arXiv subjects

Paul Vanetti

Publications and source records attributed to Paul Vanetti.

3 recordsLinked to original sources

arXiv2019

Estimating Convergence of Markov chains with L-Lag Couplings

Markov chain Monte Carlo (MCMC) methods generate samples that are asymptotically distributed from a target distribution of interest as the number of iterations goes to infinity. Various theoretical results provide upper bounds on the distance between the target and marginal distribution after a fixed number of iterations. These upper bounds are on a case by case basis and typically involve intractable quantities, which limits their use for practitioners. We introduce L-lag couplings to generate computable, non-asymptotic upper bound estimates for the total variation or the Wasserstein distance of general Markov chains. We apply L-lag couplings to the tasks of (i) determining MCMC burn-in, (ii) comparing different MCMC algorithms with the same target, and (iii) comparing exact and approximate MCMC. Lastly, we (iv) assess the bias of sequential Monte Carlo and self-normalized importance samplers.

stat.CO↗

arXiv2019

Scalable Metropolis-Hastings for Exact Bayesian Inference with Large Datasets

Bayesian inference via standard Markov Chain Monte Carlo (MCMC) methods is too computationally intensive to handle large datasets, since the cost per step usually scales like $Θ(n)$ in the number of data points $n$. We propose the Scalable Metropolis-Hastings (SMH) kernel that exploits Gaussian concentration of the posterior to require processing on average only $O(1)$ or even $O(1/\sqrt{n})$ data points per step. This scheme is based on a combination of factorized acceptance probabilities, procedures for fast simulation of Bernoulli processes, and control variate ideas. Contrary to many MCMC subsampling schemes such as fixed step-size Stochastic Gradient Langevin Dynamics, our approach is exact insofar as the invariant distribution is the true posterior and not an approximation to it. We characterise the performance of our algorithm theoretically, and give realistic and verifiable conditions under which it is geometrically ergodic. This theory is borne out by empirical results that demonstrate overall performance benefits over standard Metropolis-Hastings and various subsampling algorithms.

stat.ML↗

arXiv2018

Piecewise-Deterministic Markov Chain Monte Carlo

A novel class of non-reversible Markov chain Monte Carlo schemes relying on continuous-time piecewise-deterministic Markov Processes has recently emerged. In these algorithms, the state of the Markov process evolves according to a deterministic dynamics which is modified using a Markov transition kernel at random event times. These methods enjoy remarkable features including the ability to update only a subset of the state components while other components implicitly keep evolving and the ability to use an unbiased estimate of the gradient of the log-target while preserving the target as invariant distribution. However, they also suffer from important limitations. The deterministic dynamics used so far do not exploit the structure of the target. Moreover, exact simulation of the event times is feasible for an important yet restricted class of problems and, even when it is, it is application specific. This limits the applicability of these techniques and prevents the development of a generic software implementation of them. We introduce novel MCMC methods addressing these shortcomings. In particular, we introduce novel continuous-time algorithms relying on exact Hamiltonian flows and novel non-reversible discrete-time algorithms which can exploit complex dynamics such as approximate Hamiltonian dynamics arising from symplectic integrators while preserving the attractive features of continuous-time algorithms. We demonstrate the performance of these schemes on a variety of applications.

stat.ME↗