Is there an analog of Nesterov acceleration for gradient-based MCMC?

IF 1.5 2区数学 Q2 STATISTICS & PROBABILITY

Bernoulli Pub Date : 2021-05-01 DOI:10.3150/20-BEJ1297

Ma Yi-An, Niladri S. Chatterji, Xiang Cheng, Nicolas, Flammarion, P. Bartlett, Michael I. Jordan

引用次数: 44

Abstract

We formulate gradient-based Markov chain Monte Carlo (MCMC) sampling as optimization on the space of probability measures, with Kullback–Leibler (KL) divergence as the objective functional. We show that an underdamped form of the Langevin algorithm performs accelerated gradient descent in this metric. To characterize the convergence of the algorithm, we construct a Lyapunov functional and exploit hypocoercivity of the underdamped Langevin algorithm. As an application, we show that accelerated rates can be obtained for a class of nonconvex functions with the Langevin algorithm.

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对于基于梯度的MCMC，是否有Nesterov加速度的类似物？

我们将基于梯度的马尔可夫链蒙特卡罗（MCMC）采样公式化为概率测度空间上的优化，以Kullback–Leibler（KL）散度为目标函数。我们证明了Langevin算法的欠阻尼形式在该度量中执行加速梯度下降。为了表征该算法的收敛性，我们构造了一个李雅普诺夫函数，并利用了欠阻尼Langevin算法的次优性。作为一个应用，我们证明了用Langevin算法可以获得一类非凸函数的加速率。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Bernoulli 数学-统计学与概率论

CiteScore

3.40

自引率

0.00%

发文量

116

审稿时长

6-12 weeks

期刊介绍： BERNOULLI is the journal of the Bernoulli Society for Mathematical Statistics and Probability, issued four times per year. The journal provides a comprehensive account of important developments in the fields of statistics and probability, offering an international forum for both theoretical and applied work. BERNOULLI will publish: Papers containing original and significant research contributions: with background, mathematical derivation and discussion of the results in suitable detail and, where appropriate, with discussion of interesting applications in relation to the methodology proposed. Papers of the following two types will also be considered for publication, provided they are judged to enhance the dissemination of research: Review papers which provide an integrated critical survey of some area of probability and statistics and discuss important recent developments. Scholarly written papers on some historical significant aspect of statistics and probability.