基本上有三种不确定性量化方法(UQ):(a)强大的优化,(b)贝叶斯,(c)决策理论。尽管(a)坚固,但在准确性和数据同化方面是不利的。 (b)需要先验,通常是脆弱的,后验估计可能很慢。尽管(c)导致对最佳先验的识别,但其近似遭受了维度的诅咒,风险的概念是相对于数据分布的平均值。我们引入了第四种,它是(a),(b),(c)和假设检验之间的杂种。可以总结为在观察样本$ x $之后,(1)通过相对可能性定义了可能性区域,(2)在该区域玩Minmax游戏以定义最佳估计器及其风险。最终的方法具有几种理想的属性(a)测量数据后确定了最佳先验,并且风险概念是后部的,(b)确定最佳估计值,其风险可以降低到计算最小封闭的最小封闭式。利益图量下的可能性区域图像的球(这是快速的,不受维数的诅咒)。该方法的特征在于$ [0,1] $中的参数,该参数是在观察到的数据(相对可能性)的稀有度上被假定的下限。当该参数接近$ 1 $时,该方法会产生一个后分布,该分布集中在最大似然估计的情况下,并具有较低的置信度UQ估计值。当该参数接近$ 0 $时,该方法会产生最大风险后验分布,并具有很高的信心UQ估计值。除了导航准确性不确定性权衡外,该建议的方法还通过导航与数据同化相关的稳健性 - 准确性权衡解决了贝叶斯推断的脆弱性。
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我们在对数损失下引入条件密度估计的过程,我们调用SMP(样本Minmax预测器)。该估算器最大限度地减少了统计学习的新一般过度风险。在标准示例中,此绑定量表为$ d / n $,$ d $ d $模型维度和$ n $ sample大小,并在模型拼写条目下批判性仍然有效。作为一个不当(超出型号)的程序,SMP在模型内估算器(如最大似然估计)的内部估算器上,其风险过高的风险降低。相比,与顺序问题的方法相比,我们的界限删除了SubOltimal $ \ log n $因子,可以处理无限的类。对于高斯线性模型,SMP的预测和风险受到协变量的杠杆分数,几乎匹配了在没有条件的线性模型的噪声方差或近似误差的条件下匹配的最佳风险。对于Logistic回归,SMP提供了一种非贝叶斯方法来校准依赖于虚拟样本的概率预测,并且可以通过解决两个逻辑回归来计算。它达到了$ O的非渐近风险((d + b ^ 2r ^ 2)/ n)$,其中$ r $绑定了特征的规范和比较参数的$ B $。相比之下,在模型内估计器内没有比$ \ min达到更好的速率({b r} / {\ sqrt {n}},{d e ^ {br} / {n})$。这为贝叶斯方法提供了更实用的替代方法,这需要近似的后部采样,从而部分地解决了Foster等人提出的问题。 (2018)。
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We study a double robust Bayesian inference procedure on the average treatment effect (ATE) under unconfoundedness. Our Bayesian approach involves a correction term for prior distributions adjusted by the propensity score. We prove asymptotic equivalence of our Bayesian estimator and efficient frequentist estimators by establishing a new semiparametric Bernstein-von Mises theorem under double robustness; i.e., the lack of smoothness of conditional mean functions can be compensated by high regularity of the propensity score and vice versa. Consequently, the resulting Bayesian point estimator internalizes the bias correction as the frequentist-type doubly robust estimator, and the Bayesian credible sets form confidence intervals with asymptotically exact coverage probability. In simulations, we find that this corrected Bayesian procedure leads to significant bias reduction of point estimation and accurate coverage of confidence intervals, especially when the dimensionality of covariates is large relative to the sample size and the underlying functions become complex. We illustrate our method in an application to the National Supported Work Demonstration.
