MALA with annealed proposals: a generalization of locally and globally balanced proposal distributions

Abstract

We introduce a generalized version of the Metropolis-adjusted Langevin algorithm (MALA). The informed proposal distribution of this new sampler features two tuning parameters: the usual step size parameter $\sigma ^2$ and an interpolation parameter $\gamma$ that may be adjusted to accommodate the dimension of the target distribution. We theoretically study the efficiency of the sampler by making use of the local- and global-balance concepts introduced in Zanella (JASA 115:852–865, 2020) and provide efficient tuning guidelines that work well with a variety of target distributions. Although the usual MALA ($\gamma =1$) is shown to be optimal for infinite-dimensional targets, in practice, the generalized MALA ($1<\gamma \le 2$) remains the most appealing option, even in high-dimensional contexts. Simulation studies and numerical experiments are presented to illustrate our findings. We apply the new sampler to a Bayesian logistic regression context and show that its efficiency compares favourably to competing algorithms.