DMS-EPSRC: Fast martingales, large deviations and randomised gradients for heavy-tailed target distributions

Markov chain is a mathematical object representing a random evolution with the following property: if we know the present state of the chain, its past and future are independent (i.e. information about the past does not alter the distribution of its future states). Markov chain models are fundamental across sciences and engineering. At the centre of this project are Markov chains on multi-dimensional state spaces that arise in randomised algorithms used in statistics and machine learning. This proposal is focused on the theoretical analysis of chains arising in applications in the case when their limiting distribution has heavy tails. The analysis of the heavy-tailed phenomena is crucial for the future success of randomised algorithms for two reasons: (a) they arise naturally in many applied problems and (b) are least well understood as they violate standard assumptions made in the existing theory (e.g. asymptotic linearity of the potential of the limit distribution at infinity).

(a) Heavy-tailed limiting distributions arise naturally in many applications. For example, if the errors in a regression model are distributed according to a Cauchy distribution, the posterior density has polynomial tails. Perhaps a more startling fact is that heavy tails can arise in the posterior even though a heavy-tailed distribution does not appear in the definition of a model. If the errors in a data set are heteroscedastic, meaning that the variance of the error term varies with each observation, it is necessary to use the so-called robust regression (based on e.g. Lasso-type penalisation) in order to reduce the effect of the outliers. Again the posterior has heavy tails.

(b) The presence of a spectral gap is known to be equivalent to geometric convergence of a Markov chain. However, as pointed out recently in the queueing literature, under geometric convergence ergodic estimators may still exhibit large deviation behaviour of the heavy-tailed type. Conversely, Markov chains with heavy tailed stationary measures typically do not have a spectral gap but might nevertheless exhibit good convergence properties. The EPSRC-NSF Lead Agency agreement presents a unique opportunity to combine the US expertise in theoretical Operations Research with the UK's capability in Computational Statistics, resulting in novel methodology for the analysis of the convergence of Markov chains with heavy-tailed targets, the main focus for this project.

Our main goal is to fill the gap in the literature, best illustrated by the following baseline algorithm from applications: a random-scan Metropolis-within-Gibbs chain picks randomly a coordinate of a target distribution and moves it by a one-dimensional Metropolis step based on the conditional of the target. It is possible to prove that if ANY one-dimensional marginal of the target has heavy tails, the random-scan chain is NOT geometrically ergodic. The main goal of this proposal is to lay the theoretical foundations for the analysis of the stability of Markov chains with heavy-tailed targets, focusing on the processes that underpin many randomised algorithms used in practice. In time, this work is expected to have impact far beyond applied probability in a number of sub-areas of computational statistics and machine learning where heavy-tailed targets arise.