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机器学习通常以经典的概率理论为前提,这意味着聚集是基于期望的。现在有多种原因可以激励人们将经典概率理论作为机器学习的数学基础。我们系统地检查了一系列强大而丰富的此类替代品,即各种称为光谱风险度量,Choquet积分或Lorentz规范。我们提出了一系列的表征结果,并演示了使这个光谱家族如此特别的原因。在此过程中,我们证明了所有连贯的风险度量的自然分层,从它们通过利用重新安排不变性Banach空间理论的结果来诱导的上层概率。我们凭经验证明了这种新的不确定性方法如何有助于解决实用的机器学习问题。
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In a mixed generalized linear model, the objective is to learn multiple signals from unlabeled observations: each sample comes from exactly one signal, but it is not known which one. We consider the prototypical problem of estimating two statistically independent signals in a mixed generalized linear model with Gaussian covariates. Spectral methods are a popular class of estimators which output the top two eigenvectors of a suitable data-dependent matrix. However, despite the wide applicability, their design is still obtained via heuristic considerations, and the number of samples $n$ needed to guarantee recovery is super-linear in the signal dimension $d$. In this paper, we develop exact asymptotics on spectral methods in the challenging proportional regime in which $n, d$ grow large and their ratio converges to a finite constant. By doing so, we are able to optimize the design of the spectral method, and combine it with a simple linear estimator, in order to minimize the estimation error. Our characterization exploits a mix of tools from random matrices, free probability and the theory of approximate message passing algorithms. Numerical simulations for mixed linear regression and phase retrieval display the advantage enabled by our analysis over existing designs of spectral methods.
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我们研究只有历史数据时设计最佳学习和决策制定公式的问题。先前的工作通常承诺要进行特定的数据驱动配方,并随后尝试建立样本外的性能保证。我们以相反的方式采取了相反的方法。我们首先定义一个明智的院子棒,以测量任何数据驱动的公式的质量,然后寻求找到最佳的这种配方。在非正式的情况下,可以看到任何数据驱动的公式可以平衡估计成本与实际成本的接近度的量度,同时保证了样本外的性能水平。考虑到可接受的样本外部性能水平,我们明确地构建了一个数据驱动的配方,该配方比任何其他享有相同样本外部性能的其他配方都更接近真实成本。我们展示了三种不同的样本外绩效制度(超大型制度,指数状态和次指数制度)之间存在,最佳数据驱动配方的性质会经历相变的性质。最佳数据驱动的公式可以解释为超级稳定的公式,在指数方面是一种熵分布在熵上稳健的公式,最后是次指数制度中的方差惩罚公式。这个最终的观察揭示了这三个观察之间的令人惊讶的联系,乍一看似乎是无关的,数据驱动的配方,直到现在仍然隐藏了。
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We study distributionally robust optimization (DRO) with Sinkhorn distance -- a variant of Wasserstein distance based on entropic regularization. We provide convex programming dual reformulation for a general nominal distribution. Compared with Wasserstein DRO, it is computationally tractable for a larger class of loss functions, and its worst-case distribution is more reasonable. We propose an efficient first-order algorithm with bisection search to solve the dual reformulation. We demonstrate that our proposed algorithm finds $\delta$-optimal solution of the new DRO formulation with computation cost $\tilde{O}(\delta^{-3})$ and memory cost $\tilde{O}(\delta^{-2})$, and the computation cost further improves to $\tilde{O}(\delta^{-2})$ when the loss function is smooth. Finally, we provide various numerical examples using both synthetic and real data to demonstrate its competitive performance and light computational speed.
