A common tool in the practice of Markov Chain Monte Carlo is to use approximating transition kernels to speed up computation when the true kernel is slow to evaluate. A relatively limited set of quantitative tools exist to determine whether the performance of such approximations will be well behaved and to assess the quality of approximation. We derive a set a tools for such analysis based on the Hilbert space generated by the stationary distribution we intend to sample, @math . The focus of our work is on determining whether the approximating kernel (i.e. perturbation) will preserve the geometric ergodicity of the chain, and whether the approximating stationary distribution will be close to the original stationary distribution. Our results directly generalise the results of johndrow2015approximations from the uniformly ergodic case to the geometrically ergodic case. We then apply our results to the class of Noisy MCMC' algorithms.
Lastly, Rudolf and Schweizer @cite_6 also use the Wasserstein topology. They focus on approximate MCMC algorithms, with applications to autoregressive processes and stochastic Langevin algorithms for Gibbs random fields. Their results use the following assumptions: the original kernel is Wasserstein ergodic; a Lyapunov drift condition for perturbed kernel is given, with drift function @math ; @math has finite expectation under the initial distribution; and the perturbation operator is uniformly bounded in a @math -normalised Wasserstein norm. Their main results are: explicit bounds on the Wasserstein distance between the original and perturbed finite time sampling distributions; and explicit bounds on the Wasserstein distance between stationary distributions.
@cite_6
2120918452
Perturbation theory for Markov chains addresses the question how small differences in the transitions of Markov chains are reflected in differences between their distributions. We prove powerful and flexible bounds on the distance of the @math th step distributions of two Markov chains when one of them satisfies a Wasserstein ergodicity condition. Our work is motivated by the recent interest in approximate Markov chain Monte Carlo (MCMC) methods in the analysis of big data sets. By using an approach based on Lyapunov functions, we provide estimates for geometrically ergodic Markov chains under weak assumptions. In an autoregressive model, our bounds cannot be improved in general. We illustrate our theory by showing quantitative estimates for approximate versions of two prominent MCMC algorithms, the Metropolis-Hastings and stochastic Langevin algorithms. Discrete time analogues of ergodic stochastic differential equations (SDEs) are one of the most popular and flexible tools for sampling high-dimensional probability measures. Non-asymptotic analysis in the @math Wasserstein distance of sampling algorithms based on Euler discretisations of SDEs has been recently developed by several authors for log-concave probability distributions. In this work we replace the log-concavity assumption with a log-concavity at infinity condition. We provide novel @math convergence rates for Euler schemes, expressed explicitly in terms of problem parameters. From there we derive non-asymptotic bounds on the distance between the laws induced by Euler schemes and the invariant laws of SDEs, both for schemes with standard and with randomised (inaccurate) drifts. We also obtain bounds for the hierarchy of discretisation, which enables us to deploy a multi-level Monte Carlo estimator. Our proof relies on a novel construction of a coupling for the Markov chains that can be used to control both the @math and @math Wasserstein distances simultaneously. Finally, we provide a weak convergence analysis that covers both the standard and the randomised (inaccurate) drift case. In particular, we reveal that the variance of the randomised drift does not influence the rate of weak convergence of the Euler scheme to the SDE. We study the problem of sampling from a distribution where the negative logarithm of the target density is @math -smooth everywhere and @math -strongly convex outside a ball of radius @math , but potentially non-convex inside this ball. We study both overdamped and underdamped Langevin MCMC and prove upper bounds on the time required to obtain a sample from a distribution that is within @math of the target distribution in @math -Wasserstein distance. For the first-order method (overdamped Langevin MCMC), the time complexity is @math , where @math is the dimension of the underlying space. For the second-order method (underdamped Langevin MCMC), the time complexity is @math for some explicit positive constant @math . Surprisingly, the convergence rate is only polynomial in the dimension @math and the target accuracy @math . It is however exponential in the problem parameter @math , which is a measure of non-logconcavity of the target distribution. Stochastic Gradient Langevin Dynamics (SGLD) is a combination of a Robbins-Monro type algorithm with Langevin dynamics. In this paper, the SGLD method with fixed step size @math is considered in order to sample from a logconcave target distribution @math , known up to a normalisation factor. We assume that unbiased estimates of the gradient from possibly dependent observations are available. It is shown that, for all @math , the Wasserstein- @math distance of the @math th iterate of the SGLD algorithm from @math is dominated by @math with appropriate constants @math and @math , whereas for the case of i.i.d. data the upper bound is improved to @math .