytime Monte Carlo method for
likelihood-free models. The algorithm starts with the view that the
stochasticity of the pseudo-samples generated by the simulator can be
controlled externally by a vector of random numbers u, in such a way that the
outcome, knowing u, is deterministic. For each instantiation of u we run an
optimization procedure to minimize the distance between summary statistics of
the simulator and the data. After reweighing these samples using the prior and
the Jacobian (accounting for the change of volume in transforming from the
space of summary statistics to the space of parameters) we show that this
weighted ensemble represents a Monte Carlo estimate of the posterior
distribution. The procedure can be run embarrassingly parallel (each node
handling one sample) and anytime (by allocating resources to the worst
performing sample). The procedure is validated on six experiments.Comment: NIPS 2015 camera read
A Bayesian approach to constrained single- and multi-objective optimization
This article addresses the problem of derivative-free (single- or
multi-objective) optimization subject to multiple inequality constraints. Both
the objective and constraint functions are assumed to be smooth, non-linear and
expensive to evaluate. As a consequence, the number of evaluations that can be
used to carry out the optimization is very limited, as in complex industrial
design optimization problems. The method we propose to overcome this difficulty
has its roots in both the Bayesian and the multi-objective optimization
literatures. More specifically, an extended domination rule is used to handle
objectives and constraints in a unified way, and a corresponding expected
hyper-volume improvement sampling criterion is proposed. This new criterion is
naturally adapted to the search of a feasible point when none is available, and
reduces to existing Bayesian sampling criteria---the classical Expected
Improvement (EI) criterion and some of its constrained/multi-objective
extensions---as soon as at least one feasible point is available. The
calculation and optimization of the criterion are performed using Sequential
Monte Carlo techniques. In particular, an algorithm similar to the subset
simulation method, which is well known in the field of structural reliability,
is used to estimate the criterion. The method, which we call BMOO (for Bayesian
Multi-Objective Optimization), is compared to state-of-the-art algorithms for
single- and multi-objective constrained optimization
Sequential Gaussian Processes for Online Learning of Nonstationary Functions
Many machine learning problems can be framed in the context of estimating
functions, and often these are time-dependent functions that are estimated in
real-time as observations arrive. Gaussian processes (GPs) are an attractive
choice for modeling real-valued nonlinear functions due to their flexibility
and uncertainty quantification. However, the typical GP regression model
suffers from several drawbacks: i) Conventional GP inference scales
with respect to the number of observations; ii) updating a GP model
sequentially is not trivial; and iii) covariance kernels often enforce
stationarity constraints on the function, while GPs with non-stationary
covariance kernels are often intractable to use in practice. To overcome these
issues, we propose an online sequential Monte Carlo algorithm to fit mixtures
of GPs that capture non-stationary behavior while allowing for fast,
distributed inference. By formulating hyperparameter optimization as a
multi-armed bandit problem, we accelerate mixing for real time inference. Our
approach empirically improves performance over state-of-the-art methods for
online GP estimation in the context of prediction for simulated non-stationary
data and hospital time series data
Robust Online Hamiltonian Learning
In this work we combine two distinct machine learning methodologies,
sequential Monte Carlo and Bayesian experimental design, and apply them to the
problem of inferring the dynamical parameters of a quantum system. We design
the algorithm with practicality in mind by including parameters that control
trade-offs between the requirements on computational and experimental
resources. The algorithm can be implemented online (during experimental data
collection), avoiding the need for storage and post-processing. Most
importantly, our algorithm is capable of learning Hamiltonian parameters even
when the parameters change from experiment-to-experiment, and also when
additional noise processes are present and unknown. The algorithm also
numerically estimates the Cramer-Rao lower bound, certifying its own
performance.Comment: 24 pages, 12 figures; to appear in New Journal of Physic
Variational Sequential Monte Carlo
Many recent advances in large scale probabilistic inference rely on
variational methods. The success of variational approaches depends on (i)
formulating a flexible parametric family of distributions, and (ii) optimizing
the parameters to find the member of this family that most closely approximates
the exact posterior. In this paper we present a new approximating family of
distributions, the variational sequential Monte Carlo (VSMC) family, and show
how to optimize it in variational inference. VSMC melds variational inference
(VI) and sequential Monte Carlo (SMC), providing practitioners with flexible,
accurate, and powerful Bayesian inference. The VSMC family is a variational
family that can approximate the posterior arbitrarily well, while still
allowing for efficient optimization of its parameters. We demonstrate its
utility on state space models, stochastic volatility models for financial data,
and deep Markov models of brain neural circuits
Bayesian subset simulation
We consider the problem of estimating a probability of failure ,
defined as the volume of the excursion set of a function above a given threshold, under a given
probability measure on . In this article, we combine the popular
subset simulation algorithm (Au and Beck, Probab. Eng. Mech. 2001) and our
sequential Bayesian approach for the estimation of a probability of failure
(Bect, Ginsbourger, Li, Picheny and Vazquez, Stat. Comput. 2012). This makes it
possible to estimate when the number of evaluations of is very
limited and is very small. The resulting algorithm is called Bayesian
subset simulation (BSS). A key idea, as in the subset simulation algorithm, is
to estimate the probabilities of a sequence of excursion sets of above
intermediate thresholds, using a sequential Monte Carlo (SMC) approach. A
Gaussian process prior on is used to define the sequence of densities
targeted by the SMC algorithm, and drive the selection of evaluation points of
to estimate the intermediate probabilities. Adaptive procedures are
proposed to determine the intermediate thresholds and the number of evaluations
to be carried out at each stage of the algorithm. Numerical experiments
illustrate that BSS achieves significant savings in the number of function
evaluations with respect to other Monte Carlo approaches
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