神经塌陷是指表征类嵌入和分类器重量的几何形状的显着结构特性,当经过零训练误差以外的训练时,深网被发现。但是,这种表征仅适用于平衡数据。因此,我们在这里询问是否可以使阶级失衡不变。为此,我们采用了不受限制的功能模型(UFM),这是一种用于研究神经塌陷的最新理论模型,并引入了单纯形编码标签的插值(SELI)作为神经崩溃现象的不变特征。具体而言,我们证明了UFM的跨凝结损失和消失的正则化,无论阶级失衡如何,嵌入和分类器总是插入单纯形编码的标签矩阵,并且其单个几何形状都由同一标签矩阵矩阵矩阵的SVD因子确定。然后,我们对合成和真实数据集进行了广泛的实验,这些实验确认了与SELI几何形状的收敛。但是,我们警告说,融合会随着不平衡的增加而恶化。从理论上讲,我们通过表明与平衡的情况不同,当存在少数民族时,山脊规范化在调整几何形状中起着至关重要的作用。这定义了新的问题,并激发了对阶级失衡对一阶方法融合其渐近优先解决方案的速率的影响的进一步研究。
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标签 - 不平衡和组敏感分类中的目标是优化相关的指标,例如平衡错误和相同的机会。经典方法,例如加权交叉熵,在训练深网络到训练(TPT)的终端阶段时,这是超越零训练误差的训练。这种观察发生了最近在促进少数群体更大边值的直观机制之后开发启发式替代品的动力。与之前的启发式相比,我们遵循原则性分析,说明不同的损失调整如何影响边距。首先,我们证明,对于在TPT中训练的所有线性分类器,有必要引入乘法,而不是添加性的Logit调整,以便对杂项边缘进行适当的变化。为了表明这一点,我们发现将乘法CE修改的连接到成本敏感的支持向量机。也许是违反,我们还发现,在培训开始时,相同的乘法权重实际上可以损害少数群体。因此,虽然在TPT中,添加剂调整无效,但我们表明它们可以通过对乘法重量的初始负效应进行抗衡来加速会聚。通过这些发现的动机,我们制定了矢量缩放(VS)丢失,即捕获现有技术作为特殊情况。此外,我们引入了对群体敏感分类的VS损失的自然延伸,从而以统一的方式处理两种常见类型的不平衡(标签/组)。重要的是,我们对最先进的数据集的实验与我们的理论见解完全一致,并确认了我们算法的卓越性能。最后,对于不平衡的高斯 - 混合数据,我们执行泛化分析,揭示平衡/标准错误和相同机会之间的权衡。
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Modern deep neural networks have achieved superhuman performance in tasks from image classification to game play. Surprisingly, these various complex systems with massive amounts of parameters exhibit the same remarkable structural properties in their last-layer features and classifiers across canonical datasets. This phenomenon is known as "Neural Collapse," and it was discovered empirically by Papyan et al. \cite{Papyan20}. Recent papers have theoretically shown the global solutions to the training network problem under a simplified "unconstrained feature model" exhibiting this phenomenon. We take a step further and prove the Neural Collapse occurrence for deep linear network for the popular mean squared error (MSE) and cross entropy (CE) loss. Furthermore, we extend our research to imbalanced data for MSE loss and present the first geometric analysis for Neural Collapse under this setting.
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训练深层神经网络进行分类任务的现代策略包括优化网络的权重,即使训练错误消失了,以进一步将训练损失推向零。最近,在此训练程序中凭经验观察到了一种称为“神经崩溃”(NC)的现象。具体而言,已经表明,课堂样品的学习特征(倒数第二层的输出)融合到它们的平均值,不同类别的平均值表现出一定的紧密框架结构,这也与最后一层的重量对齐。最近的论文表明,当使用正则化交叉渗透损失优化简化的“无约束特征模型”(UFM)时,具有这种结构的最小化。在本文中,我们进一步分析并扩展了UFM。首先,我们研究了正规化MSE损失的UFM,并表明最小化器的特征比在跨膜片情况下具有更精致的结构。这也影响了权重的结构。然后,我们通过向模型添加另一层权重以及依赖非线性来扩展UFM并概括我们先前的结果。最后,我们从经验上证明了非线性扩展UFM在对实用网络发生的NC现象进行建模时的实用性。
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我们提供了通过线性激活的多渠道卷积神经网络中的$ \ ell_2 $标准来最大程度地减少$ \ ell_2 $标准而产生的功能空间表征,并经验测试了我们对使用梯度下降训练的Relu网络的假设。我们将功能空间中的诱导正规化程序定义为实现函数所需的网络权重规范的最小$ \ ell_2 $。对于具有$ C $输出频道和内核尺寸$ K $的两个层线性卷积网络,我们显示以下内容:(a)如果网络的输入是单个渠道,则任何$ k $的诱导正规器都与数字无关输出频道$ c $。此外,我们得出正常化程序是由半决赛程序(SDP)给出的规范。 (b)相比之下,对于多通道输入,仅实现所有矩阵值值线性函数而需要多个输出通道,因此归纳偏置确实取决于$ c $。但是,对于足够大的$ c $,诱导的正规化程序再次由独立于$ c $的SDP给出。特别是,$ k = 1 $和$ k = d $(输入维度)的诱导正规器以封闭形式作为核标准和$ \ ell_ {2,1} $ group-sparse Norm,线性预测指标的傅立叶系数。我们通过对MNIST和CIFAR-10数据集的实验来研究理论结果对从线性和RELU网络上梯度下降的隐式正则化的更广泛的适用性。
