  Langevin algorithms are gradient descent methods augmented with additive
noise, and are widely used in Markov Chain Monte Carlo (MCMC) sampling,
optimization, and machine learning. In recent years, the non-asymptotic
analysis of Langevin algorithms for non-convex learning has been extensively
explored. For constrained problems with non-convex losses over a compact convex
domain with IID data variables, the projected Langevin algorithm achieves a
deviation of $O(T^{-1/4} (\log T)^{1/2})$ from its target distribution [27] in
$1$-Wasserstein distance.
  In this paper, we obtain a deviation of $O(T^{-1/2} \log T)$ in
$1$-Wasserstein distance for non-convex losses with $L$-mixing data variables
and polyhedral constraints (which are not necessarily bounded). This improves
on the previous bound for constrained problems and matches the best-known bound
for unconstrained problems.
