In this technical note, we introduce an improved variant of nearest neighbors for counterfactual inference in panel data settings where multiple units are assigned multiple treatments over multiple time points, each sampled with constant probabilities. We call this estimator a doubly robust nearest neighbor estimator and provide a high probability non-asymptotic error bound for the mean parameter corresponding to each unit at each time. Our guarantee shows that the doubly robust estimator provides a (near-)quadratic improvement in the error compared to nearest neighbor estimators analyzed in prior work for these settings.
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We consider after-study statistical inference for sequentially designed experiments wherein multiple units are assigned treatments for multiple time points using treatment policies that adapt over time. Our goal is to provide inference guarantees for the counterfactual mean at the smallest possible scale -- mean outcome under different treatments for each unit and each time -- with minimal assumptions on the adaptive treatment policy. Without any structural assumptions on the counterfactual means, this challenging task is infeasible due to more unknowns than observed data points. To make progress, we introduce a latent factor model over the counterfactual means that serves as a non-parametric generalization of the non-linear mixed effects model and the bilinear latent factor model considered in prior works. For estimation, we use a non-parametric method, namely a variant of nearest neighbors, and establish a non-asymptotic high probability error bound for the counterfactual mean for each unit and each time. Under regularity conditions, this bound leads to asymptotically valid confidence intervals for the counterfactual mean as the number of units and time points grows to $\infty$.
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我们调查识别来自域中的采样点的域的边界。我们向边界引入正常矢量的新估计,指向边界的距离,以及对边界条内的点位于边界的测试。可以有效地计算估算器,并且比文献中存在的估计更准确。我们为估算者提供严格的错误估计。此外,我们使用检测到的边界点来解决Point云上PDE的边值问题。我们在点云上证明了LAPLACH和EIKONG方程的错误估计。最后,我们提供了一系列数值实验,说明了我们的边界估计器,在点云上的PDE应用程序的性能,以及在图像数据集上测试。
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我们考虑估计与I.I.D的排名$ 1 $矩阵因素的问题。高斯,排名$ 1 $的测量值,这些测量值非线性转化和损坏。考虑到非线性的两种典型选择,我们研究了从随机初始化开始的此非convex优化问题的天然交流更新规则的收敛性能。我们通过得出确定性递归,即使在高维问题中也是准确的,我们显示出算法的样本分割版本的敏锐收敛保证。值得注意的是,虽然无限样本的种群更新是非信息性的,并提示单个步骤中的精确恢复,但算法 - 我们的确定性预测 - 从随机初始化中迅速地收敛。我们尖锐的非反应分析也暴露了此问题的其他几种细粒度,包括非线性和噪声水平如何影响收敛行为。从技术层面上讲,我们的结果可以通过证明我们的确定性递归可以通过我们的确定性顺序来预测我们的确定性序列,而当每次迭代都以$ n $观测来运行时,我们的确定性顺序可以通过$ n^{ - 1/2} $的波动。我们的技术利用了源自有关高维$ m $估计文献的遗留工具,并为通过随机数据的其他高维优化问题的随机初始化而彻底地分析了高阶迭代算法的途径。
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我们为基于Kaplan-Meier的最近的邻居和内核存活率估计值建立了第一个非矩形误差界限,其中特征向量位于度量空间中。我们的边界意味着这些非参数估计器的强度速率,并且最多可与对数因子匹配有条件的CDF估计的现有下限。我们的证明策略还为纳尔逊 - 阿伦累积危害估计量的最近的邻居和内核变体提供了非矩形保证。我们在四个数据集上实验比较这些方法。我们发现,对于内核存活率估计量,核心的一个不错的选择是使用随机生存森林学习的。
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现代神经网络通常以强烈的过度构造状态运行:它们包含许多参数,即使实际标签被纯粹随机的标签代替,它们也可以插入训练集。尽管如此,他们在看不见的数据上达到了良好的预测错误:插值训练集并不会导致巨大的概括错误。此外,过度散色化似乎是有益的,因为它简化了优化景观。在这里,我们在神经切线(NT)制度中的两层神经网络的背景下研究这些现象。我们考虑了一个简单的数据模型,以及各向同性协变量的矢量,$ d $尺寸和$ n $隐藏的神经元。我们假设样本量$ n $和尺寸$ d $都很大,并且它们在多项式上相关。我们的第一个主要结果是对过份术的经验NT内核的特征结构的特征。这种表征意味着必然的表明,经验NT内核的最低特征值在$ ND \ gg n $后立即从零界限,因此网络可以在同一制度中精确插值任意标签。我们的第二个主要结果是对NT Ridge回归的概括误差的表征,包括特殊情况,最小值-ULL_2 $ NORD插值。我们证明,一旦$ nd \ gg n $,测试误差就会被内核岭回归之一相对于无限宽度内核而近似。多项式脊回归的误差依次近似后者,从而通过与激活函数的高度组件相关的“自我诱导的”项增加了正则化参数。多项式程度取决于样本量和尺寸(尤其是$ \ log n/\ log d $)。
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Many scientific and engineering challenges-ranging from personalized medicine to customized marketing recommendations-require an understanding of treatment effect heterogeneity. In this paper, we develop a non-parametric causal forest for estimating heterogeneous treatment effects that extends Breiman's widely used random forest algorithm. In the potential outcomes framework with unconfoundedness, we show that causal forests are pointwise consistent for the true treatment effect, and have an asymptotically Gaussian and centered sampling distribution. We also discuss a practical method for constructing asymptotic confidence intervals for the true treatment effect that are centered at the causal forest estimates. Our theoretical results rely on a generic Gaussian theory for a large family of random forest algorithms. To our knowledge, this is the first set of results that allows any type of random forest, including classification and regression forests, to be used for provably valid statistical inference. In experiments, we find causal forests to be substantially more powerful than classical methods based on nearest-neighbor matching, especially in the presence of irrelevant covariates.
