在机器学习和高维统计领域的有限样本理论中，恒定指定的浓度不平等至关重要。我们获得了独立亚网络随机变量总和的更清晰和常数的浓度不平等，这导致了两个尾巴的混合物：尺寸的小偏差和较大偏差的小偏差。这些界限是新的，并通过更清晰的常数改善了现有的界限。另外，如果应保留斜体，则新的子韦布尔参数。请检查整个文本。还提出了提出的，它可以为随机变量（向量）恢复紧密浓度不平等。对于统计应用，我们给出了$ \ ell_2 $ - 估计系数在负二项式回归中的估计系数时，当重尾协变量是稀疏结构分布的亚weibull时，这是负二项式回归的新结果。在应用随机矩阵时，我们得出了Bai-Yin定理的非反应版本，用于具有指数尾巴边界的亚weibull条目。最后，通过为没有第二瞬间条件的对数截断的Z-测验器演示一个子静电区域，我们讨论并定义了独立观测值的sub-weibull类型稳健估计器$ \ {x_i \} _ {i = 1 }^{n} $没有指数矩条件。

在统计和机器学习中具有重尾数据的模型开发强大的估计估计兴趣兴趣。本文提出了一个用于大家庭统计回归的日志截断的M估计，并在数据具有$ \ varepsilon \中的数据（0，1] $。随着相关风险函数的额外假设，我们获得了估计的$ \ ell_2 $ -Error绑定。我们的定理应用于建立具体回归的强大M估计。除了凸面回归等分位数回归之外广义线性模型，许多非凸回归也可以符合我们的定理，我们专注于强大的深度神经网络回归，这可以通过随机梯度下降算法解决。模拟和实际数据分析证明了日志截断估计的优越性超过标准估计。

本文研究了具有对抗性误差的强大一位压缩感应的二进制分类。假设该模型过度分配，并且感兴趣的参数有效稀疏。adaboost被考虑，并且通过其与MAX - $ \ ell_1 $ -Margin-Scressifir的关系，派生预测错误界限。开发的理论是一般的，并且允许重型的特征分布，只需要一个薄弱的时刻假设和抗浓缩条件。当特征满足小偏差下限时，示出了改善的收敛速率。特别是，结果提供了解释为什么内插对抗性噪声对于分类问题可以是无害的。模拟说明了所提出的理论。

我们证明了连续和离散时间添加功能的浓度不平等和相关的PAC界限，用于可能是多元，不可逆扩散过程的无界函数。我们的分析依赖于通过泊松方程的方法，使我们能够考虑一系列非常广泛的指数性千古过程。这些结果增加了现有的浓度不平等，用于扩散过程的加性功能，这些功能仅适用于有界函数或从明显较小的类别中的过程的无限函数。我们通过两个截然不同的区域的例子来证明这些指数不平等的力量。考虑到在稀疏性约束下可能具有高维参数非线性漂移模型，我们应用连续的时间浓度结果来验证套索估计的受限特征值条件，这对于甲骨文不平等的推导至关重要。离散添加功能的结果用于研究未经调整的Langevin MCMC算法，用于采样中等重尾密度$ \ pi $。特别是，我们为多项式增长功能$ f $的样品蒙特卡洛估计量$ \ pi（f）提供PAC边界，以量化足够的样本和阶梯尺寸，以在规定的边距内近似具有很高的可能性。

We study the fundamental task of outlier-robust mean estimation for heavy-tailed distributions in the presence of sparsity. Specifically, given a small number of corrupted samples from a high-dimensional heavy-tailed distribution whose mean $\mu$ is guaranteed to be sparse, the goal is to efficiently compute a hypothesis that accurately approximates $\mu$ with high probability. Prior work had obtained efficient algorithms for robust sparse mean estimation of light-tailed distributions. In this work, we give the first sample-efficient and polynomial-time robust sparse mean estimator for heavy-tailed distributions under mild moment assumptions. Our algorithm achieves the optimal asymptotic error using a number of samples scaling logarithmically with the ambient dimension. Importantly, the sample complexity of our method is optimal as a function of the failure probability $\tau$, having an additive $\log(1/\tau)$ dependence. Our algorithm leverages the stability-based approach from the algorithmic robust statistics literature, with crucial (and necessary) adaptations required in our setting. Our analysis may be of independent interest, involving the delicate design of a (non-spectral) decomposition for positive semi-definite matrices satisfying certain sparsity properties.

