我们基于功能分析中的分类结构开发了一种自动和符号分化的组成方法，其中衍生物是抽象向量上的线性函数，而不是限于标量，向量，矩阵或张力器，表示为多维阵列。我们表明，可以使用差分计算来实现符号和自动分化，以生成基于原始，恒定，线性和双线性函数的规则以及其顺序和并行组成的线性函数。线性函数以组合域特异性语言表示。最后，我们提供了一个微积分，用于象征性地计算衍生物的伴随，而无需使用矩阵，而矩阵过于效率低，无法在高维空间上使用。衍生物的最终符号表示保留了输入程序的数据并行操作。组合分化和计算形式的伴随的组合在行为上等同于反向模式自动分化。特别是，它为矩阵过于效率而无法表示线性功能的优化提供了机会。

Derivatives, mostly in the form of gradients and Hessians, are ubiquitous in machine learning. Automatic differentiation (AD), also called algorithmic differentiation or simply "autodiff", is a family of techniques similar to but more general than backpropagation for efficiently and accurately evaluating derivatives of numeric functions expressed as computer programs. AD is a small but established field with applications in areas including computational fluid dynamics, atmospheric sciences, and engineering design optimization. Until very recently, the fields of machine learning and AD have largely been unaware of each other and, in some cases, have independently discovered each other's results. Despite its relevance, general-purpose AD has been missing from the machine learning toolbox, a situation slowly changing with its ongoing adoption under the names "dynamic computational graphs" and "differentiable programming". We survey the intersection of AD and machine learning, cover applications where AD has direct relevance, and address the main implementation techniques. By precisely defining the main differentiation techniques and their interrelationships, we aim to bring clarity to the usage of the terms "autodiff", "automatic differentiation", and "symbolic differentiation" as these are encountered more and more in machine learning settings.

Automatic differentiation (AD) is a technique for computing the derivative of a function represented by a program. This technique is considered as the de-facto standard for computing the differentiation in many machine learning and optimisation software tools. Despite the practicality of this technique, the performance of the differentiated programs, especially for functional languages and in the presence of vectors, is suboptimal. We present an AD system for a higher-order functional array-processing language. The core functional language underlying this system simultaneously supports both source-to-source forward-mode AD and global optimisations such as loop transformations. In combination, gradient computation with forward-mode AD can be as efficient as reverse mode, and the Jacobian matrices required for numerical algorithms such as Gauss-Newton and Levenberg-Marquardt can be efficiently computed.

我们在真实和复杂的环境中介绍了前向和向后模式广告的经典无坐标形式主义。我们展示了如何从基本原理开始的许多矩阵函数正式得出前向和向后公式。

We present a new algorithm for automatically bounding the Taylor remainder series. In the special case of a scalar function $f: \mathbb{R} \mapsto \mathbb{R}$, our algorithm takes as input a reference point $x_0$, trust region $[a, b]$, and integer $k \ge 0$, and returns an interval $I$ such that $f(x) - \sum_{i=0}^k \frac {f^{(i)}(x_0)} {i!} (x - x_0)^i \in I (x - x_0)^{k+1}$ for all $x \in [a, b]$. As in automatic differentiation, the function $f$ is provided to the algorithm in symbolic form, and must be composed of known elementary functions. At a high level, our algorithm has two steps. First, for a variety of commonly-used elementary functions (e.g., $\exp$, $\log$), we derive sharp polynomial upper and lower bounds on the Taylor remainder series. We then recursively combine the bounds for the elementary functions using an interval arithmetic variant of Taylor-mode automatic differentiation. Our algorithm can make efficient use of machine learning hardware accelerators, and we provide an open source implementation in JAX. We then turn our attention to applications. Most notably, we use our new machinery to create the first universal majorization-minimization optimization algorithms: algorithms that iteratively minimize an arbitrary loss using a majorizer that is derived automatically, rather than by hand. Applied to machine learning, this leads to architecture-specific optimizers for training deep networks that converge from any starting point, without hyperparameter tuning. Our experiments show that for some optimization problems, these hyperparameter-free optimizers outperform tuned versions of gradient descent, Adam, and AdaGrad. We also show that our automatically-derived bounds can be used for verified global optimization and numerical integration, and to prove sharper versions of Jensen's inequality.

