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Supremal inequalities for convex M-estimators with applications to complete and quick convergence 凸m估计的极大不等式及其完全和快速收敛的应用
Pub Date : 2023-11-29 DOI: arxiv-2311.17623
Dietmar Ferger
We consider M-estimators and derive supremal-inequalities of exponential-orpolynomial type according as a boundedness- or a moment-condition is fulfilled.This enables us to derive rates of r-complete convergence and also to showr-qick convergence in the sense of Strasser.
考虑m -估计量,根据有界性条件或矩性条件的满足,推导出指数型或多项式型的最高不等式。这使我们能够推导出r-完全收敛的速率,并在Strasser的意义上显示出快速收敛。
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引用次数: 0
On Adaptive Stochastic Optimization for Streaming Data: A Newton's Method with O(dN) Operations 流数据的自适应随机优化:带有O(dN)运算的牛顿方法
Pub Date : 2023-11-29 DOI: arxiv-2311.17753
Antoine Godichon-BaggioniLPSM, Nicklas Werge
Stochastic optimization methods encounter new challenges in the realm ofstreaming, characterized by a continuous flow of large, high-dimensional data.While first-order methods, like stochastic gradient descent, are the naturalchoice, they often struggle with ill-conditioned problems. In contrast,second-order methods, such as Newton's methods, offer a potential solution, buttheir computational demands render them impractical. This paper introducesadaptive stochastic optimization methods that bridge the gap between addressingill-conditioned problems while functioning in a streaming context. Notably, wepresent an adaptive inversion-free Newton's method with a computationalcomplexity matching that of first-order methods, $mathcal{O}(dN)$, where $d$represents the number of dimensions/features, and $N$ the number of data.Theoretical analysis confirms their asymptotic efficiency, and empiricalevidence demonstrates their effectiveness, especially in scenarios involvingcomplex covariance structures and challenging initializations. In particular,our adaptive Newton's methods outperform existing methods, while maintainingfavorable computational efficiency.
随机优化方法在大、高维数据连续流动的流领域遇到了新的挑战。虽然一阶方法,如随机梯度下降,是自然的选择,但它们经常与病态问题作斗争。相比之下,二阶方法,如牛顿的方法,提供了一个潜在的解决方案,但它们的计算要求使它们不切实际。本文介绍了自适应随机优化方法,该方法在处理病态问题的同时在流环境中发挥作用。值得注意的是,我们提出了一种自适应无反转牛顿方法,其计算复杂度与一阶方法相匹配,$mathcal{O}(dN)$,其中$d$表示维度/特征的数量,$N$表示数据的数量。理论分析证实了它们的渐近效率,经验证据证明了它们的有效性,特别是在涉及复杂协方差结构和具有挑战性的初始化的情况下。特别是,我们的自适应牛顿方法在保持良好的计算效率的同时,优于现有的方法。
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引用次数: 0
Fully lifted random duality theory 完全解除随机对偶理论
Pub Date : 2023-11-29 DOI: arxiv-2312.00070
Mihailo Stojnic
We study a generic class of emph{random optimization problems} (rops) andtheir typical behavior. The foundational aspects of the random duality theory(RDT), associated with rops, were discussed in cite{StojnicRegRndDlt10}, whereit was shown that one can often infer rops' behavior even without actuallysolving them. Moreover, cite{StojnicRegRndDlt10} uncovered that variousquantities relevant to rops (including, for example, their typical objectivevalues) can be determined (in a large dimensional context) even completelyanalytically. The key observation was that the emph{strong deterministicduality} implies the, so-called, emph{strong random duality} and therefore thefull exactness of the analytical RDT characterizations. Here, we attackprecisely those scenarios where the strong deterministic duality is notnecessarily present and connect them to the recent progress made in studyingbilinearly indexed (bli) random processes incite{Stojnicnflgscompyx23,Stojnicsflgscompyx23}. In particular, utilizing afully lifted (fl) interpolating comparison mechanism introduced incite{Stojnicnflgscompyx23}, we establish corresponding emph{fully lifted} RDT(fl RDT). We then rely on a stationarized fl interpolation realizationintroduced in cite{Stojnicsflgscompyx23} to obtain completeemph{statitionarized} fl RDT (sfl RDT). A few well known problems are thendiscussed as illustrations of a wide range of practical applications implied bythe generality of the considered rops.
