  The Langevin Markov chain algorithms are widely deployed methods to sample
from distributions in challenging high-dimensional and non-convex statistics
and machine learning applications. Despite this, current bounds for the
Langevin algorithms are slower than those of competing algorithms in many
important situations, for instance when sampling from weakly log-concave
distributions, or when sampling or optimizing non-convex log-densities. In this
paper, we obtain improved bounds in many of these situations, showing that the
Metropolis-adjusted Langevin algorithm (MALA) is faster than the best bounds
for its competitor algorithms when the target distribution satisfies weak
third- and fourth- order regularity properties associated with the input data.
In many settings, our regularity conditions are weaker than the usual Euclidean
operator norm regularity properties, allowing us to show faster bounds for a
much larger class of distributions than would be possible with the usual
Euclidean operator norm approach, including in statistics and machine learning
applications where the data satisfy a certain incoherence condition. In
particular, we show that using our regularity conditions one can obtain faster
bounds for applications which include sampling problems in Bayesian logistic
regression with weakly convex priors, and the nonconvex optimization problem of
learning linear classifiers with zero-one loss functions.
  Our main technical contribution in this paper is our analysis of the
Metropolis acceptance probability of MALA in terms of its "energy-conservation
error," and our bound for this error in terms of third- and fourth- order
regularity conditions. Our combination of this higher-order analysis of the
energy conservation error with the conductance method is key to obtaining
bounds which have a sub-linear dependence on the dimension $d$ in the
non-strongly logconcave setting.
