了解深度神经网络的泛化是深度学习中最重要的任务之一。虽然已经取得了很大进展，但理论错误界限仍然往往与经验观察结果不同。在这项工作中，我们开发基于保证金的泛化界，其中边距是在从训练分布中采样的独立随机子集之间的最佳运输成本标准化。特别地，最佳运输成本可以被解释为方差的概念，其捕获学习特征空间的结构特性。我们的界限强大地预测了在大规模数据集上给定培训数据和网络参数的泛化误差。从理论上讲，我们表明特征的浓度和分离在泛化中起着至关重要的作用，支持文献中的经验结果。该代码可用于\ url {https:/github.com/chingyaoc/kv-margin}。

Existing generalization bounds fail to explain crucial factors that drive generalization of modern neural networks. Since such bounds often hold uniformly over all parameters, they suffer from over-parametrization, and fail to account for the strong inductive bias of initialization and stochastic gradient descent. As an alternative, we propose a novel optimal transport interpretation of the generalization problem. This allows us to derive instance-dependent generalization bounds that depend on the local Lipschitz regularity of the earned prediction function in the data space. Therefore, our bounds are agnostic to the parametrization of the model and work well when the number of training samples is much smaller than the number of parameters. With small modifications, our approach yields accelerated rates for data on low-dimensional manifolds, and guarantees under distribution shifts. We empirically analyze our generalization bounds for neural networks, showing that the bound values are meaningful and capture the effect of popular regularization methods during training.

This paper studies the distribution estimation of contaminated data by the MoM-GAN method, which combines generative adversarial net (GAN) and median-of-mean (MoM) estimation. We use a deep neural network (DNN) with a ReLU activation function to model the generator and discriminator of the GAN. Theoretically, we derive a non-asymptotic error bound for the DNN-based MoM-GAN estimator measured by integral probability metrics with the $b$-smoothness H\"{o}lder class. The error bound decreases essentially as $n^{-b/p}\vee n^{-1/2}$, where $n$ and $p$ are the sample size and the dimension of input data. We give an algorithm for the MoM-GAN method and implement it through two real applications. The numerical results show that the MoM-GAN outperforms other competitive methods when dealing with contaminated data.

我们研究了广义熵的连续性属性作为潜在的概率分布的函数，用动作空间和损失函数定义，并使用此属性来回答统计学习理论中的基本问题：各种学习方法的过度风险分析。我们首先在几种常用的F分歧，Wassersein距离的熵差异导出了两个分布的熵差，这取决于动作空间的距离和损失函数，以及由熵产生的Bregman发散，这也诱导了两个分布之间的欧几里德距离方面的界限。对于每个一般结果的讨论给出了示例，使用现有的熵差界进行比较，并且基于新结果导出新的相互信息上限。然后，我们将熵差异界限应用于统计学习理论。结果表明，两种流行的学习范式，频繁学习和贝叶斯学习中的过度风险都可以用不同形式的广义熵的连续性研究。然后将分析扩展到广义条件熵的连续性。扩展为贝叶斯决策提供了不匹配的分布来提供性能范围。它也会导致第三个划分的学习范式的过度风险范围，其中决策规则是在经验分布的预定分布家族的预测下进行最佳设计。因此，我们通过广义熵的连续性建立了统计学习三大范式的过度风险分析的统一方法。

我们提出了Pac-Bayes风格的概括结合，该结合可以用各种积分概率指标（IPM）替换KL-Divergence。我们提供了这种结合的实例，IPM是总变异度量和Wasserstein距离。获得的边界的一个显着特征是，它们在最坏的情况下（当前和后距离彼此远距离时）在经典均匀收敛边界之间自然插值，并且在更好的情况下（后验和先验都关闭时）优选界限。这说明了使用算法和数据依赖性组件加强经典概括界限的可能性，从而使它们更适合分析使用大假设空间的算法。

生成的对抗网络后面的数学力量提高了具有挑战性的理论问题。通过表征产生的分布的几何特性的重要问题，我们在有限的样本和渐近制度中对Wassersein Gans（WGAN）进行了彻底分析。我们研究了潜伏空间是单变量的特定情况，并且不管输出空间的尺寸如何有效。我们特别地显示出用于固定的样本大小，最佳WGAN与连接路径紧密相连，最小化采样点之间的平方欧几里德距离的总和。我们还强调了WGAN能够接近的事实（对于1-Wasserstein距离）目标分布，因为样本大小趋于无穷大，在给定的会聚速率下，并且提供了生成的Lipschitz函数的家族适当地增长。我们在半离散环境中获得了在最佳运输理论上传递新结果。

