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We consider Markov chain Monte Carlo (MCMC) algorithms for Bayesianhigh-dimensional regression with continuous shrinkage priors. A commonchallenge with these algorithms is the choice of the number of iterations toperform. This is critical when each iteration is expensive, as is the case whendealing with modern data sets, such as genome-wide association studies withthousands of rows and up to hundred of thousands of columns. We developcoupling techniques tailored to the setting of high-dimensional regression withshrinkage priors, which enable practical, non-asymptotic diagnostics ofconvergence without relying on traceplots or long-run asymptotics. Byestablishing geometric drift and minorization conditions for the algorithmunder consideration, we prove that the proposed couplings have finite expectedmeeting time. Focusing on a class of shrinkage priors which includes the'Horseshoe', we empirically demonstrate the scalability of the proposedcouplings. A highlight of our findings is that less than 1000 iterations can beenough for a Gibbs sampler to reach stationarity in a regression on 100,000covariates. The numerical results also illustrate the impact of the prior onthe computational efficiency of the coupling, and suggest the use of priorswhere the local precisions are Half-t distributed with degree of freedom largerthan one.

Coupling-based convergence assessment of some Gibbs samplers for high-dimensional Bayesian regression with shrinkage priors