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This paper provides estimation and inference methods for a conditional average treatment effects (CATE) characterized by a high-dimensional parameter in both homogeneous cross-sectional and unit-heterogeneous dynamic panel data settings. In our leading example, we model CATE by interacting the base treatment variable with explanatory variables. The first step of our procedure is orthogonalization, where we partial out the controls and unit effects from the outcome and the base treatment and take the cross-fitted residuals. This step uses a novel generic cross-fitting method we design for weakly dependent time series and panel data. This method "leaves out the neighbors" when fitting nuisance components, and we theoretically power it by using Strassen's coupling. As a result, we can rely on any modern machine learning method in the first step, provided it learns the residuals well enough. Second, we construct an orthogonal (or residual) learner of CATE -- the Lasso CATE -- that regresses the outcome residual on the vector of interactions of the residualized treatment with explanatory variables. If the complexity of CATE function is simpler than that of the first-stage regression, the orthogonal learner converges faster than the single-stage regression-based learner. Third, we perform simultaneous inference on parameters of the CATE function using debiasing. We also can use ordinary least squares in the last two steps when CATE is low-dimensional. In heterogeneous panel data settings, we model the unobserved unit heterogeneity as a weakly sparse deviation from Mundlak (1978)'s model of correlated unit effects as a linear function of time-invariant covariates and make use of L1-penalization to estimate these models. We demonstrate our methods by estimating price elasticities of groceries based on scanner data. We note that our results are new even for the cross-sectional (i.i.d) case.
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We consider the problem of estimating a multivariate function $f_0$ of bounded variation (BV), from noisy observations $y_i = f_0(x_i) + z_i$ made at random design points $x_i \in \mathbb{R}^d$, $i=1,\ldots,n$. We study an estimator that forms the Voronoi diagram of the design points, and then solves an optimization problem that regularizes according to a certain discrete notion of total variation (TV): the sum of weighted absolute differences of parameters $\theta_i,\theta_j$ (which estimate the function values $f_0(x_i),f_0(x_j)$) at all neighboring cells $i,j$ in the Voronoi diagram. This is seen to be equivalent to a variational optimization problem that regularizes according to the usual continuum (measure-theoretic) notion of TV, once we restrict the domain to functions that are piecewise constant over the Voronoi diagram. The regression estimator under consideration hence performs (shrunken) local averaging over adaptively formed unions of Voronoi cells, and we refer to it as the Voronoigram, following the ideas in Koenker (2005), and drawing inspiration from Tukey's regressogram (Tukey, 1961). Our contributions in this paper span both the conceptual and theoretical frontiers: we discuss some of the unique properties of the Voronoigram in comparison to TV-regularized estimators that use other graph-based discretizations; we derive the asymptotic limit of the Voronoi TV functional; and we prove that the Voronoigram is minimax rate optimal (up to log factors) for estimating BV functions that are essentially bounded.