比较概率分布是许多机器学习算法的关键。最大平均差异（MMD）和最佳运输距离（OT）是在过去几年吸引丰富的关注的概率措施之间的两类距离。本文建立了一些条件，可以通过MMD规范控制Wassersein距离。我们的作品受到压缩统计学习（CSL）理论的推动，资源有效的大规模学习的一般框架，其中训练数据总结在单个向量（称为草图）中，该训练数据捕获与所考虑的学习任务相关的信息。在CSL中的现有结果启发，我们介绍了H \“较旧的较低限制的等距属性（H \”较旧的LRIP）并表明这家属性具有有趣的保证对压缩统计学习。基于MMD与Wassersein距离之间的关系，我们通过引入和研究学习任务的Wassersein可读性的概念来提供压缩统计学习的保证，即概率分布之间的某些特定于特定的特定度量，可以由Wassersein界定距离。

生成的对抗网络（GAN）在无监督学习方面取得了巨大的成功。尽管具有显着的经验表现，但关于gan的统计特性的理论研究有限。本文提供了gan的近似值和统计保证，以估算具有H \“ {o} lder空间密度的数据分布。我们的主要结果表明，如果正确选择了生成器和鉴别器网络架构，则gan是一致的估计器在较强的差异指标下的数据分布（例如Wasserstein-1距离。 ，这不受环境维度的诅咒。我们对低维数据的分析基于具有Lipschitz连续性保证的神经网络的通用近似理论，这可能具有独立的兴趣。

Empirical risk minimization (ERM) and distributionally robust optimization (DRO) are popular approaches for solving stochastic optimization problems that appear in operations management and machine learning. Existing generalization error bounds for these methods depend on either the complexity of the cost function or dimension of the uncertain parameters; consequently, the performance of these methods is poor for high-dimensional problems with objective functions under high complexity. We propose a simple approach in which the distribution of uncertain parameters is approximated using a parametric family of distributions. This mitigates both sources of complexity; however, it introduces a model misspecification error. We show that this new source of error can be controlled by suitable DRO formulations. Our proposed parametric DRO approach has significantly improved generalization bounds over existing ERM / DRO methods and parametric ERM for a wide variety of settings. Our method is particularly effective under distribution shifts. We also illustrate the superior performance of our approach on both synthetic and real-data portfolio optimization and regression tasks.

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。

We study the ability of foundation models to learn representations for classification that are transferable to new, unseen classes. Recent results in the literature show that representations learned by a single classifier over many classes are competitive on few-shot learning problems with representations learned by special-purpose algorithms designed for such problems. We offer an explanation for this phenomenon based on the concept of class-features variability collapse, which refers to the training dynamics of deep classification networks where the feature embeddings of samples belonging to the same class tend to concentrate around their class means. More specifically, we examine the few-shot error of the learned feature map, which is the classification error of the nearest class-center classifier using centers learned from a small number of random samples from each class. Assuming that the classes appearing in the data are selected independently from a distribution, we show that the few-shot error generalizes from the training data to unseen test data, and we provide an upper bound on the expected few-shot error for new classes (selected from the same distribution) using the average few-shot error for the source classes. Additionally, we show that the few-shot error on the training data can be upper bounded using the degree of class-features variability collapse. This suggests that foundation models can provide feature maps that are transferable to new downstream tasks even with limited data available.

