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DMS-EPSRC: Fast martingales, large deviations and randomised gradients for heavy-tailed target distributions

DMS-EPSRC：重尾目标分布的快速鞅、大偏差和随机梯度

基本信息

批准号：
EP/V009478/1
负责人：
Aleksandar Mijatovic
金额：
$ 84.43万
依托单位：
University of Warwick
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2021
资助国家：
英国
起止时间：
2021 至无数据
项目状态：
未结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FV009478%2F1
关键词：
DMS EPSRC Fast martingales large

项目摘要

Markov chain is a mathematical object representing a random evolution with the following property: if we know the present state of the chain, its past and future are independent (i.e. information about the past does not alter the distribution of its future states). Markov chain models are fundamental across sciences and engineering. At the centre of this project are Markov chains on multi-dimensional state spaces that arise in randomised algorithms used in statistics and machine learning. This proposal is focused on the theoretical analysis of chains arising in applications in the case when their limiting distribution has heavy tails. The analysis of the heavy-tailed phenomena is crucial for the future success of randomised algorithms for two reasons: (a) they arise naturally in many applied problems and (b) are least well understood as they violate standard assumptions made in the existing theory (e.g. asymptotic linearity of the potential of the limit distribution at infinity).(a) Heavy-tailed limiting distributions arise naturally in many applications. For example, if the errors in a regression model are distributed according to a Cauchy distribution, the posterior density has polynomial tails. Perhaps a more startling fact is that heavy tails can arise in the posterior even though a heavy-tailed distribution does not appear in the definition of a model. If the errors in a data set are heteroscedastic, meaning that the variance of the error term varies with each observation, it is necessary to use the so-called robust regression (based on e.g. Lasso-type penalisation) in order to reduce the effect of the outliers. Again the posterior has heavy tails. (b) The presence of a spectral gap is known to be equivalent to geometric convergence of a Markov chain. However, as pointed out recently in the queueing literature, under geometric convergence ergodic estimators may still exhibit large deviation behaviour of the heavy-tailed type. Conversely, Markov chains with heavy tailed stationary measures typically do not have a spectral gap but might nevertheless exhibit good convergence properties. The EPSRC-NSF Lead Agency agreement presents a unique opportunity to combine the US expertise in theoretical Operations Research with the UK's capability in Computational Statistics, resulting in novel methodology for the analysis of the convergence of Markov chains with heavy-tailed targets, the main focus for this project.Our main goal is to fill the gap in the literature, best illustrated by the following baseline algorithm from applications: a random-scan Metropolis-within-Gibbs chain picks randomly a coordinate of a target distribution and moves it by a one-dimensional Metropolis step based on the conditional of the target. It is possible to prove that if ANY one-dimensional marginal of the target has heavy tails, the random-scan chain is NOT geometrically ergodic. The main goal of this proposal is to lay the theoretical foundations for the analysis of the stability of Markov chains with heavy-tailed targets, focusing on the processes that underpin many randomised algorithms used in practice. In time, this work is expected to have impact far beyond applied probability in a number of sub-areas of computational statistics and machine learning where heavy-tailed targets arise.

马尔可夫链是一个数学对象，表示具有以下性质的随机演化：如果我们知道链的当前状态，则其过去和未来是独立的（即关于过去的信息不会改变其未来状态的分布）。马尔可夫链模型是科学和工程的基础。该项目的核心是多维状态空间上的马尔可夫链，它出现在统计和机器学习中使用的随机算法中。这个建议的重点是理论分析的情况下，他们的极限分布有重尾的应用程序中出现的链。重尾现象的分析对于随机算法未来的成功至关重要，原因有二：（a）它们在许多应用问题中自然出现;（B）由于它们违反了现有理论中的标准假设（例如，极限分布在无穷远处的势函数的渐近线性），因此最不容易理解。(a)重尾极限分布在许多应用中自然出现。例如，如果回归模型中的误差根据柯西分布，则后验密度具有多项式尾部。也许一个更令人吃惊的事实是，即使模型的定义中没有出现重尾分布，重尾分布也会出现在后验中。如果数据集中的误差是异方差的，这意味着误差项的方差随每个观测值而变化，则有必要使用所谓的稳健回归（基于例如Lasso型惩罚），以减少离群值的影响。再一次，尾部有很重的尾巴。(b)已知谱隙的存在等价于马尔可夫链的几何收敛。然而，正如最近的研究文献所指出的那样，在几何收敛下，遍历估计仍然可能表现出重尾型的大偏差行为。相反，具有重尾平稳测度的马尔可夫链通常不具有谱间隙，但可能仍然表现出良好的收敛性。EPSRC-NSF牵头机构协议提供了一个独特的机会，将美国在理论运筹学方面的专业知识与英国在计算统计方面的能力相结合，从而产生用于分析具有重尾目标的马尔可夫链收敛性的新方法，这是该项目的主要重点。我们的主要目标是填补文献中差距，最好的说明是以下应用中的基线算法：随机扫描Metropolis-within-Gibbs链随机挑选目标分布的坐标，并基于目标的条件将其移动一维大都会步长。可以证明，如果目标的任何一维边缘具有重尾，则随机扫描链不是几何遍历的。该提案的主要目标是为分析具有重尾目标的马尔可夫链的稳定性奠定理论基础，重点是支持实践中使用的许多随机算法的过程。随着时间的推移，这项工作预计将在计算统计和机器学习的一些子领域产生远远超出应用概率的影响，这些子领域出现了重尾目标。