论文标题
用L混合的外部随机变量约束的Langevin算法
Constrained Langevin Algorithms with L-mixing External Random Variables
论文作者
论文摘要
Langevin算法是增强噪声增强的梯度下降方法，并广泛用于Markov Chain Monte Carlo（MCMC）采样，优化和机器学习。近年来，已广泛探索了非凸学习算法的非反应分析。对于带有IID数据变量的紧凑型凸域上非凸损失的受限问题，预计的langevin算法实现了$ O（t^{ - 1/4}（\ log t）^{1/2}）$的偏差（t^{ - 1/4}（\ log t）^{1/2}）$在$ -WASSERSTEIN距离中的目标分布[27]。 在本文中，我们获得了$ O（t^{ - 1/2} \ log t）$ $ 1 $ -WASSERSTEIN距离的偏差，用于非convex损失，具有$ L $ - 混合的数据变量和多面体约束（不一定是边界）。这改善了以前的问题的界限，并匹配了无约束问题的最著名界限。
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.