Deep neural networks are vulnerable to adversarial attacks. Ideally, a robust model shall perform well on both the perturbed training data and the unseen perturbed test data. It is found empirically that fitting perturbed training data is not hard, but generalizing to perturbed test data is quite difficult. To better understand adversarial generalization, it is of great interest to study the adversarial Rademacher complexity (ARC) of deep neural networks. However, how to bound ARC in multi-layers cases is largely unclear due to the difficulty of analyzing adversarial loss in the definition of ARC. There have been two types of attempts of ARC. One is to provide the upper bound of ARC in linear and one-hidden layer cases. However, these approaches seem hard to extend to multi-layer cases. Another is to modify the adversarial loss and provide upper bounds of Rademacher complexity on such surrogate loss in multi-layer cases. However, such variants of Rademacher complexity are not guaranteed to be bounds for meaningful robust generalization gaps (RGG). In this paper, we provide a solution to this unsolved problem. Specifically, we provide the first bound of adversarial Rademacher complexity of deep neural networks. Our approach is based on covering numbers. We provide a method to handle the robustify function classes of DNNs such that we can calculate the covering numbers. Finally, we provide experiments to study the empirical implication of our bounds and provide an analysis of poor adversarial generalization.

Deep nets generalize well despite having more parameters than the number of training samples. Recent works try to give an explanation using PAC-Bayes and Margin-based analyses, but do not as yet result in sample complexity bounds better than naive parameter counting. The current paper shows generalization bounds that're orders of magnitude better in practice. These rely upon new succinct reparametrizations of the trained net -a compression that is explicit and efficient. These yield generalization bounds via a simple compression-based framework introduced here. Our results also provide some theoretical justification for widespread empirical success in compressing deep nets.Analysis of correctness of our compression relies upon some newly identified "noise stability"properties of trained deep nets, which are also experimentally verified. The study of these properties and resulting generalization bounds are also extended to convolutional nets, which had eluded earlier attempts on proving generalization.

Wasserstein distributionally robust optimization (DRO) has found success in operations research and machine learning applications as a powerful means to obtain solutions with favourable out-of-sample performances. Two compelling explanations for the success are the generalization bounds derived from Wasserstein DRO and the equivalency between Wasserstein DRO and the regularization scheme commonly applied in machine learning. Existing results on generalization bounds and the equivalency to regularization are largely limited to the setting where the Wasserstein ball is of a certain type and the decision criterion takes certain forms of an expected function. In this paper, we show that by focusing on Wasserstein DRO problems with affine decision rules, it is possible to obtain generalization bounds and the equivalency to regularization in a significantly broader setting where the Wasserstein ball can be of a general type and the decision criterion can be a general measure of risk, i.e., nonlinear in distributions. This allows for accommodating many important classification, regression, and risk minimization applications that have not been addressed to date using Wasserstein DRO. Our results are strong in that the generalization bounds do not suffer from the curse of dimensionality and the equivalency to regularization is exact. As a byproduct, our regularization results broaden considerably the class of Wasserstein DRO models that can be solved efficiently via regularization formulations.

过度装备数据是与生成模型的众所周知的现象，其模拟太紧密（或准确）的特定数据实例，因此可能无法可靠地预测未来的观察。在实践中，这种行为是由各种 - 有时启发式的 - 正则化技术控制，这是通过将上限发展到泛化误差的激励。在这项工作中，我们研究依赖于在跨熵损失的随机编码上依赖于随机编码的泛化误差，这通常用于深度学习进行分类问题。我们导出界定误差，示出存在根据编码分布随机生成的输入特征和潜在空间中的相应表示之间的相互信息界定的制度。我们的界限提供了对所谓的各种变分类分类中的概括的信息理解，其由Kullback-Leibler（KL）发散项进行规则化。这些结果为变分推理方法提供了高度流行的KL术语的理论理由，这些方法已经认识到作为正则化罚款有效行动。我们进一步观察了具有良好研究概念的连接，例如变形自动化器，信息丢失，信息瓶颈和Boltzmann机器。最后，我们对Mnist和CiFar数据集进行了数值实验，并表明相互信息确实高度代表了泛化误差的行为。

转移学习或域适应性与机器学习问题有关，在这些问题中，培训和测试数据可能来自可能不同的概率分布。在这项工作中，我们在Russo和Xu发起的一系列工作之后，就通用错误和转移学习算法的过量风险进行了信息理论分析。我们的结果也许表明，也许正如预期的那样，kullback-leibler（kl）Divergence $ d（\ mu || \ mu'）$在$ \ mu $和$ \ mu'$表示分布的特征中起着重要作用。培训数据和测试测试。具体而言，我们为经验风险最小化（ERM）算法提供了概括误差上限，其中两个分布的数据在训练阶段都可用。我们进一步将分析应用于近似的ERM方法，例如Gibbs算法和随机梯度下降方法。然后，我们概括了与$ \ phi $ -Divergence和Wasserstein距离绑定的共同信息。这些概括导致更紧密的范围，并且在$ \ mu $相对于$ \ mu' $的情况下，可以处理案例。此外，我们应用了一套新的技术来获得替代的上限，该界限为某些学习问题提供了快速（最佳）的学习率。最后，受到派生界限的启发，我们提出了Infoboost算法，其中根据信息测量方法对源和目标数据的重要性权重进行了调整。经验结果表明了所提出的算法的有效性。