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对于高维和非参数统计模型,速率最优估计器平衡平方偏差和方差是一种常见的现象。虽然这种平衡被广泛观察到,但很少知道是否存在可以避免偏差和方差之间的权衡的方法。我们提出了一般的策略,以获得对任何估计方差的下限,偏差小于预先限定的界限。这表明偏差差异折衷的程度是不可避免的,并且允许量化不服从其的方法的性能损失。该方法基于许多抽象的下限,用于涉及关于不同概率措施的预期变化以及诸如Kullback-Leibler或Chi-Sque-diversence的信息措施的变化。其中一些不平等依赖于信息矩阵的新概念。在该物品的第二部分中,将抽象的下限应用于几种统计模型,包括高斯白噪声模型,边界估计问题,高斯序列模型和高维线性回归模型。对于这些特定的统计应用,发生不同类型的偏差差异发生,其实力变化很大。对于高斯白噪声模型中集成平方偏置和集成方差之间的权衡,我们将较低界限的一般策略与减少技术相结合。这允许我们将原始问题与估计的估计器中的偏差折衷联动,以更简单的统计模型中具有额外的对称性属性。在高斯序列模型中,发生偏差差异的不同相位转换。虽然偏差和方差之间存在非平凡的相互作用,但是平方偏差的速率和方差不必平衡以实现最小估计速率。
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变性推理(VI)为基于传统的采样方法提供了一种吸引人的替代方法,用于实施贝叶斯推断,因为其概念性的简单性,统计准确性和计算可扩展性。然而,常见的变分近似方案(例如平均场(MF)近似)需要某些共轭结构以促进有效的计算,这可能会增加不必要的限制对可行的先验分布家族,并对变异近似族对差异进行进一步的限制。在这项工作中,我们开发了一个通用计算框架,用于实施MF-VI VIA WASSERSTEIN梯度流(WGF),这是概率度量空间上的梯度流。当专门针对贝叶斯潜在变量模型时,我们将分析基于时间消化的WGF交替最小化方案的算法收敛,用于实现MF近似。特别是,所提出的算法类似于EM算法的分布版本,包括更新潜在变量变异分布的E step以及在参数的变异分布上进行最陡峭下降的m step。我们的理论分析依赖于概率度量空间中的最佳运输理论和细分微积分。我们证明了时间限制的WGF的指数收敛性,以最大程度地减少普通大地测量学严格的凸度的通用物镜功能。我们还提供了通过使用时间限制的WGF的固定点方程从MF近似获得的变异分布的指数收缩的新证明。我们将方法和理论应用于两个经典的贝叶斯潜在变量模型,即高斯混合模型和回归模型的混合物。还进行了数值实验,以补充这两个模型下的理论发现。
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We introduce and study a novel model-selection strategy for Bayesian learning, based on optimal transport, along with its associated predictive posterior law: the Wasserstein population barycenter of the posterior law over models. We first show how this estimator, termed Bayesian Wasserstein barycenter (BWB), arises naturally in a general, parameter-free Bayesian model-selection framework, when the considered Bayesian risk is the Wasserstein distance. Examples are given, illustrating how the BWB extends some classic parametric and non-parametric selection strategies. Furthermore, we also provide explicit conditions granting the existence and statistical consistency of the BWB, and discuss some of its general and specific properties, providing insights into its advantages compared to usual choices, such as the model average estimator. Finally, we illustrate how this estimator can be computed using the stochastic gradient descent (SGD) algorithm in Wasserstein space introduced in a companion paper arXiv:2201.04232v2 [math.OC], and provide a numerical example for experimental validation of the proposed method.
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主题模型为学习,提取和发现大型文本语料库中的潜在结构提供了有用的文本挖掘工具。尽管已经为主题建模提出了大量方法,但文献缺乏是对潜在主题估计的统计识别性和准确性的正式理论研究。在本文中,我们提出了一个基于特定的集成可能性的潜在主题的最大似然估计量(MLE),该主题自然地与该概念相连,在计算几何学中,体积最小化。我们的理论介绍了主题模型可识别性的一组新几何条件,这些条件比常规的可分离性条件弱,这些条件通常依赖于纯主题文档或锚定词的存在。较弱的条件允许更广泛的调查,因此可能会更加富有成果的研究。我们对拟议的估计器进行有限样本误差分析,并讨论我们的结果与先前研究的结果之间的联系。我们以使用模拟和真实数据集的实证研究结论。
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我们在分布式框架中得出最小值测试错误,其中数据被分成多个机器,并且它们与中央机器的通信仅限于$ b $位。我们研究了高斯白噪声下的$ d $ - 和无限维信号检测问题。我们还得出达到理论下限的分布式测试算法。我们的结果表明,分布式测试受到从根本上不同的现象,这些现象在分布式估计中未观察到。在我们的发现中,我们表明,可以访问共享随机性的测试协议在某些制度中的性能比不进行的测试协议可以更好地表现。我们还观察到,即使仅使用单个本地计算机上可用的信息,一致的非参数分布式测试始终是可能的,即使只有$ 1 $的通信和相应的测试优于最佳本地测试。此外,我们还得出了自适应非参数分布测试策略和相应的理论下限。
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Uncertainty is prevalent in engineering design, statistical learning, and decision making broadly. Due to inherent risk-averseness and ambiguity about assumptions, it is common to address uncertainty by formulating and solving conservative optimization models expressed using measure of risk and related concepts. We survey the rapid development of risk measures over the last quarter century. From its beginning in financial engineering, we recount their spread to nearly all areas of engineering and applied mathematics. Solidly rooted in convex analysis, risk measures furnish a general framework for handling uncertainty with significant computational and theoretical advantages. We describe the key facts, list several concrete algorithms, and provide an extensive list of references for further reading. The survey recalls connections with utility theory and distributionally robust optimization, points to emerging applications areas such as fair machine learning, and defines measures of reliability.
