  Langevin algorithms are gradient descent methods with additive noise. They
have been used for decades in Markov chain Monte Carlo (MCMC) sampling,
optimization, and learning. Their convergence properties for unconstrained
non-convex optimization and learning problems have been studied widely in the
last few years. Other work has examined projected Langevin algorithms for
sampling from log-concave distributions restricted to convex compact sets. For
learning and optimization, log-concave distributions correspond to convex
losses. In this paper, we analyze the case of non-convex losses with compact
convex constraint sets and IID external data variables. We term the resulting
method the projected stochastic gradient Langevin algorithm (PSGLA). We show
the algorithm achieves a deviation of $O(T^{-1/4}(\log T)^{1/2})$ from its
target distribution in 1-Wasserstein distance. For optimization and learning,
we show that the algorithm achieves $\epsilon$-suboptimal solutions, on
average, provided that it is run for a time that is polynomial in
$\epsilon^{-1}$ and slightly super-exponential in the problem dimension.