研究了一类一般的emph{随机优化问题}及其典型行为。与随机对偶理论(RDT)相关的基本方面在cite{StojnicRegRndDlt10}中进行了讨论,其中表明即使没有实际解决它们,人们也可以经常推断出随机对偶理论的行为。此外,cite{StojnicRegRndDlt10}发现,与绳索相关的各种数量(包括,例如,它们的典型客观值)甚至可以完全分析地确定(在大维度上下文中)。关键的观察是,emph{强确定性二象性}意味着,所谓的emph{强随机二象性},因此分析RDT表征的完全准确性。在这里,我们精确地攻击那些不一定存在强确定性对偶性的场景,并将它们与cite{Stojnicnflgscompyx23,Stojnicsflgscompyx23}中研究双线性索引(bli)随机过程的最新进展联系起来。特别地,利用cite{Stojnicnflgscompyx23}中介绍的全提升(fl)插值比较机制,我们建立了相应的emph{全提升}RDT(fl RDT)。然后,我们依靠cite{Stojnicsflgscompyx23}中介绍的平稳化fl插值实现来获得emph{完全验证化}fl RDT (sfl RDT)。然后讨论一些众所周知的问题,以说明所考虑的绳索的普遍性所隐含的广泛的实际应用。
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引用次数: 0
Are ensembles getting better all the time? 乐团是否一直在变得越来越好?
Pub Date : 2023-11-29 DOI: arxiv-2311.17885
Pierre-Alexandre Mattei, Damien Garreau
Ensemble methods combine the predictions of several base models. We studywhether or not including more models in an ensemble always improve its averageperformance. Such a question depends on the kind of ensemble considered, aswell as the predictive metric chosen. We focus on situations where all membersof the ensemble are a priori expected to perform as well, which is the case ofseveral popular methods like random forests or deep ensembles. In this setting,we essentially show that ensembles are getting better all the time if, and onlyif, the considered loss function is convex. More precisely, in that case, theaverage loss of the ensemble is a decreasing function of the number of models.When the loss function is nonconvex, we show a series of results that can besummarised by the insight that ensembles of good models keep getting better,and ensembles of bad models keep getting worse. To this end, we prove a newresult on the monotonicity of tail probabilities that may be of independentinterest. We illustrate our results on a simple machine learning problem(diagnosing melanomas using neural nets).
集合方法结合了几个基本模型的预测。我们研究了是否在一个集成中包含更多的模型总是提高它的平均性能。这样的问题取决于所考虑的集成类型,以及所选择的预测度量。我们关注的情况是,集合的所有成员都被先验地期望表现良好,这是几个流行的方法,如随机森林或深度集合的情况。在这种情况下,我们基本上表明,当且仅当所考虑的损失函数是凸的时,集成系统一直在变得更好。更准确地说,在这种情况下,整体的平均损失是模型数量的递减函数。当损失函数是非凸时,我们展示了一系列结果,这些结果可以总结为好的模型的集成越来越好,而坏模型的集成越来越差。为此,我们证明了一个关于尾概率单调性的新结果。我们在一个简单的机器学习问题(使用神经网络诊断黑色素瘤)上说明了我们的结果。
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引用次数: 0
The Bayesian approach to inverse Robin problems 逆Robin问题的贝叶斯方法
Pub Date : 2023-11-29 DOI: arxiv-2311.17542
Aksel Kaastrup Rasmussen, Fanny Seizilles, Mark Girolami, Ieva Kazlauskaite
In this paper we investigate the Bayesian approach to inverse Robin problems.These are problems for certain elliptic boundary value problems of determininga Robin coefficient on a hidden part of the boundary from Cauchy data on theobservable part. Such a nonlinear inverse problem arises naturally in theinitialisation of large-scale ice sheet models that are crucial in climate andsea-level predictions. We motivate the Bayesian approach for a prototypicalRobin inverse problem by showing that the posterior mean converges inprobability to the data-generating ground truth as the number of observationsincreases. Related to the stability theory for inverse Robin problems, weestablish a logarithmic convergence rate for Sobolev-regular Robincoefficients, whereas for analytic coefficients we can attain an algebraicrate. The use of rescaled analytic Gaussian priors in posterior consistency fornonlinear inverse problems is new and may be of separate interest in otherinverse problems. Our numerical results illustrate the convergence property intwo observation settings.