对于子伽马噪声下的差异隐私，我们通过常规链路函数推导出一类具有二进制值的网络模型的渐近属性。在本文中，我们根据具有离散拉普拉斯机制的一般嘈杂机制来释放二元网络的程度序列。我们建立渐近结果，包括参数估计器的一致性和渐近常态，当参数的数量进入无限远在一类网络模型中时。提供模拟和实际数据示例以说明渐近结果。

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。

我们获得了具有重尾分布的独立和相同分布的随机变量的总和。我们的浓度结果与随机变量有关，其分布满足$ \ mathbb {p}（x> t）\ leq {\ rm e}^{ - i（t）} $，其中$ i：\ mathbb {r} \ rightarrow\ mathbb {r} $是一个增加的功能，$ i（t）/t \ rightArrow \ alpha \ in [0，\ infty）$ as $ t \ rightArrow \ rightArrow \ infty $。我们的主要定理不仅可以恢复一些现有结果，例如亚韦伯随机变量的总和的浓度，而且还可以为带有较重尾巴的随机变量的总和产生新的结果。我们表明，我们获得的浓度不平等足以为独立随机变量的总和提供较大的偏差结果。我们的基于标准截断参数的分析简化，统一和推广有关重尾随机变量的浓度和较大偏差的现有结果。

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.

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。

在本文中，我们研究了经验$ \ ell_2 $最小化（erm）的估计性能（标准）阶段检索（NPR），由$ y_k = | \ alpha_k^*x_0 |^2+\ eta_k $，或嘈杂的广义阶段检索（NGPR）以$ y_k = x_0^*a_kx_0 + \ eta_k $，其中$ x_0 \ in \ mathbb {k}^d $是所需的信号，$ n $是样本大小，$ \ eta =（\ eta_1，...，\ eta_n）^\ top $是噪声向量。我们在不同的噪声模式下建立了新的错误界限，我们的证明对$ \ mathbb {k} = \ mathbb {r} $和$ \ mathbb {k} = \ mathbb {c} $有效。在任意噪声向量$ \ eta $下的NPR中，我们得出了一个新的错误$ o \ big（\ | \ eta \ | _ \ | _ \ infty \ sqrt {\ frac {d} {1}^\ top \ eta |} {n} \ big）$，它比当前已知的一个$ o \ big（\ frac {\ | \ eTa \ |} {\ sqrt {\ sqrt {n}} \ big big ）$在许多情况下。在NGPR中，我们显示了$ o \ big（\ | \ eta \ | \ frac {\ sqrt {d}}} {n} {n} \ big）$ for nutary $ \ eta $。在这两个问题上，任意噪声的范围立即引起$ \ tilde {o}（\ sqrt {\ frac {d} {n}}}}）$，用于次高斯或次指数随机噪声，带有一些常规但不可吻的去除或削弱的假设（例如，独立或均值均值的条件）。此外，我们首次尝试在假定$ l $ -th时刻的重尾随机噪声下进行ERM。为了实现偏见和差异之间的权衡，我们截断了响应并提出了相应的稳健ERM估计器，该估计量具有保证$ \ tilde {o} \ big（\ big [\ sqrt {\ frac {\ frac {d}） {n}} \ big]^{1-1/l} \ big）$在NPR，NGPR中。所有错误都直接扩展到等级$ r $矩阵恢复的更普遍的问题，这些结果得出的结论是，全级框架$ \ {a_k \} _ {k = 1}^n $ in ngpr是比级别1帧$ \ {\ alpha_k \ alpha_k^*\} _ {k = 1}^n $在npr中更强大。提出了广泛的实验结果，以说明我们的理论发现。