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生成对抗网络(GAN)在数据生成方面取得了巨大成功。但是,其统计特性尚未完全理解。在本文中,我们考虑了GAN的一般$ f $ divergence公式的统计行为,其中包括Kullback- Leibler Divergence与最大似然原理密切相关。我们表明,对于正确指定的参数生成模型,在适当的规律性条件下,所有具有相同歧视类别类别的$ f $ divergence gans均在渐近上等效。 Moreover, with an appropriately chosen local discriminator, they become equivalent to the maximum likelihood estimate asymptotically.对于被误解的生成模型,具有不同$ f $ -Divergences {收敛到不同估计器}的gan,因此无法直接比较。但是,结果表明,对于某些常用的$ f $ -Diverences,原始的$ f $ gan并不是最佳的,因为当更换原始$ f $ gan配方中的判别器培训时,可以实现较小的渐近方差通过逻辑回归。结果估计方法称为对抗梯度估计(年龄)。提供了实证研究来支持该理论,并证明了年龄的优势,而不是模型错误的原始$ f $ gans。
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套索是一种高维回归的方法,当时,当协变量$ p $的订单数量或大于观测值$ n $时,通常使用它。由于两个基本原因,经典的渐近态性理论不适用于该模型:$(1)$正规风险是非平滑的; $(2)$估算器$ \ wideHat {\ boldsymbol {\ theta}} $与true参数vector $ \ boldsymbol {\ theta}^*$无法忽略。结果,标准的扰动论点是渐近正态性的传统基础。另一方面,套索估计器可以精确地以$ n $和$ p $大,$ n/p $的订单为一。这种表征首先是在使用I.I.D的高斯设计的情况下获得的。协变量:在这里,我们将其推广到具有非偏差协方差结构的高斯相关设计。这是根据更简单的``固定设计''模型表示的。我们在两个模型中各种数量的分布之间的距离上建立了非反应界限,它们在合适的稀疏类别中均匀地固定在信号上$ \ boldsymbol {\ theta}^*$。作为应用程序,我们研究了借助拉索的分布,并表明需要校正程度对于计算有效的置信区间是必要的。
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鉴于$ n $ i.i.d.从未知的分发$ P $绘制的样本,何时可以生成更大的$ n + m $ samples,这些标题不能与$ n + m $ i.i.d区别区别。从$ p $绘制的样品?(AXELROD等人2019)将该问题正式化为样本放大问题,并为离散分布和高斯位置模型提供了最佳放大程序。然而,这些程序和相关的下限定制到特定分布类,对样本扩增的一般统计理解仍然很大程度上。在这项工作中,我们通过推出通常适用的放大程序,下限技术和与现有统计概念的联系来放置对公司统计基础的样本放大问题。我们的技术适用于一大类分布,包括指数家庭,并在样本放大和分配学习之间建立严格的联系。
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在因果推理和强盗文献中,基于观察数据的线性功能估算线性功能的问题是规范的。我们分析了首先估计治疗效果函数的广泛的两阶段程序,然后使用该数量来估计线性功能。我们证明了此类过程的均方误差上的非反应性上限:这些边界表明,为了获得非反应性最佳程序,应在特定加权$ l^2 $中最大程度地估算治疗效果的误差。 -规范。我们根据该加权规范的约束回归分析了两阶段的程序,并通过匹配非轴突局部局部最小值下限,在有限样品中建立了实例依赖性最优性。这些结果表明,除了取决于渐近效率方差之外,最佳的非质子风险除了取决于样本量支持的最富有函数类别的真实结果函数与其近似类别之间的加权规范距离。
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Classical asymptotic theory for statistical inference usually involves calibrating a statistic by fixing the dimension $d$ while letting the sample size $n$ increase to infinity. Recently, much effort has been dedicated towards understanding how these methods behave in high-dimensional settings, where $d$ and $n$ both increase to infinity together. This often leads to different inference procedures, depending on the assumptions about the dimensionality, leaving the practitioner in a bind: given a dataset with 100 samples in 20 dimensions, should they calibrate by assuming $n \gg d$, or $d/n \approx 0.2$? This paper considers the goal of dimension-agnostic inference; developing methods whose validity does not depend on any assumption on $d$ versus $n$. We introduce an approach that uses variational representations of existing test statistics along with sample splitting and self-normalization to produce a new test statistic with a Gaussian limiting distribution, regardless of how $d$ scales with $n$. The resulting statistic can be viewed as a