  There is a tension between robustness and efficiency when designing Markov
chain Monte Carlo (MCMC) sampling algorithms. Here we focus on robustness with
respect to tuning parameters, showing that more sophisticated algorithms tend
to be more sensitive to the choice of step-size parameter and less robust to
heterogeneity of the distribution of interest. We characterise this phenomenon
by studying the behaviour of spectral gaps as an increasingly poor step-size is
chosen for the algorithm. Motivated by these considerations, we propose a novel
and simple gradient-based MCMC algorithm, inspired by the classical Barker
accept-reject rule, with improved robustness properties. Extensive theoretical
results, dealing with robustness to tuning, geometric ergodicity and scaling
with dimension, suggest that the novel scheme combines the robustness of simple
schemes with the efficiency of gradient-based ones. We show numerically that
this type of robustness is particularly beneficial in the context of adaptive
MCMC, giving examples where our proposed scheme significantly outperforms
state-of-the-art alternatives.
