Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userFast MCMC sampling algorithms on polytopes
214
0
0.0
(
0
)
Added by
Raaz Dwivedi
Publication date
2017
fields
Mathematical Statistics
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
We propose and analyze two new MCMC sampling algorithms, the Vaidya walk and the John walk, for generating samples from the uniform distribution over a polytope. Both random walks are sampling algorithms derived from interior point methods. The former is based on volumetric-logarithmic barrier introduced by Vaidya whereas the latter uses Johns ellipsoids. We show that the Vaidya walk mixes in significantly fewer steps than the logarithmic-barrier based Dikin walk studied in past work. For a polytope in $mathbb{R}^d$ defined by $n >d$ linear constraints, we show that the mixing time from a warm start is bounded as $mathcal{O}(n^{0.5}d^{1.5})$, compared to the $mathcal{O}(nd)$ mixing time bound for the Dikin walk. The cost of each step of the Vaidya walk is of the same order as the Dikin walk, and at most twice as large in terms of constant pre-factors. For the John walk, we prove an $mathcal{O}(d^{2.5}cdotlog^4(n/d))$ bound on its mixing time and conjecture that an improved variant of it could achieve a mixing time of $mathcal{O}(d^2cdottext{polylog}(n/d))$. Additionally, we propose variants of the Vaidya and John walks that mix in polynomial time from a deterministic starting point. The speed-up of the Vaidya walk over the Dikin walk are illustrated in numerical examples.
rate research
Read More
We consider the problem of sampling from a strongly log-concave density in $mathbb{R}^d$, and prove a non-asymptotic upper bound on the mixing time of the Metropolis-adjusted Langevin algorithm (MALA). The method draws samples by simulating a Markov chain obtained from the discretization of an appropriate Langevin diffusion, combined with an accept-reject step. Relative to known guarantees for the unadjusted Langevin algorithm (ULA), our bounds show that the use of an accept-reject step in MALA leads to an exponentially improved dependence on the error-tolerance. Concretely, in order to obtain samples with TV error at most $delta$ for a density with condition number $kappa$, we show that MALA requires $mathcal{O} big(kappa d log(1/delta) big)$ steps, as compared to the $mathcal{O} big(kappa^2 d/delta^2 big)$ steps established in past work on ULA. We also demonstrate the gains of MALA over ULA for weakly log-concave densities. Furthermore, we derive mixing time bounds for the Metropolized random walk (MRW) and obtain $mathcal{O}(kappa)$ mixing time slower than MALA. We provide numerical examples that support our theoretical findings, and demonstrate the benefits of Metropolis-Hastings adjustment for Langevin-type sampling algorithms.
We study the problem of sampling from the power posterior distribution in Bayesian Gaussian mixture models, a robust version of the classical posterior. This power posterior is known to be non-log-concave and multi-modal, which leads to exponential mixing times for some standard MCMC algorithms. We introduce and study the Reflected Metropolis-Hastings Random Walk (RMRW) algorithm for sampling. For symmetric two-component Gaussian mixtures, we prove that its mixing time is bounded as $d^{1.5}(d + Vert theta_{0} Vert^2)^{4.5}$ as long as the sample size $n$ is of the order $d (d + Vert theta_{0} Vert^2)$. Notably, this result requires no conditions on the separation of the two means. En route to proving this bound, we establish some new results of possible independent interest that allow for combining Poincar{e} inequalities for conditional and marginal densities.
Markov chain Monte Carlo algorithms are used to simulate from complex statistical distributions by way of a local exploration of these distributions. This local feature avoids heavy requests on understanding the nature of the target, but it also potentially induces a lengthy exploration of this target, with a requirement on the number of simulations that grows with the dimension of the problem and with the complexity of the data behind it. Several techniques are available towards accelerating the convergence of these Monte Carlo algorithms, either at the exploration level (as in tempering, Hamiltonian Monte Carlo and partly deterministic methods) or at the exploitation level (with Rao-Blackwellisation and scalable methods).
Energy-Based Models (EBMs) present a flexible and appealing way to represent uncertainty. Despite recent advances, training EBMs on high-dimensional data remains a challenging problem as the state-of-the-art approaches are costly, unstable, and require considerable tuning and domain expertise to apply successfully. In this work, we present a simple method for training EBMs at scale which uses an entropy-regularized generator to amortize the MCMC sampling typically used in EBM training. We improve upon prior MCMC-based entropy regularization methods with a fast variational approximation. We demonstrate the effectiveness of our approach by using it to train tractable likelihood models. Next, we apply our estimator to the recently proposed Joint Energy Model (JEM), where we match the original performance with faster and stable training. This allows us to extend JEM models to semi-supervised classification on tabular data from a variety of continuous domains.
Gaussian process (GP) models form a core part of probabilistic machine learning. Considerable research effort has been made into attacking three issues with GP models: how to compute efficiently when the number of data is large; how to approximate the posterior when the likelihood is not Gaussian and how to estimate covariance function parameter posteriors. This paper simultaneously addresses these, using a variational approximation to the posterior which is sparse in support of the function but otherwise free-form. The result is a Hybrid Monte-Carlo sampling scheme which allows for a non-Gaussian approximation over the function values and covariance parameters simultaneously, with efficient computations based on inducing-point sparse GPs. Code to replicate each experiment in this paper will be available shortly.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