本文研究了逆Robin问题的贝叶斯方法。这是用可观测部分的柯西数据确定边界隐藏部分的罗宾系数的椭圆边值问题。这种非线性逆问题在大尺度冰盖模型的初始化中自然出现,而大尺度冰盖模型对气候和海平面的预测至关重要。我们通过表明随着观测数量的增加,后验均值不概率地收敛于数据生成的基础真值,从而激发了典型robin反问题的贝叶斯方法。结合逆Robin问题的稳定性理论,我们建立了sobolev -正则Robin系数的对数收敛速率,而对于解析系数,我们可以得到一个代数的收敛速率。在非线性逆问题的后验一致性中使用重标解析高斯先验是一种新的方法,在其他逆问题中可能会有单独的兴趣。我们的数值结果说明了在两种观测条件下的收敛性。
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引用次数: 0
Confidence Regions for Filamentary Structures 丝状结构的置信区域
Pub Date : 2023-11-29 DOI: arxiv-2311.17831
Wanli Qiao
Filamentary structures, also called ridges, generalize the concept of modesof density functions and provide low-dimensional representations of pointclouds. Using kernel type plug-in estimators, we give asymptotic confidenceregions for filamentary structures based on two bootstrap approaches:multiplier bootstrap and empirical bootstrap. Our theoretical frameworkrespects the topological structure of ridges by allowing the possible existenceof intersections. Different asymptotic behaviors of the estimators are analyzeddepending on how flat the ridges are, and our confidence regions are shown tobe asymptotically valid in different scenarios in a unified form. As a criticalstep in the derivation, we approximate the suprema of the relevant empiricalprocesses by those of Gaussian processes, which are degenerate in our problemand are handled by anti-concentration inequalities for Gaussian processes thatdo not require positive infimum variance.
丝状结构,也称为脊,推广了密度函数模式的概念,并提供了点云的低维表示。利用核型插件估计器,基于乘子自举和经验自举两种自举方法,给出了丝状结构的渐近置信区域。我们的理论框架通过允许可能存在的交叉来尊重脊的拓扑结构。根据脊的平坦程度分析了估计量的不同渐近行为,并以统一的形式证明了我们的置信区域在不同情况下是渐近有效的。作为推导的关键一步,我们用高斯过程的经验过程逼近相关经验过程的上界,在我们的问题中,高斯过程是退化的,用不需要正最小方差的高斯过程的反集中不等式来处理。
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引用次数: 0
Inference of Sample Complier Average Causal Effects in Completely Randomized Experiments 完全随机实验中样本编译器平均因果效应的推断
Pub Date : 2023-11-29 DOI: arxiv-2311.17476
Zhen Zhong, Per Johansson, Junni L. Zhang
In randomized experiments with non-compliance scholars have argued that thecomplier average causal effect (CACE) ought to be the main causal estimand. Theliterature on inference of the complier average treatment effect (CACE) hasfocused on inference about the population CACE. However, in general individualsin the experiments are volunteers. This means that there is a risk thatindividuals partaking in a given experiment differ in important ways from apopulation of interest. It is thus of interest to focus on the sample at handand have easy to use and correct procedures for inference about the sampleCACE. We consider a more general setting than in the previous literature andconstruct a confidence interval based on the Wald estimator in the form of afinite closed interval that is familiar to practitioners. Furthermore, with theaccess of pre-treatment covariates, we propose a new regression adjustmentestimator and associated methods for constructing confidence intervals. Finitesample performance of the methods is examined through a Monte Carlo simulationand the methods are used in an application to a job training experiment.
在不服从的随机实验中,学者们认为编译者平均因果效应(CACE)应该是主要的因果估计。关于编译器平均治疗效应(CACE)推断的文献主要集中在对总体CACE的推断上。然而,总的来说,参与实验的个体都是志愿者。这意味着,参与特定实验的个体在重要方面与感兴趣的群体存在差异,这是有风险的。因此,关注手头的样本并具有易于使用和正确的程序来推断样本是有意义的。我们考虑了一个比以往文献更一般的设置,并以从业者熟悉的有限闭区间形式基于Wald估计构造了一个置信区间。此外,利用预处理协变量,我们提出了一种新的回归调整估计量和构造置信区间的相关方法。通过蒙特卡罗仿真验证了该方法的有限样本性能,并将该方法应用于一个岗位培训实验。
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引用次数: 0
Detection of an Arbitrary Number of Communities in a Block Spin Ising Model 块自旋Ising模型中任意数量社团的检测
Pub Date : 2023-11-29 DOI: arxiv-2311.18112
Miguel Ballesteros, Ramsés H. Mena, José Luis Pérez, Gabor Toth
We study the problem of community detection in a general version of the blockspin Ising model featuring M groups, a model inspired by the Curie-Weiss modelof ferromagnetism in statistical mechanics. We solve the general problem ofidentifying any number of groups with any possible coupling constants. Up tonow, the problem was only solved for the specific situation with two groups ofidentical size and identical interactions. Our results can be applied to themost realistic situations, in which there are many groups of different sizesand different interactions. In addition, we give an explicit algorithm thatpermits the reconstruction of the structure of the model from a sample ofobservations based on the comparison of empirical correlations of the spinvariables, thus unveiling easy applications of the model to real-world votingdata and communities in biology.