十年自2010年以来，人工智能成功一直处于计算机科学和技术的最前沿，传染媒介空间模型已经巩固了人工智能最前沿的位置。与此同时，量子计算机已经变得更加强大，主要进步的公告经常在新闻中。这些区域的基础的数学技术比有时意识到更多的共同之处。传染媒介空间在20世纪30年代的量子力学的公理心脏上采取了位置，这一采用是从矢量空间的线性几何形状推导逻辑和概率的关键动机。粒子之间的量子相互作用是使用张量产品进行建模的，其也用于表达人工神经网络中的物体和操作。本文介绍了这些常见的数学区域中的一些，包括如何在人工智能（AI）中使用的示例，特别是在自动推理和自然语言处理（NLP）中。讨论的技术包括矢量空间，标量产品，子空间和含义，正交投影和否定，双向矩阵，密度矩阵，正算子和张量产品。应用领域包括信息检索，分类和含义，建模字传感和歧义，知识库的推断和语义构成。其中一些方法可能会在量子硬件上实现。该实施中的许多实际步骤都处于早期阶段，其中一些已经实现了。解释一些常见的数学工具可以帮助AI和量子计算中的研究人员进一步利用这些重叠，识别和沿途探索新方向。

We propose a notation for tensors with named axes, which relieves the author, reader, and future implementers of machine learning models from the burden of keeping track of the order of axes and the purpose of each. The notation makes it easy to lift operations on low-order tensors to higher order ones, for example, from images to minibatches of images, or from an attention mechanism to multiple attention heads. After a brief overview and formal definition of the notation, we illustrate it through several examples from modern machine learning, from building blocks like attention and convolution to full models like Transformers and LeNet. We then discuss differential calculus in our notation and compare with some alternative notations. Our proposals build on ideas from many previous papers and software libraries. We hope that our notation will encourage more authors to use named tensors, resulting in clearer papers and more precise implementations.

我们使用fr \'echet演算介绍了前馈神经网络梯度的推导，这比文献中通常呈现的梯度更紧凑。我们首先得出了在矢量数据上工作的普通神经网络的梯度，并展示如何使用这些派生公式来得出一种简单有效的算法来计算神经网络梯度。随后，我们展示了我们的分析如何推广到更通用的神经网络架构，包括但不限于卷积网络。

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。

本文通过引入几何深度学习（GDL）框架来构建通用馈电型型模型与可区分的流形几何形状兼容的通用馈电型模型，从而解决了对非欧国人数据进行处理的需求。我们表明，我们的GDL模型可以在受控最大直径的紧凑型组上均匀地近似任何连续目标函数。我们在近似GDL模型的深度上获得了最大直径和上限的曲率依赖性下限。相反，我们发现任何两个非分类紧凑型歧管之间始终都有连续的函数，任何“局部定义”的GDL模型都不能均匀地近似。我们的最后一个主要结果确定了数据依赖性条件，确保实施我们近似的GDL模型破坏了“维度的诅咒”。我们发现，任何“现实世界”（即有限）数据集始终满足我们的状况，相反，如果目标函数平滑，则任何数据集都满足我们的要求。作为应用，我们确认了以下GDL模型的通用近似功能：Ganea等。 （2018）的双波利馈电网络，实施Krishnan等人的体系结构。 （2015年）的深卡尔曼 - 滤波器和深度玛克斯分类器。我们构建了：Meyer等人的SPD-Matrix回归剂的通用扩展/变体。 （2011）和Fletcher（2003）的Procrustean回归剂。在欧几里得的环境中，我们的结果暗示了Kidger和Lyons（2020）的近似定理和Yarotsky和Zhevnerchuk（2019）无估计近似率的数据依赖性版本的定量版本。

