在神经网络中,与任务相关的信息由神经元组共同表示。但是,对信息分布在单个神经元之间的特定方式尚不清楚:虽然部分只能从特定的单个神经元中获得,但其他部分是由多个神经元冗余或协同携带的。我们展示了部分信息分解(PID)是信息理论的最新扩展,可以解散这些贡献。由此,我们介绍了“代表性复杂性”的度量,该量度量化了访问跨多个神经元信息的难度。我们展示了这种复杂性如何直接适用于较小的层。对于较大的层,我们提出了子采样和粗粒程序,并证明了后者的相应边界。从经验上讲,为了量化解决MNIST任务的深度神经网络,我们观察到,代表性复杂性通过连续的隐藏层和过度训练都会降低。总体而言,我们建议代表性复杂性作为分析神经表示结构的原则且可解释的摘要统计量。
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Selecting a minimal feature set that is maximally informative about a target variable is a central task in machine learning and statistics. Information theory provides a powerful framework for formulating feature selection algorithms -- yet, a rigorous, information-theoretic definition of feature relevancy, which accounts for feature interactions such as redundant and synergistic contributions, is still missing. We argue that this lack is inherent to classical information theory which does not provide measures to decompose the information a set of variables provides about a target into unique, redundant, and synergistic contributions. Such a decomposition has been introduced only recently by the partial information decomposition (PID) framework. Using PID, we clarify why feature selection is a conceptually difficult problem when approached using information theory and provide a novel definition of feature relevancy and redundancy in PID terms. From this definition, we show that the conditional mutual information (CMI) maximizes relevancy while minimizing redundancy and propose an iterative, CMI-based algorithm for practical feature selection. We demonstrate the power of our CMI-based algorithm in comparison to the unconditional mutual information on benchmark examples and provide corresponding PID estimates to highlight how PID allows to quantify information contribution of features and their interactions in feature-selection problems.
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Two approaches to AI, neural networks and symbolic systems, have been proven very successful for an array of AI problems. However, neither has been able to achieve the general reasoning ability required for human-like intelligence. It has been argued that this is due to inherent weaknesses in each approach. Luckily, these weaknesses appear to be complementary, with symbolic systems being adept at the kinds of things neural networks have trouble with and vice-versa. The field of neural-symbolic AI attempts to exploit this asymmetry by combining neural networks and symbolic AI into integrated systems. Often this has been done by encoding symbolic knowledge into neural networks. Unfortunately, although many different methods for this have been proposed, there is no common definition of an encoding to compare them. We seek to rectify this problem by introducing a semantic framework for neural-symbolic AI, which is then shown to be general enough to account for a large family of neural-symbolic systems. We provide a number of examples and proofs of the application of the framework to the neural encoding of various forms of knowledge representation and neural network. These, at first sight disparate approaches, are all shown to fall within the framework's formal definition of what we call semantic encoding for neural-symbolic AI.
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Understanding the functional principles of information processing in deep neural networks continues to be a challenge, in particular for networks with trained and thus non-random weights. To address this issue, we study the mapping between probability distributions implemented by a deep feed-forward network. We characterize this mapping as an iterated transformation of distributions, where the non-linearity in each layer transfers information between different orders of correlation functions. This allows us to identify essential statistics in the data, as well as different information representations that can be used by neural networks. Applied to an XOR task and to MNIST, we show that correlations up to second order predominantly capture the information processing in the internal layers, while the input layer also extracts higher-order correlations from the data. This analysis provides a quantitative and explainable perspective on classification.
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Quantifying which neurons are important with respect to the classification decision of a trained neural network is essential for understanding their inner workings. Previous work primarily attributed importance to individual neurons. In this work, we study which groups of neurons contain synergistic or redundant information using a multivariate mutual information method called the O-information. We observe the first layer is dominated by redundancy suggesting general shared features (i.e. detecting edges) while the last layer is dominated by synergy indicating local class-specific features (i.e. concepts). Finally, we show the O-information can be used for multi-neuron importance. This can be demonstrated by re-training a synergistic sub-network, which results in a minimal change in performance. These results suggest our method can be used for pruning and unsupervised representation learning.