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在许多应用中,我们获得了流畅的函数的嘈杂模态样本的访问,其目标是鲁棒地解开样本,即估计该功能的原始样本。在最近的工作中,Cucuringu和Tyagi通过首先将它们代表在单元复杂圆上,然后解决平滑度规则化最小二乘问题 - Laplacian的平滑度适用的Proximity Graph的平滑度$ G $ - ON单位圆的产品歧管。这个问题是二次受约束的二次程序(QCQP),其是非凸显的,因此提出解决其球形放松导致信任区域子问题(TRS)。就理论担保而言,派生$ \ ell_2 $错误界限(trs)。然而,这些界限通常弱,并且没有真正证明由(TRS)进行的去噪。在这项工作中,我们分析(TRS)以及(QCQP)的不受约束的放松。对于这些估算器,我们在高斯噪声的设置中提供了一种精致的分析,并导出了噪音制度,其中他们可否证明模数观察W.R.T $ \ ell_2 $常规。分析在$ G $是任何连接的图形中的常规设置中进行。
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本文研究了在潜在的结果框架中使用深神经网络(DNN)的平均治疗效果(ATE)的估计和推理。在一些规则性条件下,观察到的响应可以作为与混杂变量和治疗指标作为自变量的平均回归问题的响应。使用这种配方,我们研究了通过使用特定网络架构的DNN回归基于估计平均回归函数的两种尝试估计和推断方法。我们表明ATE的两个DNN估计在底层真正的均值回归模型上的一些假设下与无维一致性率一致。我们的模型假设可容纳观察到的协变量的潜在复杂的依赖结构,包括治疗指标和混淆变量之间的潜在因子和非线性相互作用。我们还基于采样分裂的思想,确保精确推理和不确定量化,建立了我们估计的渐近常态。仿真研究和实际数据应用证明了我们的理论调查结果,支持我们的DNN估计和推理方法。
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加权最近的邻居(WNN)估计量通常用作平均回归估计的灵活且易于实现的非参数工具。袋装技术是一种优雅的方式,可以自动生成最近邻居的重量的WNN估计器;我们将最终的估计量命名为分布最近的邻居(DNN),以便于参考。然而,这种估计器缺乏分布结果,从而将其应用于统计推断。此外,当平均回归函数具有高阶平滑度时,DNN无法达到最佳的非参数收敛率,这主要是由于偏差问题。在这项工作中,我们对DNN提供了深入的技术分析,我们建议通过线性将两个DNN估计量与不同的子采样量表进行线性相结合,从而提出了DNN估计量的偏差方法,从而导致新型的两尺度DNN(TDNN(TDNN) )估计器。两尺度的DNN估计量具有等效的WNN表示,重量承认明确形式,有些则是负面的。我们证明,由于使用负权重,两尺度DNN估计器在四阶平滑度条件下估算回归函数时享有最佳的非参数收敛速率。我们进一步超出了估计,并确定DNN和两个规模的DNN均无渐进地正常,因为亚次采样量表和样本量差异到无穷大。对于实际实施,我们还使用二尺度DNN的Jacknife和Bootstrap技术提供方差估计器和分配估计器。可以利用这些估计器来构建有效的置信区间,以用于回归函数的非参数推断。建议的两尺度DNN方法的理论结果和吸引人的有限样本性能用几个数值示例说明了。
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我们提供匹配的Under $ \ sigma ^ 2 / \ log(d / n)$的匹配的上下界限为最低$ \ ell_1 $ -norm插值器,a.k.a.基础追踪。我们的结果紧紧达到可忽略的术语,而且是第一个暗示噪声最小范围内插值的渐近一致性,因为各向同性特征和稀疏的地面真理。我们的工作对最低$ \ ell_2 $ -norm插值的“良性接收”进行了补充文献,其中才能在特征有效地低维时实现渐近一致性。
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我们为梯度下降提供了收敛分析,以解决高斯分布中不可知的问题。与研究零偏差的设置的先前工作不同,我们考虑了当relu函数的偏见非零时更具挑战性的情况。我们的主要结果确定,从随机初始化开始,从多项式迭代梯度下降输出中,具有很高的概率,与最佳relu函数的误差相比,可以实现竞争错误保证。我们还提供有限的样本保证,这些技术将其推广到高斯以外的更广泛的边际分布。
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In this paper, we study the trace regression when a matrix of parameters B* is estimated via the convex relaxation of a rank-regularized regression or via regularized non-convex optimization. It is known that these estimators satisfy near-optimal error bounds under assumptions on the rank, coherence, and spikiness of B*. We start by introducing a general notion of spikiness for B* that provides a generic recipe to prove the restricted strong convexity of the sampling operator of the trace regression and obtain near-optimal and non-asymptotic error bounds for the estimation error. Similar to the existing literature, these results require the regularization parameter to be above a certain theory-inspired threshold that depends on observation noise that may be unknown in practice. Next, we extend the error bounds to cases where the regularization parameter is chosen via cross-validation. This result is significant in that existing theoretical results on cross-validated estimators (Kale et al., 2011; Kumar et al., 2013; Abou-Moustafa and Szepesvari, 2017) do not apply to our setting since the estimators we study are not known to satisfy their required notion of stability. Finally, using simulations on synthetic and real data, we show that the cross-validated estimator selects a near-optimal penalty parameter and outperforms the theory-inspired approach of selecting the parameter.
