Constrained Langevin Algorithms with L-mixing External Random Variables

Langevin algorithms are gradient descent methods augmented with additive noise, and are widely used in Markov Chain Monte Carlo (MCMC) sampling, optimization, and learning. In recent years, the non-asymptotic analysis of Langevin algorithms for non-convex optimization learning has been extensively explored. For constrained problems with non-convex losses over compact convex domain in the case of IID data variables, Langevin algorithm achieves a deviation of $O(T^{-1/4} (\log T)^{1/2})$ from its target distribution [22]. 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 deviation indicates that our convergence rate is faster than those in the previous works on constrained Langevin algorithms for non-convex optimization.