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这是一门专门针对STEM学生开发的介绍性机器学习课程。我们的目标是为有兴趣的读者提供基础知识,以在自己的项目中使用机器学习,并将自己熟悉术语作为进一步阅读相关文献的基础。在这些讲义中,我们讨论受监督,无监督和强化学习。注释从没有神经网络的机器学习方法的说明开始,例如原理分析,T-SNE,聚类以及线性回归和线性分类器。我们继续介绍基本和先进的神经网络结构,例如密集的进料和常规神经网络,经常性的神经网络,受限的玻尔兹曼机器,(变性)自动编码器,生成的对抗性网络。讨论了潜在空间表示的解释性问题,并使用梦和对抗性攻击的例子。最后一部分致力于加强学习,我们在其中介绍了价值功能和政策学习的基本概念。
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每个已知的人工深神经网络(DNN)都对应于规范Grothendieck的拓扑中的一个物体。它的学习动态对应于此拓扑中的形态流动。层中的不变结构(例如CNNS或LSTMS)对应于Giraud的堆栈。这种不变性应该是对概括属性的原因,即从约束下的学习数据中推断出来。纤维代表语义前类别(Culioli,Thom),在该类别上定义了人工语言,内部逻辑,直觉主义者,古典或线性(Girard)。网络的语义功能是其能够用这种语言表达理论的能力,以回答输出数据中有关输出的问题。语义信息的数量和空间是通过类比与2015年香农和D.Bennequin的Shannon熵的同源解释来定义的。他们概括了Carnap和Bar-Hillel(1952)发现的措施。令人惊讶的是,上述语义结构通过封闭模型类别的几何纤维对象进行了分类,然后它们产生了DNNS及其语义功能的同位不变。故意类型的理论(Martin-Loef)组织了这些物体和它们之间的纤维。 Grothendieck的导数分析了信息内容和交流。
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迄今为止,通信系统主要旨在可靠地交流位序列。这种方法提供了有效的工程设计,这些设计对消息的含义或消息交换所旨在实现的目标不可知。但是,下一代系统可以通过将消息语义和沟通目标折叠到其设计中来丰富。此外,可以使这些系统了解进行交流交流的环境,从而为新颖的设计见解提供途径。本教程总结了迄今为止的努力,从早期改编,语义意识和以任务为导向的通信开始,涵盖了基础,算法和潜在的实现。重点是利用信息理论提供基础的方法,以及学习在语义和任务感知通信中的重要作用。
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代表学习算法为讨论有关滋扰因素的输入数据的不变表示提供了机会。许多作者利用此类策略来学习公平表示,即删除有关敏感属性的信息的向量。这些方法很有吸引力,因为它们可以解释为最大程度地减少神经层的激活与敏感属性之间的相互信息。但是,这种方法的理论基础依赖于无限准确的对手的计算或最小化相互信息估计的变异上限。在本文中,我们提出了一种直接计算神经层和敏感属性之间相互信息的方法。我们采用随机激活的二进制神经网络,使我们可以将神经元视为随机变量。然后,我们能够在层和敏感属性之间计算(不绑定)相互信息,并在梯度下降期间使用此信息作为正则化因子。我们表明,该方法与公平表示学习中的艺术状态相比,与完全精确的神经网络相比,学习的表示形式显示出更高的不变性水平。
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Deep-learning of artificial neural networks (ANNs) is creating highly functional tools that are, unfortunately, as hard to interpret as their natural counterparts. While it is possible to identify functional modules in natural brains using technologies such as fMRI, we do not have at our disposal similarly robust methods for artificial neural networks. Ideally, understanding which parts of an artificial neural network perform what function might help us to address a number of vexing problems in ANN research, such as catastrophic forgetting and overfitting. Furthermore, revealing a network's modularity could improve our trust in them by making these black boxes more transparent. Here we introduce a new information-theoretic concept that proves useful in understanding and analyzing a network's functional modularity: the relay information $I_R$. The relay information measures how much information groups of neurons that participate in a particular function (modules) relay from inputs to outputs. Combined with a greedy search algorithm, relay information can be used to {\em identify} computational modules in neural networks. We also show that the functionality of modules correlates with the amount of relay information they carry.
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Deep Neural Networks (DNNs) are analyzed via the theoretical framework of the information bottleneck (IB) principle. We first show that any DNN can be quantified by the mutual information between the layers and the input and output variables. Using this representation we can calculate the optimal information theoretic limits of the DNN and obtain finite sample generalization bounds. The advantage of getting closer to the theoretical limit is quantifiable both by the generalization bound and by the network's simplicity. We argue that both the optimal architecture, number of layers and features/connections at each layer, are related to the bifurcation points of the information bottleneck tradeoff, namely, relevant compression of the input layer with respect to the output layer. The hierarchical representations at the layered network naturally correspond to the structural phase transitions along the information curve. We believe that this new insight can lead to new optimality bounds and deep learning algorithms.