High-dimensional data can often display heterogeneity due to heteroscedastic variance or inhomogeneous covariate effects. Penalized quantile and expectile regression methods offer useful tools to detect heteroscedasticity in high-dimensional data. The former is computationally challenging due to the non-smooth nature of the check loss, and the latter is sensitive to heavy-tailed error distributions. In this paper, we propose and study (penalized) robust expectile regression (retire), with a focus on iteratively reweighted $\ell_1$-penalization which reduces the estimation bias from $\ell_1$-penalization and leads to oracle properties. Theoretically, we establish the statistical properties of the retire estimator under two regimes: (i) low-dimensional regime in which $d \ll n$; (ii) high-dimensional regime in which $s\ll n\ll d$ with $s$ denoting the number of significant predictors. In the high-dimensional setting, we carefully characterize the solution path of the iteratively reweighted $\ell_1$-penalized retire estimation, adapted from the local linear approximation algorithm for folded-concave regularization. Under a mild minimum signal strength condition, we show that after as many as $\log(\log d)$ iterations the final iterate enjoys the oracle convergence rate. At each iteration, the weighted $\ell_1$-penalized convex program can be efficiently solved by a semismooth Newton coordinate descent algorithm. Numerical studies demonstrate the competitive performance of the proposed procedure compared with either non-robust or quantile regression based alternatives.

在本文中，我们提出了一种均匀抖动的一位量化方案，以进行高维统计估计。该方案包含截断，抖动和量化，作为典型步骤。作为规范示例，量化方案应用于三个估计问题：稀疏协方差矩阵估计，稀疏线性回归和矩阵完成。我们研究了高斯和重尾政权，假定重尾数据的基本分布具有有限的第二或第四刻。对于每个模型，我们根据一位量化的数据提出新的估计器。在高斯次级政权中，我们的估计器达到了对数因素的最佳最小速率，这表明我们的量化方案几乎没有额外的成本。在重尾状态下，虽然我们的估计量基本上变慢，但这些结果是在这种单位量化和重型尾部设置中的第一个结果，或者比现有可比结果表现出显着改善。此外，我们为一位压缩传感和一位矩阵完成的问题做出了巨大贡献。具体而言，我们通过凸面编程将一位压缩感传感扩展到次高斯甚至是重尾传感向量。对于一位矩阵完成，我们的方法与标准似然方法基本不同，并且可以处理具有未知分布的预量化随机噪声。提出了有关合成数据的实验结果，以支持我们的理论分析。

In many modern applications of deep learning the neural network has many more parameters than the data points used for its training. Motivated by those practices, a large body of recent theoretical research has been devoted to studying overparameterized models. One of the central phenomena in this regime is the ability of the model to interpolate noisy data, but still have test error lower than the amount of noise in that data. arXiv:1906.11300 characterized for which covariance structure of the data such a phenomenon can happen in linear regression if one considers the interpolating solution with minimum $\ell_2$-norm and the data has independent components: they gave a sharp bound on the variance term and showed that it can be small if and only if the data covariance has high effective rank in a subspace of small co-dimension. We strengthen and complete their results by eliminating the independence assumption and providing sharp bounds for the bias term. Thus, our results apply in a much more general setting than those of arXiv:1906.11300, e.g., kernel regression, and not only characterize how the noise is damped but also which part of the true signal is learned. Moreover, we extend the result to the setting of ridge regression, which allows us to explain another interesting phenomenon: we give general sufficient conditions under which the optimal regularization is negative.