我们研究了具有M群的块自旋Ising模型的一般版本中的群体检测问题,该模型是受统计力学中的Curie-Weiss铁磁模型的启发而建立的。我们解决了识别具有任意可能耦合常数的任意数量的群的一般问题。到目前为止,这个问题只解决了两个规模相同、相互作用相同的群体的特定情况。我们的结果可以应用于最现实的情况,其中有许多不同规模和不同互动的群体。此外,我们给出了一个明确的算法,该算法允许基于不变量的经验相关性比较的观察样本重建模型的结构,从而揭示了该模型在现实世界的投票数据和生物群落中的简单应用。
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引用次数: 0
In search of the perfect fit: interpretation, flexible modelling, and the existing generalisations of the normal distribution 寻找完美的拟合:解释,灵活的建模,以及正态分布的现有概括
Pub Date : 2023-11-29 DOI: arxiv-2311.17962
Andriette Bekker, Matthias Wagener, Muhammad Arashi
Many generalised distributions exist for modelling data with vastly diversecharacteristics. However, very few of these generalisations of the normaldistribution have shape parameters with clear roles that determine, forinstance, skewness and tail shape. In this chapter, we review existing skewingmechanisms and their properties in detail. Using the knowledge acquired, we adda skewness parameter to the body-tail generalised normal distributioncite{BTGN}, that yields the ac{FIN} with parameters for location, scale,body-shape, skewness, and tail weight. Basic statistical properties of theac{FIN} are provided, such as the ac{PDF}, cumulative distribution function,moments, and likelihood equations. Additionally, the ac{FIN} ac{PDF} isextended to a multivariate setting using a student t-copula, yielding theac{MFIN}. The ac{MFIN} is applied to stock returns data, where it outperformsthe t-copula multivariate generalised hyperbolic, Azzalini skew-t, hyperbolic,and normal inverse Gaussian distributions.
存在许多广义分布,用于建模具有大量不同特征的数据。然而,很少有正态分布的这些概括具有具有明确作用的形状参数,例如,决定偏度和尾部形状。在本章中,我们详细回顾了现有的偏斜机制及其性质。利用所获得的知识,我们将偏度参数添加到体尾广义正态分布cite{BTGN}中,从而得到包含位置、规模、体型、偏度和尾重参数的ac{FIN}。提供了ac{FIN}的基本统计特性,如ac{PDF}、累积分布函数、矩和似然方程。此外,使用学生t-copula将ac{FIN}ac{PDF}扩展到多变量设置,从而得到ac{MFIN}。ac{MFIN}应用于股票收益数据,它优于t-copula多元广义双曲分布、Azzalini偏t分布、双曲分布和正态反高斯分布。
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引用次数: 0
Stochastic orderings between two finite mixture models with inverted-Kumaraswamy distributed components 具有逆kumaraswamy分布分量的两个有限混合模型之间的随机排序
Pub Date : 2023-11-29 DOI: arxiv-2311.17568
Raju Bhakta, Pradip Kundu, Suchandan Kayal
In this paper, we consider two finite mixture models (FMMs), withinverted-Kumaraswamy distributed components' lifetimes. Several stochasticordering results between the FMMs have been obtained. Mainly, we focus on threedifferent cases in terms of the heterogeneity of parameters. The usualstochastic order between the FMMs have been established when heterogeneitypresents in one parameter as well as two parameters. In addition, we have alsostudied ageing faster order in terms of the reversed hazard rate between twoFMMs when heterogeneity is in two parameters. For the case of heterogeneity inthree parameters, we obtain the comparison results based on reversed hazardrate and likelihood ratio orders. The theoretical developments have beenillustrated using several examples and counterexamples.
本文考虑了两种具有逆kumaraswamy分布组件寿命的有限混合模型(fmm)。得到了几个fmm之间的随机排序结果。在参数异质性方面,我们主要关注三种不同情况。在单参数和双参数均存在异质性的情况下,建立了fmm之间通常的随机顺序。此外,当两个参数存在异质性时,我们还研究了两个ofmm之间的反向危险率老化更快的顺序。对于三种参数均存在异质性的情况,我们根据风险和似然比的倒序得到了比较结果。用几个例子和反例说明了理论的发展。
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引用次数: 0
期刊
arXiv - MATH - Statistics Theory
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