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这项调查的目的是介绍对深神经网络的近似特性的解释性回顾。具体而言,我们旨在了解深神经网络如何以及为什么要优于其他经典线性和非线性近似方法。这项调查包括三章。在第1章中,我们回顾了深层网络及其组成非线性结构的关键思想和概念。我们通过在解决回归和分类问题时将其作为优化问题来形式化神经网络问题。我们简要讨论用于解决优化问题的随机梯度下降算法以及用于解决优化问题的后传播公式,并解决了与神经网络性能相关的一些问题,包括选择激活功能,成本功能,过度适应问题和正则化。在第2章中,我们将重点转移到神经网络的近似理论上。我们首先介绍多项式近似中的密度概念,尤其是研究实现连续函数的Stone-WeierStrass定理。然后,在线性近似的框架内,我们回顾了馈电网络的密度和收敛速率的一些经典结果,然后在近似Sobolev函数中进行有关深网络复杂性的最新发展。在第3章中,利用非线性近似理论,我们进一步详细介绍了深度和近似网络与其他经典非线性近似方法相比的近似优势。
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作为一种强大的建模方法,分段线性神经网络(PWLNNS)已在各个领域都被证明是成功的,最近在深度学习中。为了应用PWLNN方法,长期以来一直研究了表示和学习。 1977年,规范表示率先通过增量设计学到了浅层PWLNN的作品,但禁止使用大规模数据的应用。 2010年,纠正的线性单元(RELU)提倡在深度学习中PWLNN的患病率。从那以后,PWLNNS已成功地应用于广泛的任务并实现了有利的表现。在本引物中,我们通过将作品分组为浅网络和深层网络来系统地介绍PWLNNS的方法。首先,不同的PWLNN表示模型是由详细示例构建的。使用PWLNNS,提出了学习数据的学习算法的演变,并且基本理论分析遵循深入的理解。然后,将代表性应用与讨论和前景一起引入。
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We consider the algorithmic problem of finding the optimal weights and biases for a two-layer fully connected neural network to fit a given set of data points. This problem is known as empirical risk minimization in the machine learning community. We show that the problem is $\exists\mathbb{R}$-complete. This complexity class can be defined as the set of algorithmic problems that are polynomial-time equivalent to finding real roots of a polynomial with integer coefficients. Furthermore, we show that arbitrary algebraic numbers are required as weights to be able to train some instances to optimality, even if all data points are rational. Our results hold even if the following restrictions are all added simultaneously. $\bullet$ There are exactly two output neurons. $\bullet$ There are exactly two input neurons. $\bullet$ The data has only 13 different labels. $\bullet$ The number of hidden neurons is a constant fraction of the number of data points. $\bullet$ The target training error is zero. $\bullet$ The ReLU activation function is used. This shows that even very simple networks are difficult to train. The result explains why typical methods for $\mathsf{NP}$-complete problems, like mixed-integer programming or SAT-solving, cannot train neural networks to global optimality, unless $\mathsf{NP}=\exists\mathbb{R}$. We strengthen a recent result by Abrahamsen, Kleist and Miltzow [NeurIPS 2021].
