Enhanced gradient-based MCMC in discrete spaces

The recent introduction of gradient-based Markov chain Monte Carlo (MCMC) for discrete spaces holds great promise, and comes with the tantalising possibility of new discrete counterparts to celebrated continuous methods such as the Metropolis-adjusted Langevin algorithm (MALA). 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’.