通过定义和上限，通过定义和上限，分析了贝叶斯学习的最佳成绩性能，通过限定了最小的过度风险（MER）：通过从数据学习和最低预期损失可以实现的最低预期损失之间的差距认识到了。 MER的定义提供了一种原则状的方式来定义贝叶斯学习中的不同概念的不确定性，包括炼膜不确定性和最小的认知不确定性。提出了用于衍生MER的上限的两种方法。第一方法，通常适用于具有参数生成模型的贝叶斯学习，通过在模型参数之间的条件互信息和所观察到的数据预测的量之间的条件相互信息。它允许我们量化MER衰减随着更多数据可用而衰减为零的速率。在可实现的模型中，该方法还将MER与生成函数类的丰富性涉及，特别是二进制分类中的VC维度。具有参数预测模型的第二种方法，特别适用于贝叶斯学习，将MER与来自数据的模型参数的最小估计误差相关联。它明确地说明了模型参数估计中的不确定性如何转化为MER和最终预测不确定性。我们还将MER的定义和分析扩展到具有多个模型系列的设置以及使用非参数模型的设置。沿着讨论，我们在贝叶斯学习中的MER与频繁学习的过度风险之间建立了一些比较。

我们研究了基础模型的能力，以了解可转让给新的看不见的课程的分类的表现。文献中最近的结果表明，单个分类器在许多课程中学到的表示在少量学习问题上具有竞争力，这些问题是由专为这些问题设计的特殊用途算法学习的表示。在本文中，我们基于最近观察到的现象提供了对这种行为的解释，即通过共同计量的分类网络学习的特征显示有趣的聚类属性，称为神经崩溃。理论上，我们在理论上展示了神经崩溃的展示给来自培训类的新样本，更重要的是 - 对于新课程，允许基础模型提供在转移学习中良好工作的特征地图，具体地，少量拍摄设置。

Classically, data interpolation with a parametrized model class is possible as long as the number of parameters is larger than the number of equations to be satisfied. A puzzling phenomenon in deep learning is that models are trained with many more parameters than what this classical theory would suggest. We propose a partial theoretical explanation for this phenomenon. We prove that for a broad class of data distributions and model classes, overparametrization is necessary if one wants to interpolate the data smoothly. Namely we show that smooth interpolation requires $d$ times more parameters than mere interpolation, where $d$ is the ambient data dimension. We prove this universal law of robustness for any smoothly parametrized function class with polynomial size weights, and any covariate distribution verifying isoperimetry. In the case of two-layers neural networks and Gaussian covariates, this law was conjectured in prior work by Bubeck, Li and Nagaraj. We also give an interpretation of our result as an improved generalization bound for model classes consisting of smooth functions.

我们观察到，给定两个（兼容的）函数类别$ \ MATHCAL {f} $和$ \ MATHCAL {h} $，具有较小的容量，按其均匀覆盖的数字测量，组成类$ \ Mathcal {H} \ Circ \ Mathcal {f} $可能会变得非常大，甚至无限。然后，我们证明，在用$ \ Mathcal {h} $构成$ \ Mathcal {f} $的输出中，添加少量高斯噪声可以有效地控制$ \ Mathcal {H} \ Circ \ Mathcal { F} $，提供模块化设计的一般配方。为了证明我们的结果，我们定义了均匀覆盖随机函数数量的新概念，相对于总变异和瓦斯坦斯坦距离。我们将结果实例化，以实现多层Sigmoid神经​​网络。 MNIST数据集的初步经验结果表明，在现有统一界限上改善所需的噪声量在数值上可以忽略不计（即，元素的I.I.D. I.I.D.高斯噪声，具有标准偏差$ 10^{ - 240} $）。源代码可从https://github.com/fathollahpour/composition_noise获得。