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在当前的嘈杂中间尺度量子(NISQ)时代,量子机学习正在成为基于程序门的量子计算机的主要范式。在量子机学习中,对量子电路的门进行了参数化,并且参数是根据数据和电路输出的测量来通过经典优化来调整的。参数化的量子电路(PQC)可以有效地解决组合优化问题,实施概率生成模型并进行推理(分类和回归)。该专着为具有概率和线性代数背景的工程师的观众提供了量子机学习的独立介绍。它首先描述了描述量子操作和测量所必需的必要背景,概念和工具。然后,它涵盖了参数化的量子电路,变异量子本质层以及无监督和监督的量子机学习公式。
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可解释的人工智能(XAI)的新兴领域旨在为当今强大但不透明的深度学习模型带来透明度。尽管本地XAI方法以归因图的形式解释了个体预测,从而确定了重要特征的发生位置(但没有提供有关其代表的信息),但全局解释技术可视化模型通常学会的编码的概念。因此,两种方法仅提供部分见解,并留下将模型推理解释的负担。只有少数当代技术旨在将本地和全球XAI背后的原则结合起来,以获取更多信息的解释。但是,这些方法通常仅限于特定的模型体系结构,或对培训制度或数据和标签可用性施加其他要求,这实际上使事后应用程序成为任意预训练的模型。在这项工作中,我们介绍了概念相关性传播方法(CRP)方法,该方法结合了XAI的本地和全球观点,因此允许回答“何处”和“ where”和“什么”问题,而没有其他约束。我们进一步介绍了相关性最大化的原则,以根据模型对模型的有用性找到代表性的示例。因此,我们提高了对激活最大化及其局限性的共同实践的依赖。我们证明了我们方法在各种环境中的能力,展示了概念相关性传播和相关性最大化导致了更加可解释的解释,并通过概念图表,概念组成分析和概念集合和概念子区和概念子区和概念子集和定量研究对模型的表示和推理提供了深刻的见解。它们在细粒度决策中的作用。
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我们研究了重整化组(RG)和深神经网络之间的类比,其中随后的神经元层类似于沿RG的连续步骤。特别地,我们通过在抽取RG下明确计算在DIMIMATION RG下的一个和二维insing模型中的相对熵或kullback-leibler发散,以及作为深度的函数的前馈神经网络中的相对熵或kullback-leibler发散。我们观察到单调增加到参数依赖性渐近值的定性相同的行为。在量子场理论方面,单调增加证实了相对熵和C定理之间的连接。对于神经网络,渐近行为可能对机器学习中的各种信息最大化方法以及解开紧凑性和概括性具有影响。此外,虽然我们考虑的二维误操作模型和随机神经网络都表现出非差异临界点,但是对任何系统的相位结构的相对熵看起来不敏感。从这个意义上讲,需要更精细的探针以充分阐明这些模型中的信息流。
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The success of machine learning algorithms generally depends on data representation, and we hypothesize that this is because different representations can entangle and hide more or less the different explanatory factors of variation behind the data. Although specific domain knowledge can be used to help design representations, learning with generic priors can also be used, and the quest for AI is motivating the design of more powerful representation-learning algorithms implementing such priors. This paper reviews recent work in the area of unsupervised feature learning and deep learning, covering advances in probabilistic models, auto-encoders, manifold learning, and deep networks. This motivates longer-term unanswered questions about the appropriate objectives for learning good representations, for computing representations (i.e., inference), and the geometrical connections between representation learning, density estimation and manifold learning.
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即使机器学习算法已经在数据科学中发挥了重要作用,但许多当前方法对输入数据提出了不现实的假设。由于不兼容的数据格式,或数据集中的异质,分层或完全缺少的数据片段,因此很难应用此类方法。作为解决方案,我们提出了一个用于样本表示,模型定义和培训的多功能,统一的框架,称为“ Hmill”。我们深入审查框架构建和扩展的机器学习的多个范围范式。从理论上讲,为HMILL的关键组件的设计合理,我们将通用近似定理的扩展显示到框架中实现的模型所实现的所有功能的集合。本文还包含有关我们实施中技术和绩效改进的详细讨论,该讨论将在MIT许可下发布供下载。该框架的主要资产是其灵活性,它可以通过相同的工具对不同的现实世界数据源进行建模。除了单独观察到每个对象的一组属性的标准设置外,我们解释了如何在框架中实现表示整个对象系统的图表中的消息推断。为了支持我们的主张,我们使用框架解决了网络安全域的三个不同问题。第一种用例涉及来自原始网络观察结果的IoT设备识别。在第二个问题中,我们研究了如何使用以有向图表示的操作系统的快照可以对恶意二进制文件进行分类。最后提供的示例是通过网络中实体之间建模域黑名单扩展的任务。在所有三个问题中,基于建议的框架的解决方案可实现与专业方法相当的性能。
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我们建议出现的定量和客观概念。我们的建议使用算法信息理论作为一个客观框架的基础,其中某个字符串编码观测数据。这种字符串的Kolmogorov结构功能中有多个滴剂被视为出现的标志。我们的定义除了扩展了粗粒和边界条件的概念外,还提供了一些理论上的结果。最后,我们面对对动态系统和热力学的应用。
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