本文涉及根N的可行性和手段，始终估算高维，大约稀疏回归的线性，均方连续功能。这些对象包括各种有趣的参数，例如回归系数，平均衍生物和平均治疗效果。我们给出了回归斜率和平均导数的估计量的收敛速率的下限，并发现这些界限大大比低维，半参数设置大。我们还提供了依据的机器学习者，这些学习者在最小的稀疏条件或速率双重鲁棒性下是一致的。这些估计值对在先前已知的更一般条件下保持root-n一致的现有估计值有所改善。

我们研究了称为“乐观速率”（Panchenko 2002; Srebro等，2010）的统一收敛概念，用于与高斯数据的线性回归。我们的精致分析避免了现有结果中的隐藏常量和对数因子，这已知在高维设置中至关重要，特别是用于了解插值学习。作为一个特殊情况，我们的分析恢复了Koehler等人的保证。（2021年），在良性过度的过度条件下，严格地表征了低规范内插器的人口风险。但是，我们的乐观速度绑定还分析了具有任意训练错误的预测因子。这使我们能够在随机设计下恢复脊和套索回归的一些经典统计保障，并有助于我们在过度参数化制度中获得精确了解近端器的过度风险。

在负面的感知问题中，我们给出了$ n $数据点$（{\ boldsymbol x} _i，y_i）$，其中$ {\ boldsymbol x} _i $是$ d $ -densional vector和$ y_i \ in \ { + 1，-1 \} $是二进制标签。数据不是线性可分离的，因此我们满足自己的内容，以找到最大的线性分类器，具有最大的\ emph {否定}余量。换句话说，我们想找到一个单位常规矢量$ {\ boldsymbol \ theta} $，最大化$ \ min_ {i \ le n} y_i \ langle {\ boldsymbol \ theta}，{\ boldsymbol x} _i \ rangle $ 。这是一个非凸优化问题（它相当于在Polytope中找到最大标准矢量），我们在两个随机模型下研究其典型属性。我们考虑比例渐近，其中$ n，d \ to \ idty $以$ n / d \ to \ delta $，并在最大边缘$ \ kappa _ {\ text {s}}（\ delta）上证明了上限和下限）$或 - 等效 - 在其逆函数$ \ delta _ {\ text {s}}（\ kappa）$。换句话说，$ \ delta _ {\ text {s}}（\ kappa）$是overparametization阈值：以$ n / d \ le \ delta _ {\ text {s}}（\ kappa） - \ varepsilon $一个分类器实现了消失的训练错误，具有高概率，而以$ n / d \ ge \ delta _ {\ text {s}}（\ kappa）+ \ varepsilon $。我们在$ \ delta _ {\ text {s}}（\ kappa）$匹配，以$ \ kappa \ to - \ idty $匹配。然后，我们分析了线性编程算法来查找解决方案，并表征相应的阈值$ \ delta _ {\ text {lin}}（\ kappa）$。我们观察插值阈值$ \ delta _ {\ text {s}}（\ kappa）$和线性编程阈值$ \ delta _ {\ text {lin {lin}}（\ kappa）$之间的差距，提出了行为的问题其他算法。

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.

This paper studies the quantization of heavy-tailed data in some fundamental statistical estimation problems, where the underlying distributions have bounded moments of some order. We propose to truncate and properly dither the data prior to a uniform quantization. Our major standpoint is that (near) minimax rates of estimation error are achievable merely from the quantized data produced by the proposed scheme. In particular, concrete results are worked out for covariance estimation, compressed sensing, and matrix completion, all agreeing that the quantization only slightly worsens the multiplicative factor. Besides, we study compressed sensing where both covariate (i.e., sensing vector) and response are quantized. Under covariate quantization, although our recovery program is non-convex because the covariance matrix estimator lacks positive semi-definiteness, all local minimizers are proved to enjoy near optimal error bound. Moreover, by the concentration inequality of product process and covering argument, we establish near minimax uniform recovery guarantee for quantized compressed sensing with heavy-tailed noise.

我们考虑与高斯数据的高维线性回归中的插值学习，并在类高斯宽度方面证明了任意假设类别中的内插器的泛化误差。将通用绑定到欧几里德常规球恢复了Bartlett等人的一致性结果。（2020）对于最小规范内插器，并确认周等人的预测。（2020）在高斯数据的特殊情况下，对于近乎最小常态的内插器。我们通过将其应用于单位来证明所界限的一般性，从而获得最小L1-NORM Interpoolator（基础追踪）的新型一致性结果。我们的结果表明，基于规范的泛化界限如何解释并用于分析良性过度装备，至少在某些设置中。