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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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Popular iterative algorithms such as boosting methods and coordinate descent on linear models converge to the maximum $\ell_1$-margin classifier, a.k.a. sparse hard-margin SVM, in high dimensional regimes where the data is linearly separable. Previous works consistently show that many estimators relying on the $\ell_1$-norm achieve improved statistical rates for hard sparse ground truths. We show that surprisingly, this adaptivity does not apply to the maximum $\ell_1$-margin classifier for a standard discriminative setting. In particular, for the noiseless setting, we prove tight upper and lower bounds for the prediction error that match existing rates of order $\frac{\|\wgt\|_1^{2/3}}{n^{1/3}}$ for general ground truths. To complete the picture, we show that when interpolating noisy observations, the error vanishes at a rate of order $\frac{1}{\sqrt{\log(d/n)}}$. We are therefore first to show benign overfitting for the maximum $\ell_1$-margin classifier.
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由于在数据稀缺的设置中,交叉验证的性能不佳,我们提出了一个新颖的估计器,以估计数据驱动的优化策略的样本外部性能。我们的方法利用优化问题的灵敏度分析来估计梯度关于数据中噪声量的最佳客观值,并利用估计的梯度将策略的样本中的表现为依据。与交叉验证技术不同,我们的方法避免了为测试集牺牲数据,在训练和因此非常适合数据稀缺的设置时使用所有数据。我们证明了我们估计量的偏见和方差范围,这些问题与不确定的线性目标优化问题,但已知的,可能是非凸的,可行的区域。对于更专业的优化问题,从某种意义上说,可行区域“弱耦合”,我们证明结果更强。具体而言,我们在估算器的错误上提供明确的高概率界限,该估计器在策略类别上均匀地保持,并取决于问题的维度和策略类的复杂性。我们的边界表明,在轻度条件下,随着优化问题的尺寸的增长,我们的估计器的误差也会消失,即使可用数据的量仍然很小且恒定。说不同的是,我们证明我们的估计量在小型数据中的大规模政权中表现良好。最后,我们通过数值将我们提出的方法与最先进的方法进行比较,通过使用真实数据调度紧急医疗响应服务的案例研究。我们的方法提供了更准确的样本外部性能估计,并学习了表现更好的政策。
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我们研究了随机近似程序,以便基于观察来自ergodic Markov链的长度$ n $的轨迹来求近求解$ d -dimension的线性固定点方程。我们首先表现出$ t _ {\ mathrm {mix}} \ tfrac {n}} \ tfrac {n}} \ tfrac {d}} \ tfrac {d} {n} $的非渐近性界限。$ t _ {\ mathrm {mix $是混合时间。然后,我们证明了一种在适当平均迭代序列上的非渐近实例依赖性,具有匹配局部渐近最小的限制的领先术语,包括对参数$的敏锐依赖(d,t _ {\ mathrm {mix}}) $以高阶术语。我们将这些上限与非渐近Minimax的下限补充,该下限是建立平均SA估计器的实例 - 最优性。我们通过Markov噪声的政策评估导出了这些结果的推导 - 覆盖了所有$ \ lambda \中的TD($ \ lambda $)算法,以便[0,1)$ - 和线性自回归模型。我们的实例依赖性表征为HyperParameter调整的细粒度模型选择程序的设计开放了门(例如,在运行TD($ \ Lambda $)算法时选择$ \ lambda $的值)。
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本文讨论了ERD \ H {O} S-R \'enyi图的图形匹配或网络对齐问题,可以将其视为图同构问题的嘈杂平均案例版本。令$ g $和$ g'$ be $ g(n,p)$ erd \ h {o} s--r \'enyi略微图形,并用其邻接矩阵识别。假设$ g $和$ g'$是相关的,因此$ \ mathbb {e} [g_ {ij} g'_ {ij}] = p(1- \ alpha)$。对于置换$ \ pi $,代表$ g $和$ g'$之间的潜在匹配,用$ g^\ pi $表示从$ \ pi $的$ g $的顶点获得的图表。观察$ g^\ pi $和$ g'$,我们的目标是恢复匹配的$ \ pi $。在这项工作中,我们证明,在(0,1] $中,每$ \ varepsilon \ in(0,1] $,都有$ n_0> 0 $,具体取决于$ \ varepsilon $和绝对常数$ \ alpha_0,r> 0 $,带有以下属性。令$ n \ ge n_0 $,$(1+ \ varepsilon)\ log n \ le np \ le n^{\ frac {1} {r \ log \ log \ log n}} $ (\ alpha_0,\ varepsilon/4)$。有一个多项式时算法$ f $,因此$ \ m athbb {p} \ {f(g^\ pi,g')= \ pi \} = 1-o (1)$。这是第一种多项式时算法,它恢复了相关的ERD \ H {O} S-r \'enyi图与具有恒定相关性的相关性图与高概率相关性的确切匹配。该算法是基于比较的比较与图形顶点关联的分区树。
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