12,697 research outputs found
A Cubic Algorithm for Computing Gaussian Volume
We present randomized algorithms for sampling the standard Gaussian
distribution restricted to a convex set and for estimating the Gaussian measure
of a convex set, in the general membership oracle model. The complexity of
integration is while the complexity of sampling is for
the first sample and for every subsequent sample. These bounds
improve on the corresponding state-of-the-art by a factor of . Our
improvement comes from several aspects: better isoperimetry, smoother
annealing, avoiding transformation to isotropic position and the use of the
"speedy walk" in the analysis.Comment: 23 page
On the Computational Complexity of MCMC-based Estimators in Large Samples
In this paper we examine the implications of the statistical large sample
theory for the computational complexity of Bayesian and quasi-Bayesian
estimation carried out using Metropolis random walks. Our analysis is motivated
by the Laplace-Bernstein-Von Mises central limit theorem, which states that in
large samples the posterior or quasi-posterior approaches a normal density.
Using the conditions required for the central limit theorem to hold, we
establish polynomial bounds on the computational complexity of general
Metropolis random walks methods in large samples. Our analysis covers cases
where the underlying log-likelihood or extremum criterion function is possibly
non-concave, discontinuous, and with increasing parameter dimension. However,
the central limit theorem restricts the deviations from continuity and
log-concavity of the log-likelihood or extremum criterion function in a very
specific manner.
Under minimal assumptions required for the central limit theorem to hold
under the increasing parameter dimension, we show that the Metropolis algorithm
is theoretically efficient even for the canonical Gaussian walk which is
studied in detail. Specifically, we show that the running time of the algorithm
in large samples is bounded in probability by a polynomial in the parameter
dimension , and, in particular, is of stochastic order in the leading
cases after the burn-in period. We then give applications to exponential
families, curved exponential families, and Z-estimation of increasing
dimension.Comment: 36 pages, 2 figure
Escaping the Local Minima via Simulated Annealing: Optimization of Approximately Convex Functions
We consider the problem of optimizing an approximately convex function over a
bounded convex set in using only function evaluations. The
problem is reduced to sampling from an \emph{approximately} log-concave
distribution using the Hit-and-Run method, which is shown to have the same
complexity as sampling from log-concave distributions. In
addition to extend the analysis for log-concave distributions to approximate
log-concave distributions, the implementation of the 1-dimensional sampler of
the Hit-and-Run walk requires new methods and analysis. The algorithm then is
based on simulated annealing which does not relies on first order conditions
which makes it essentially immune to local minima.
We then apply the method to different motivating problems. In the context of
zeroth order stochastic convex optimization, the proposed method produces an
-minimizer after noisy function
evaluations by inducing a -approximately log concave
distribution. We also consider in detail the case when the "amount of
non-convexity" decays towards the optimum of the function. Other applications
of the method discussed in this work include private computation of empirical
risk minimizers, two-stage stochastic programming, and approximate dynamic
programming for online learning.Comment: 27 page
Optimal Convergence Rate of Hamiltonian Monte Carlo for Strongly Logconcave Distributions
We study Hamiltonian Monte Carlo (HMC) for sampling from a strongly logconcave density proportional to e^{-f} where f:R^d -> R is mu-strongly convex and L-smooth (the condition number is kappa = L/mu). We show that the relaxation time (inverse of the spectral gap) of ideal HMC is O(kappa), improving on the previous best bound of O(kappa^{1.5}); we complement this with an example where the relaxation time is Omega(kappa). When implemented using a nearly optimal ODE solver, HMC returns an epsilon-approximate point in 2-Wasserstein distance using O~((kappa d)^{0.5} epsilon^{-1}) gradient evaluations per step and O~((kappa d)^{1.5}epsilon^{-1}) total time
Simple Monte Carlo and the Metropolis Algorithm
We study the integration of functions with respect to an unknown density. We
compare the simple Monte Carlo method (which is almost optimal for a certain
large class of inputs) and compare it with the Metropolis algorithm (based on a
suitable ball walk).
Using MCMC we prove (for certain classes of inputs) that adaptive methods are
much better than nonadaptive ones. Actually, the curse of dimension (for
nonadaptive methods) can be broken by adaption.Comment: Journal of Complexity, to appea
On the computational complexity of MCMC-based estimators in large samples
In this paper we examine the implications of the statistical large sample theory for the computational complexity of Bayesian and quasi-Bayesian estimation carried out using Metropolis random walks. Our analysis is motivated by the Laplace-Bernstein-Von Mises central limit theorem, which states that in large samples the posterior or quasi-posterior approaches a normal density. Using this observation, we establish polynomial bounds on the computational complexity of general Metropolis random walks methods in large samples. Our analysis covers cases, where the underlying log-likelihood or extremum criterion function is possibly nonconcave, discontinuous, and of increasing dimension. However, the central limit theorem restricts the deviations from continuity and log-concavity of the log-likelihood or extremum criterion function in a very specific manner.
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