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过度参数化模型即使与传统的减轻失衡技术结合使用,在存在数据失衡的情况下也无法很好地概括。本文着重于分类数据集,其中一小部分人口(少数​​)可能包含与类标签相关的功能。对于跨凝结损失修饰和代表性高斯混合模型的参数家族,我们在最严重的组误差上得出了非反应泛化的边界,该误差揭示了不同的超参数的作用。具体而言,我们证明,在适当调整后,最近提出的VS-Loss学会了一个模型,即使伪造的特征很强,也对少数群体也是公平的。另一方面,替代性启发式方法,例如加权CE和LA-loss,可能会急剧失败。与以前的作品相比,我们的界限适用于更多的通用模型,它们是非吸血管的,即使在极端不平衡的情况下,它们也适用。
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Autoencoders are a popular model in many branches of machine learning and lossy data compression. However, their fundamental limits, the performance of gradient methods and the features learnt during optimization remain poorly understood, even in the two-layer setting. In fact, earlier work has considered either linear autoencoders or specific training regimes (leading to vanishing or diverging compression rates). Our paper addresses this gap by focusing on non-linear two-layer autoencoders trained in the challenging proportional regime in which the input dimension scales linearly with the size of the representation. Our results characterize the minimizers of the population risk, and show that such minimizers are achieved by gradient methods; their structure is also unveiled, thus leading to a concise description of the features obtained via training. For the special case of a sign activation function, our analysis establishes the fundamental limits for the lossy compression of Gaussian sources via (shallow) autoencoders. Finally, while the results are proved for Gaussian data, numerical simulations on standard datasets display the universality of the theoretical predictions.
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成功的深度学习模型往往涉及培训具有比训练样本数量更多的参数的神经网络架构。近年来已经广泛研究了这种超分子化的模型,并且通过双下降现象和通过优化景观的结构特性,从统计的角度和计算视角都建立了过分统计化的优点。尽管在过上分层的制度中深入学习架构的显着成功,但也众所周知,这些模型对其投入中的小对抗扰动感到高度脆弱。即使在普遍培训的情况下,它们在扰动输入(鲁棒泛化)上的性能也会比良性输入(标准概括)的最佳可达到的性能更糟糕。因此,必须了解如何从根本上影响稳健性的情况下如何影响鲁棒性。在本文中,我们将通过专注于随机特征回归模型(具有随机第一层权重的两层神经网络)来提供超分度化对鲁棒性的作用的精确表征。我们考虑一个制度,其中样本量,输入维度和参数的数量彼此成比例地生长,并且当模型发生前列地训练时,可以为鲁棒泛化误差导出渐近精确的公式。我们的发达理论揭示了过分统计化对鲁棒性的非竞争效果,表明对于普遍训练的随机特征模型,高度公正化可能会损害鲁棒泛化。
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教师 - 学生模型提供了一个框架,其中可以以封闭形式描述高维监督学习的典型情况。高斯I.I.D的假设然而,可以认为典型教师 - 学生模型的输入数据可以被认为过于限制,以捕获现实数据集的行为。在本文中,我们介绍了教师和学生可以在不同的空格上行动的模型的高斯协变态概括,以固定的,而是通用的特征映射。虽然仍处于封闭形式的仍然可解决,但这种概括能够捕获广泛的现实数据集的学习曲线,从而兑现师生框架的潜力。我们的贡献是两倍:首先,我们证明了渐近培训损失和泛化误差的严格公式。其次,我们呈现了许多情况,其中模型的学习曲线捕获了使用内​​核回归和分类学习的现实数据集之一,其中盒出开箱特征映射,例如随机投影或散射变换,或者与散射变换预先学习的 - 例如通过培训多层神经网络学到的特征。我们讨论了框架的权力和局限性。
