Bregman Proximal Langevin Monte Carlo via Bregman-Moreau Envelopes
Bregman Proximal Langevin Monte Carlo via Bregman-Moreau Envelopes
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DOI：
10.48550/arxiv.2207.04387
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发表时间：
2022-07
期刊：
影响因子：
--
通讯作者：
Tim Tsz-Kit Lau;Han Liu
中科院分区：
文献类型：
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作者：
Tim Tsz-Kit Lau;Han Liu
对于具有连续可微函数和可能非光滑函数之和的非光滑凸复合势，我们提出了一种有效的朗之万蒙特卡罗算法。我们设计了这样的算法，利用了涉及Bregman发散的凸分析和优化方法的最新进展，即Bregman-Moreau包络和Bregman邻近算子，以及让人想起镜像下降的朗之万蒙特卡罗算法。所提出的算法在两个方面扩展了现有的朗之万蒙特卡罗算法--使用镜面下降算法对非光滑分布进行采样的能力，以及使用更一般的Bregman-Moreau包络代替Moreau包络作为势的非光滑部分的光滑近似。所提出方案的一个特殊情况让人想起Bregman近端梯度算法。已有的朗之万蒙特卡罗方法在不同的抽样任务中表现不佳，说明了该方法的有效性。
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.