  The recent introduction of gradient-based MCMC for discrete spaces holds
great promise, and comes with the tantalising possibility of new discrete
counterparts to celebrated continuous methods such as MALA and HMC. Towards
this goal, we introduce several discrete Metropolis-Hastings samplers that are
conceptually-inspired by MALA, and demonstrate their strong empirical
performance across a range of challenging sampling problems in Bayesian
inference and energy-based modelling. Methodologically, we identify why
discrete analogues to preconditioned MALA are generally intractable, motivating
us to introduce a new kind of preconditioning based on auxiliary variables and
the `Gaussian integral trick'.