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批准方法,例如批处理[Ioffe和Szegedy,2015],体重[Salimansand Kingma,2016],实例[Ulyanov等,2016]和层归一化[Baet al。,2016]已广泛用于现代机器学习中。在这里,我们研究了体重归一化方法(WN)方法[Salimans和Kingma,2016年],以及一种称为重扎式投影梯度下降(RPGD)的变体,用于过多散热性最小二乘回归。 WN和RPGD用比例G和一个单位向量W重新绘制权重,因此目标函数变为非convex。我们表明,与原始目标的梯度下降相比,这种非凸式配方具有有益的正则化作用。这些方法适应性地使重量正规化并收敛于最小L2规范解决方案,即使初始化远非零。对于G和W的某些步骤,我们表明它们可以收敛于最小规范解决方案。这与梯度下降的行为不同,梯度下降的行为仅在特征矩阵范围内的一个点开始时才收敛到最小规范解,因此对初始化更敏感。
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强大的机器学习模型的开发中的一个重要障碍是协变量的转变,当训练和测试集的输入分布时发生的分配换档形式在条件标签分布保持不变时发生。尽管现实世界应用的协变量转变普遍存在,但在现代机器学习背景下的理论理解仍然缺乏。在这项工作中,我们检查协变量的随机特征回归的精确高尺度渐近性,并在该设置中提出了限制测试误差,偏差和方差的精确表征。我们的结果激发了一种自然部分秩序,通过协变速转移,提供足够的条件来确定何时何时损害(甚至有助于)测试性能。我们发现,过度分辨率模型表现出增强的协会转变的鲁棒性,为这种有趣现象提供了第一个理论解释之一。此外,我们的分析揭示了分销和分发外概率性能之间的精确线性关系,为这一令人惊讶的近期实证观察提供了解释。
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在深度学习中的优化分析是连续的,专注于(变体)梯度流动,或离散,直接处理(变体)梯度下降。梯度流程可符合理论分析,但是风格化并忽略计算效率。它代表梯度下降的程度是深度学习理论的一个开放问题。目前的论文研究了这个问题。将梯度下降视为梯度流量初始值问题的近似数值问题,发现近似程度取决于梯度流动轨迹周围的曲率。然后,我们表明,在具有均匀激活的深度神经网络中,梯度流动轨迹享有有利的曲率,表明它们通过梯度下降近似地近似。该发现允许我们将深度线性神经网络的梯度流分析转换为保证梯度下降,其几乎肯定会在随机初始化下有效地收敛到全局最小值。实验表明,在简单的深度神经网络中,具有传统步长的梯度下降确实接近梯度流。我们假设梯度流动理论将解开深入学习背后的奥秘。
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尽管过度参数化的模型已经在许多机器学习任务上表现出成功,但与培训不同的测试分布的准确性可能会下降。这种准确性下降仍然限制了在野外应用机器学习的限制。同时,重要的加权是一种处理分配转移的传统技术,已被证明在经验和理论上对过度参数化模型的影响较小甚至没有影响。在本文中,我们提出了重要的回火来改善决策界限,并为过度参数化模型取得更好的结果。从理论上讲,我们证明在标签移位和虚假相关设置下,组温度的选择可能不同。同时,我们还证明正确选择的温度可以解脱出少数群体崩溃的分类不平衡。从经验上讲,我们使用重要性回火来实现最严重的小组分类任务的最新结果。
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Network data are ubiquitous in modern machine learning, with tasks of interest including node classification, node clustering and link prediction. A frequent approach begins by learning an Euclidean embedding of the network, to which algorithms developed for vector-valued data are applied. For large networks, embeddings are learned using stochastic gradient methods where the sub-sampling scheme can be freely chosen. Despite the strong empirical performance of such methods, they are not well understood theoretically. Our work encapsulates representation methods using a subsampling approach, such as node2vec, into a single unifying framework. We prove, under the assumption that the graph is exchangeable, that the distribution of the learned embedding vectors asymptotically decouples. Moreover, we characterize the asymptotic distribution and provided rates of convergence, in terms of the latent parameters, which includes the choice of loss function and the embedding dimension. This provides a theoretical foundation to understand what the embedding vectors represent and how well these methods perform on downstream tasks. Notably, we observe that typically used loss functions may lead to shortcomings, such as a lack of Fisher consistency.
