Convergence rates for optimised adaptive importance samplers

Author:

Akyildiz Ömer DenizORCID,Míguez Joaquín

Abstract

AbstractAdaptive importance samplers are adaptive Monte Carlo algorithms to estimate expectations with respect to some target distribution which adapt themselves to obtain better estimators over a sequence of iterations. Although it is straightforward to show that they have the same $$\mathcal {O}(1/\sqrt{N})$$ O ( 1 / N ) convergence rate as standard importance samplers, where N is the number of Monte Carlo samples, the behaviour of adaptive importance samplers over the number of iterations has been left relatively unexplored. In this work, we investigate an adaptation strategy based on convex optimisation which leads to a class of adaptive importance samplers termed optimised adaptive importance samplers (OAIS). These samplers rely on the iterative minimisation of the $$\chi ^2$$ χ 2 -divergence between an exponential family proposal and the target. The analysed algorithms are closely related to the class of adaptive importance samplers which minimise the variance of the weight function. We first prove non-asymptotic error bounds for the mean squared errors (MSEs) of these algorithms, which explicitly depend on the number of iterations and the number of samples together. The non-asymptotic bounds derived in this paper imply that when the target belongs to the exponential family, the $$L_2$$ L 2 errors of the optimised samplers converge to the optimal rate of $$\mathcal {O}(1/\sqrt{N})$$ O ( 1 / N ) and the rate of convergence in the number of iterations are explicitly provided. When the target does not belong to the exponential family, the rate of convergence is the same but the asymptotic $$L_2$$ L 2 error increases by a factor $$\sqrt{\rho ^\star } > 1$$ ρ > 1 , where $$\rho ^\star - 1$$ ρ - 1 is the minimum $$\chi ^2$$ χ 2 -divergence between the target and an exponential family proposal.

Funder

Engineering and Physical Sciences Research Council

Agencia Estatal de Investigación

Office of Naval Research Global

Publisher

Springer Science and Business Media LLC

Subject

Computational Theory and Mathematics,Statistics, Probability and Uncertainty,Statistics and Probability,Theoretical Computer Science

Reference32 articles.

1. Agapiou, S., Papaspiliopoulos, O., Sanz-Alonso, D., Stuart, A.: Importance sampling: intrinsic dimension and computational cost. Stat. Sci. 32(3), 405–431 (2017)

2. Akyildiz, ÖD., Sabanis, S.: Nonasymptotic analysis of Stochastic Gradient Hamiltonian Monte Carlo under local conditions for nonconvex optimization. (2020). arXiv preprint arXiv:2002.05465

3. Arouna, B.: Adaptative monte carlo method, a variance reduction technique. Monte Carlo Methods Appl. 10(1), 1–24 (2004a)

4. Arouna, B.: Robbins-Monro algorithms and variance reduction in finance. J. Comput. Finance 7(2), 35–62 (2004b)

5. Bottou, L., Curtis, F.E., Nocedal, J.: Optimization methods for large-scale machine learning. (2016). arXiv:1606.04838

Cited by 13 articles. 订阅此论文施引文献 订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献

同舟云学术

1.学者识别学者识别

2.学术分析学术分析

3.人才评估人才评估

"同舟云学术"是以全球学者为主线,采集、加工和组织学术论文而形成的新型学术文献查询和分析系统,可以对全球学者进行文献检索和人才价值评估。用户可以通过关注某些学科领域的顶尖人物而持续追踪该领域的学科进展和研究前沿。经过近期的数据扩容,当前同舟云学术共收录了国内外主流学术期刊6万余种,收集的期刊论文及会议论文总量共计约1.5亿篇,并以每天添加12000余篇中外论文的速度递增。我们也可以为用户提供个性化、定制化的学者数据。欢迎来电咨询!咨询电话:010-8811{复制后删除}0370

www.globalauthorid.com

TOP

Copyright © 2019-2024 北京同舟云网络信息技术有限公司
京公网安备11010802033243号  京ICP备18003416号-3