一类非平滑实践优化问题可以写成，以最大程度地减少平滑且部分平滑的功能。我们考虑了这种结构化问题，这些问题也取决于参数矢量，并研究了将其解决方案映射相对于参数的问题，该参数在灵敏度分析和参数学习选择材料问题中具有很大的应用。我们表明，在部分平滑度和其他温和假设下，近端分裂算法产生的序列的自动分化（AD）会收敛于溶液映射的衍生物。对于一种自动分化的变体，我们称定点自动分化（FPAD），我们纠正了反向模式AD的内存开销问题，此外，理论上提供了更快的收敛。我们从数值上说明了套索和组套索问题的AD和FPAD的收敛性和收敛速率，并通过学习正则化项来证明FPAD在原型实用图像deoise问题上的工作。

我们提供了概率分布的Riemannian歧管上的经典力学的信息几何公式，该分布是具有双翼连接的仿射歧管。在非参数形式主义中，我们考虑了有限的样本空间上的全套正概率函数，并以统计歧管上的切线和cotangent空间为特定的表达式提供了一种，就希尔伯特束结构而言，我们称之统计捆绑包。在这种情况下，我们使用规范双对的平行传输来计算一维统计模型的速度和加速度，并在束上定义了Lagrangian和Hamiltonian力学的连贯形式主义。最后，在一系列示例中，我们展示了我们的形式主义如何为概率单纯性加速自然梯度动力学提供一个一致的框架，为在优化，游戏理论和神经网络中的直接应用铺平了道路。

Recent advances in operator learning theory have improved our knowledge about learning maps between infinite dimensional spaces. However, for large-scale engineering problems such as concurrent multiscale simulation for mechanical properties, the training cost for the current operator learning methods is very high. The article presents a thorough analysis on the mathematical underpinnings of the operator learning paradigm and proposes a kernel learning method that maps between function spaces. We first provide a survey of modern kernel and operator learning theory, as well as discuss recent results and open problems. From there, the article presents an algorithm to how we can analytically approximate the piecewise constant functions on R for operator learning. This implies the potential feasibility of success of neural operators on clustered functions. Finally, a k-means clustered domain on the basis of a mechanistic response is considered and the Lippmann-Schwinger equation for micro-mechanical homogenization is solved. The article briefly discusses the mathematics of previous kernel learning methods and some preliminary results with those methods. The proposed kernel operator learning method uses graph kernel networks to come up with a mechanistic reduced order method for multiscale homogenization.

神经网络的经典发展主要集中在有限维欧基德空间或有限组之间的学习映射。我们提出了神经网络的概括，以学习映射无限尺寸函数空间之间的运算符。我们通过一类线性积分运算符和非线性激活函数的组成制定运营商的近似，使得组合的操作员可以近似复杂的非线性运算符。我们证明了我们建筑的普遍近似定理。此外，我们介绍了四类运算符参数化：基于图形的运算符，低秩运算符，基于多极图形的运算符和傅里叶运算符，并描述了每个用于用每个计算的高效算法。所提出的神经运营商是决议不变的：它们在底层函数空间的不同离散化之间共享相同的网络参数，并且可以用于零击超分辨率。在数值上，与现有的基于机器学习的方法，达西流程和Navier-Stokes方程相比，所提出的模型显示出卓越的性能，而与传统的PDE求解器相比，与现有的基于机器学习的方法有关的基于机器学习的方法。

矢量值随机变量的矩序列可以表征其定律。我们通过使用所谓的稳健签名矩来研究路径值随机变量（即随机过程）的类似问题。这使我们能够为随机过程定律得出最大平均差异类型的度量，并研究其在随机过程定律方面引起的拓扑。可以使用签名内核对该度量进行内核，从而有效地计算它。作为应用程序，我们为随机过程定律提供了非参数的两样本假设检验。

Normalizing flows provide a general mechanism for defining expressive probability distributions, only requiring the specification of a (usually simple) base distribution and a series of bijective transformations. There has been much recent work on normalizing flows, ranging from improving their expressive power to expanding their application. We believe the field has now matured and is in need of a unified perspective. In this review, we attempt to provide such a perspective by describing flows through the lens of probabilistic modeling and inference. We place special emphasis on the fundamental principles of flow design, and discuss foundational topics such as expressive power and computational trade-offs. We also broaden the conceptual framing of flows by relating them to more general probability transformations. Lastly, we summarize the use of flows for tasks such as generative modeling, approximate inference, and supervised learning.

