论文标题
Bregman近端Langevin Monte Carlo通过Bregman- Moreau信封
Bregman Proximal Langevin Monte Carlo via Bregman--Moreau Envelopes
论文作者
论文摘要
我们提出了有效的Langevin Monte Carlo算法，用于采样分布，具有非平滑凸复合电势，这是连续可区分的函数和可能非滑动函数的总和。我们设计了涉及Bregman Diverences的凸入分析和优化方法的最新进展的算法，即Bregman-Moreau信封和Bregman接近运营商，以及Langevin Monte Carlo Carlo Algorithms in Mirror Serments carlo algorithms。所提出的算法将现有的Langevin Monte Carlo算法分为两个方面 - 能够用镜下下降的算法对非平滑分布进行采样，并使用更通用的Bregman- moreau-Moreau Invelope代替Moreau Inlevelope，以代替莫罗（Moreau）信封，作为非潜在部分的平滑近似。提出的方案的一个特殊情况是让人联想到布雷格曼近端梯度算法。通过各种抽样任务说明了所提出的方法的效率，其中现有的Langevin Monte Carlo方法的性能较差。
We propose efficient Langevin Monte Carlo algorithms for sampling distributions with nonsmooth convex composite potentials, which is the sum of a continuously differentiable function and a possibly nonsmooth function. We devise such algorithms leveraging recent advances in convex analysis and optimization methods involving Bregman divergences, namely the Bregman--Moreau envelopes and the Bregman proximity operators, and in the Langevin Monte Carlo algorithms reminiscent of mirror descent. The proposed algorithms extend existing Langevin Monte Carlo algorithms in two aspects -- the ability to sample nonsmooth distributions with mirror descent-like algorithms, and the use of the more general Bregman--Moreau envelope in place of the Moreau envelope as a smooth approximation of the nonsmooth part of the potential. A particular case of the proposed scheme is reminiscent of the Bregman proximal gradient algorithm. The efficiency of the proposed methodology is illustrated with various sampling tasks at which existing Langevin Monte Carlo methods are known to perform poorly.