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过度分化的深网络的泛化神秘具有有动力的努力,了解梯度下降(GD)如何收敛到概括井的低损耗解决方案。现实生活中的神经网络从小随机值初始化,并以分类的“懒惰”或“懒惰”或“NTK”的训练训练,分析更成功,以及最近的结果序列(Lyu和Li ,2020年; Chizat和Bach,2020; Ji和Telgarsky,2020)提供了理论证据,即GD可以收敛到“Max-ramin”解决方案,其零损失可能呈现良好。但是,仅在某些环境中证明了余量的全球最优性,其中神经网络无限或呈指数级宽。目前的纸张能够为具有梯度流动训练的两层泄漏的Relu网,无论宽度如何,都能为具有梯度流动的双层泄漏的Relu网建立这种全局最优性。分析还为最近的经验研究结果(Kalimeris等,2019)给出了一些理论上的理由,就GD的所谓简单的偏见为线性或其他“简单”的解决方案,特别是在训练中。在悲观方面,该论文表明这种结果是脆弱的。简单的数据操作可以使梯度流量会聚到具有次优裕度的线性分类器。
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Iterative regularization is a classic idea in regularization theory, that has recently become popular in machine learning. On the one hand, it allows to design efficient algorithms controlling at the same time numerical and statistical accuracy. On the other hand it allows to shed light on the learning curves observed while training neural networks. In this paper, we focus on iterative regularization in the context of classification. After contrasting this setting with that of regression and inverse problems, we develop an iterative regularization approach based on the use of the hinge loss function. More precisely we consider a diagonal approach for a family of algorithms for which we prove convergence as well as rates of convergence. Our approach compares favorably with other alternatives, as confirmed also in numerical simulations.
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We introduce a tunable loss function called $\alpha$-loss, parameterized by $\alpha \in (0,\infty]$, which interpolates between the exponential loss ($\alpha = 1/2$), the log-loss ($\alpha = 1$), and the 0-1 loss ($\alpha = \infty$), for the machine learning setting of classification. Theoretically, we illustrate a fundamental connection between $\alpha$-loss and Arimoto conditional entropy, verify the classification-calibration of $\alpha$-loss in order to demonstrate asymptotic optimality via Rademacher complexity generalization techniques, and build-upon a notion called strictly local quasi-convexity in order to quantitatively characterize the optimization landscape of $\alpha$-loss. Practically, we perform class imbalance, robustness, and classification experiments on benchmark image datasets using convolutional-neural-networks. Our main practical conclusion is that certain tasks may benefit from tuning $\alpha$-loss away from log-loss ($\alpha = 1$), and to this end we provide simple heuristics for the practitioner. In particular, navigating the $\alpha$ hyperparameter can readily provide superior model robustness to label flips ($\alpha > 1$) and sensitivity to imbalanced classes ($\alpha < 1$).
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随着Papyan等人最近对“神经崩溃(NC)”现象的观察,已经采取了各种努力来对其进行建模和分析。神经崩溃描述,在深层分类器网络中,与训练数据相关的最终隐藏层的类特征倾向于崩溃到各自的类功能均值。因此,将最后一层分类器的行为简化为最近级中心决策规则的行为。在这项工作中,我们分析了有助于从头开始对这种现象进行建模的原理,并展示他们如何建立对试图解释NC的最近提出的模型的共同理解。我们希望我们的分析对建模NC和有助于与神经网络的概括能力建立联系的多方面观点。最后,我们通过讨论进一步研究的途径并提出潜在的研究问题来得出结论。
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Efforts to understand the generalization mystery in deep learning have led to the belief that gradient-based optimization induces a form of implicit regularization, a bias towards models of low "complexity." We study the implicit regularization of gradient descent over deep linear neural networks for matrix completion and sensing, a model referred to as deep matrix factorization. Our first finding, supported by theory and experiments, is that adding depth to a matrix factorization enhances an implicit tendency towards low-rank solutions, oftentimes leading to more accurate recovery. Secondly, we present theoretical and empirical arguments questioning a nascent view by which implicit regularization in matrix factorization can be captured using simple mathematical norms. Our results point to the possibility that the language of standard regularizers may not be rich enough to fully encompass the implicit regularization brought forth by gradient-based optimization.
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The affine rank minimization problem consists of finding a matrix of minimum rank that satisfies a given system of linear equality constraints. Such problems have appeared in the literature of a diverse set of fields including system identification and control, Euclidean embedding, and collaborative filtering. Although specific instances can often be solved with specialized algorithms, the general affine rank minimization problem is NP-hard, because it contains vector cardinality minimization as a special case.In this paper, we show that if a certain restricted isometry property holds for the linear transformation defining the constraints, the minimum rank solution can be recovered by solving a convex optimization problem, namely the minimization of the nuclear norm over the given affine space. We present several random ensembles of equations where the restricted isometry property holds with overwhelming probability, provided the codimension of the subspace is Ω(r(m + n) log mn), where m, n are the dimensions of the matrix, and r is its rank.The techniques used in our analysis have strong parallels in the compressed sensing framework. We discuss how affine rank minimization generalizes this pre-existing concept and outline a dictionary relating concepts from cardinality minimization to those of rank minimization. We also discuss several algorithmic approaches to solving the norm minimization relaxations, and illustrate our results with numerical examples.
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