careful modification of degenerate U-statistics, dropping diagonal blocks and retaining off-diagonal blocks. We exemplify our technique for some classical problems including one-sample mean and covariance testing, and show that our tests have minimax rate-optimal power against appropriate local alternatives. In most settings, our cross U-statistic matches the high-dimensional power of the corresponding (degenerate) U-statistic up to a $\sqrt{2}$ factor.
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In the classical setting of self-selection, the goal is to learn $k$ models, simultaneously from observations $(x^{(i)}, y^{(i)})$ where $y^{(i)}$ is the output of one of $k$ underlying models on input $x^{(i)}$. In contrast to mixture models, where we observe the output of a randomly selected model, here the observed model depends on the outputs themselves, and is determined by some known selection criterion. For example, we might observe the highest output, the smallest output, or the median output of the $k$ models. In known-index self-selection, the identity of the observed model output is observable; in unknown-index self-selection, it is not. Self-selection has a long history in Econometrics and applications in various theoretical and applied fields, including treatment effect estimation, imitation learning, learning from strategically reported data, and learning from markets at disequilibrium. In this work, we present the first computationally and statistically efficient estimation algorithms for the most standard setting of this problem where the models are linear. In the known-index case, we require poly$(1/\varepsilon, k, d)$ sample and time complexity to estimate all model parameters to accuracy $\varepsilon$ in $d$ dimensions, and can accommodate quite general selection criteria. In the more challenging unknown-index case, even the identifiability of the linear models (from infinitely many samples) was not known. We show three results in this case for the commonly studied $\max$ self-selection criterion: (1) we show that the linear models are indeed identifiable, (2) for general $k$ we provide an algorithm with poly$(d) \exp(\text{poly}(k))$ sample and time complexity to estimate the regression parameters up to error $1/\text{poly}(k)$, and (3) for $k = 2$ we provide an algorithm for any error $\varepsilon$ and poly$(d, 1/\varepsilon)$ sample and time complexity.
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了解现代机器学习设置中的概括一直是统计学习理论的主要挑战之一。在这种情况下,近年来见证了各种泛化范围的发展,表明了不同的复杂性概念,例如数据样本和算法输出之间的相互信息,假设空间的可压缩性以及假设空间的分形维度。尽管这些界限从不同角度照亮了手头的问题,但它们建议的复杂性概念似乎似乎无关,从而限制了它们的高级影响。在这项研究中,我们通过速率理论的镜头证明了新的概括界定,并明确地将相互信息,可压缩性和分形维度的概念联系起来。我们的方法包括(i)通过使用源编码概念来定义可压缩性的广义概念,(ii)表明“压缩错误率”可以与预期和高概率相关。我们表明,在“无损压缩”设置中,我们恢复并改善了现有的基于信息的界限,而“有损压缩”方案使我们能够将概括与速率延伸维度联系起来,这是分形维度的特定概念。我们的结果为概括带来了更统一的观点,并打开了几个未来的研究方向。
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