  We introduce a gradient-based learning method to automatically adapt Markov
chain Monte Carlo (MCMC) proposal distributions to intractable targets. We
define a maximum entropy regularised objective function, referred to as
generalised speed measure, which can be robustly optimised over the parameters
of the proposal distribution by applying stochastic gradient optimisation. An
advantage of our method compared to traditional adaptive MCMC methods is that
the adaptation occurs even when candidate state values are rejected. This is a
highly desirable property of any adaptation strategy because the adaptation
starts in early iterations even if the initial proposal distribution is far
from optimum. We apply the framework for learning multivariate random walk
Metropolis and Metropolis-adjusted Langevin proposals with full covariance
matrices, and provide empirical evidence that our method can outperform other
MCMC algorithms, including Hamiltonian Monte Carlo schemes.