The emergence of variational quantum applications has led to the development of automatic differentiation techniques in quantum computing. Recently, Zhu et al. (PLDI 2020) have formulated differentiable quantum programming with bounded loops, providing a framework for scalable gradient calculation by quantum means for training quantum variational applications. However, promising parameterized quantum applications, e.g., quantum walk and unitary implementation, cannot be trained in the existing framework due to the natural involvement of unbounded loops. To fill in the gap, we provide the first differentiable quantum programming framework with unbounded loops, including a newly designed differentiation rule, code transformation, and their correctness proof. Technically, we introduce a randomized estimator for derivatives to deal with the infinite sum in the differentiation of unbounded loops, whose applicability in classical and probabilistic programming is also discussed. We implement our framework with Python and Q#, and demonstrate a reasonable sample efficiency. Through extensive case studies, we showcase an exciting application of our framework in automatically identifying close-to-optimal parameters for several parameterized quantum applications.

我们介绍了Netket的版本3，机器学习工具箱适用于许多身体量子物理学。Netket围绕神经网络量子状态构建，并为其评估和优化提供有效的算法。这个新版本是基于JAX的顶部，一个用于Python编程语言的可差分编程和加速的线性代数框架。最重要的新功能是使用机器学习框架的简明符号来定义纯Python代码中的任意神经网络ANS \“凝固的可能性，这允许立即编译以及渐变的隐式生成自动化。Netket 3还带来了GPU和TPU加速器的支持，对离散对称组的高级支持，块以缩放多程度的自由度，Quantum动态应用程序的驱动程序，以及改进的模块化，允许用户仅使用部分工具箱是他们自己代码的基础。

量子哈密顿学习和量子吉布斯采样的双重任务与物理和化学中的许多重要问题有关。在低温方案中，这些任务的算法通常会遭受施状能力，例如因样本或时间复杂性差而遭受。为了解决此类韧性，我们将量子自然梯度下降的概括引入了参数化的混合状态，并提供了稳健的一阶近似算法，即量子 - 固定镜下降。我们使用信息几何学和量子计量学的工具证明了双重任务的数据样本效率，因此首次将经典Fisher效率的开创性结果推广到变异量子算法。我们的方法扩展了以前样品有效的技术，以允许模型选择的灵活性，包括基于量子汉密尔顿的量子模型，包括基于量子的模型，这些模型可能会规避棘手的时间复杂性。我们的一阶算法是使用经典镜下降二元性的新型量子概括得出的。两种结果都需要特殊的度量选择，即Bogoliubov-Kubo-Mori度量。为了从数值上测试我们提出的算法，我们将它们的性能与现有基准进行了关于横向场ISING模型的量子Gibbs采样任务的现有基准。最后，我们提出了一种初始化策略，利用几何局部性来建模状态的序列（例如量子 - 故事过程）的序列。我们从经验上证明了它在实际和想象的时间演化的经验上，同时定义了更广泛的潜在应用。

神经切线核（NTK），定义为$ \ theta_ \ theta^f（x_1，x_2）= \ left [\ partial f（\ theta，x_1）\ big/\ big/\ partial \ partial \ theta \ theta \ the f（\ theta，x_2）\ big/\ partial \ theta \ right]^t $ where $ \ weft [\ partial f（\ theta，\ cdot）\ big/\ big/\ partial \ theta \ right] $是一个神经网络（nn）雅各布（Jacobian）已成为深度学习研究的核心研究对象。在无限宽度极限中，有时可以通过分析计算NTK，对于理解NN体系结构的训练和概括很有用。在有限的宽度下，NTK还用于更好地初始化NN，比较跨模型，执行体系结构搜索并进行元学习。不幸的是，众所周知，有限的宽度NTK计算昂贵，这严重限制了其实际实用程序。我们对有限宽度网络中NTK计算的计算和内存需求进行了第一个深入分析。利用神经网络的结构，我们进一步提出了两种新颖的算法，这些算法改变了有限宽度NTK的计算和内存要求的指数，从而极大地提高了效率。我们的算法可以以黑匣子方式应用于任何可区分功能，包括实现神经网络的功能。我们在https://github.com/google/neural-tangents的神经切线包（ARXIV：1912.02803）中开放我们的实现。